topological resolutions in k(2)-local homotopy theory …pgoerss/papers/xtheoremnew.pdf ·...

TOPOLOGICAL RESOLUTIONS IN K(2)-LOCAL HOMOTOPY

THEORY AT THE PRIME 2

IRINA BOBKOVA AND PAUL G GOERSS

Abstract We provide a topological duality resolution for the spectrum EhS122

which itself can be used to build the K(2)-local sphere The resolution is built

from spectra of the form EhF2 where E2 is the Morava spectrum for the formal

group of a supersingular curve at the prime 2 and F is a finite subgroup of the

automorphisms of that formal group The results are in complete analogy with

the resolutions of [GHMR05] at the prime 3 but the methods are of necessityvery different As in the prime 3 case the main difficulty is in identifying the

top fiber to do this we make calculations using Hennrsquos centralizer resolution

Chromatic stable homotopy theory uses the algebraic geometry of smooth one-parameter formal groups to organize calculations and the search for large scalephenomena In particular the chromatic filtration on the category of the p-localfinite spectra corresponds to the height filtration for formal groups The layers of thechromatic filtration are given by localization with respect to the Morava K-theoriesK(n) with n ge 0 Thus to understand the homotopy type of a finite spectrumX we begin by addressing LK(n)X for all prime numbers p and all 0 le n lt infinA useful and inspirational guide to this point of view can be found in the table insection 2 of [HG94]

If n = 0 K(0) = HQ and L0X is the rational homotopy type of X For n ge 1 thebasic computational tool in K(n)-local homotopy theory is the K(n)-local Adams-Novikov Spectral Sequence

Hs(Gn (En)tX) =rArr πtminussLK(n)X

Here Gn is the automorphism group of a pair (FqΓn) where Fq is a finite field ofcharacteristic p and Γn is a chosen formal group of height n over Fq Then En isthe Morava (or Lubin-Tate) E-theory defined by (FqΓn) We will give more detailsand make precise choices in section 1

First suppose p is large with respect to n (To be precise we may take 2pminus 2 gtmaxn2 2n+ 2 or p gt 3 if n=2) Then the Adams-Novikov Spectral Sequence forX = S0 collapses and will have no extensions so the problem becomes algebraicalthough by no means easy See for example [SY95] [Beh12] or [Lad13] for thecase n = 2 and p gt 3 However if p is small with respect to n the group Gnhas finite subgroups of p-power order and the spectral sequence will usually havedifferentials and extensions so the problem is no longer purely algebraic At this

The second author was partially supported by the National Science Foundation A portionof the work was done at the Hausdorff Institute of Mathematics during the Trimester ProgramldquoHomotopy Theory Manifolds and Field Theoriesrdquo

1

2 IRINA BOBKOVA AND PAUL G GOERSS

point topological resolutions become a useful way to organize the contributions ofthe finite subgroups The key to unlocking this idea is the Hopkins-Miller theoremwhich implies that Gn and hence all of its finite subgroups act on En and thatLK(n)S

0 EhGnn See [DH04] for this and more

The prototypical example is at n = 1 and p = 2 Adams and Baird [Bou79] andRavenel [Rav84] showed that here we have a fiber sequence

LK(1)S0 rarr KO

ψ3minus1minusminusminusrarr KO

where KO is 2-complete real K-theory For a suitable choice of a height one formalgroup Γ1 we can take E1 = K where K is 2-complete complex K-theory ThenG1 = Aut(Γ1F2) sim= Ztimes2 is the units in the 2-adic integers and C2 = plusmn1 sube Ztimes2acts through complex conjugation We can then rewrite this fiber sequence as

LK(1)S0 = EhG1

1 minusrarr EhC21

ψ3minus1minusminusminusrarr EhC21

and ψ3 is a topological generator of Ztimes2 plusmn1 Z2

For higher heights the topological resolutions will not be simple fiber sequencesbut finite towers of fibrations with the successive fibers built from EhFi where Firuns over various finite subgroups of Gn

In [GHMR05] the authors generalized the fiber sequence of the K(1)-local caseto the case n = 2 and p = 3 One way to say what they proved is the following Letus write E = E2 to simplify notation First we have a split short exact sequence ofgroups

1 rarr G12

G2N Z3 rarr 1

where N is obtained from a determinant (See (17)) and hence a fiber sequence

LK(2)S0minusrarr EhG

12ψminus1minusminusminusrarr EhG

12

where ψ is any element of G2 which maps by N to a topological generator of Z3

Then second there exists a resolution of EhG12

EhG12 rarr EhG24 rarr Σ8EhSD16 rarr Σ40EhSD16 rarr Σ48EhG24

Here resolution means each successive composition is null-homotopic and all possibleToda brackets are zero modulo indeterminacy thus the sequence refines to a towerof fibrations

EhG12 X2

X1 EhG24

Σ45EhG24

OO

Σ38EhSD16

OO

Σ7EhSD16

OO

The maximal finite subgroup of G2 of 3-power order is a cyclic group C3 oforder 3 it is unique up to conjugation in G1

2 The group G24 is the maximal finitesubgroup of G2 containing C3 The subgroup SD16 is the semidihedral group oforder 16 Because of the symmetry of this resolution and because G1

2 is a virtualPoincare duality group of dimension 3 this is called a duality resolution Thisresolution and related resolutions were instrumental in exploring the K(2)-localcategory at primes p gt 2 See [GHM04] [HKM13] [Beh06] [GHM14] [GHMR15]

TOPOLOGICAL RESOLUTIONS IN K(2)-LOCAL HOMOTOPY THEORY 3

[GH16] and [Lad13] The latter paper makes a thorough exploration of whathappens at p gt 3

The main theorem of this paper provides an analog of the duality resolution atn = 2 and p = 2 The prime 2 is much harder for a number of reasons First themaximal finite 2-subgroup of G2 is not cyclic but isomorphic to Q8 the quaterniongroup of order 8 Second every finite subgroup of G2 that we consider containsthe central C2 = plusmn1 of G2 therefore the homotopy groups of the relevant fixedpoint spectra EhF are much more complicated This means that the strategy ofproof of [GHMR05] wonrsquot work and we need to find another way

We now state our main result As our chosen formal group Γ2 we will use theformal group of a supersingular elliptic curve over F4 The curve will be definedover F2 so that G2

sim= S2 o Gal(F4F2) where S2 = Aut(Γ2F4) is the group ofautomorphisms of Γ2 over F4 Once again we have fiber sequence



12

We have EhG12 = (EhS

12)hGal where Gal = Gal(F4F2) and S1

2 = S2 capG12 For many

computational applications the difference between EhG12 and EhS

12 is innocuous

See Lemma 137 Then our main result is this

Theorem 01 There exists a resolution of EhS12 in the K(2)-local category at the

prime 2

EhS12 rarr EhG24 rarr EhC6 rarr Σ48EhC6 rarr Σ48EhG24

The spectrum EhC6 is 48-periodic so EhC6 Σ48EhC6 this suspension is thereonly to emphasize the symmetry in the resolution

Once again resolution means each successive composition is null-homotopic andall possible Toda brackets are zero modulo indeterminacy thus the sequence againrefines to a tower of fibrations This result has already had applications it is aningredient in Agnes Beaudryrsquos analysis of the Chromatic Splitting Conjecture atp = n = 2 See [Bea17]

The apparent similarity of Theorem 01 with the prime 3 analog is tantalizingespecially the suspension factor on the last term but we as yet have no conceptualexplanation The proof at the prime 2 can be adapted to the prime 3 but in bothcases it comes down to a very specific prime dependent calculation

A very satisfying feature of chromatic height 2 is the connection with the theoryof elliptic curves The subgroup G24 sube S2 is the automorphism group of our chosensupersingular curve inside the automorphisms of its formal group Appealing toStrickland [Str] we can use this to get formulas for the action of G24 on Elowast anecessary beginning to group cohomology calculations Furthermore we know that

EhG48 (EhG24)hGal LK(2)Tmf

where Tmf is the global sections of the sheaf of Einfin-ring spectra on the com-pactified stack of generalized elliptic curves provided by Hopkins and Miller See[DFHH14] Similarly EhC6 is the localization of global sections of the similar sheaffor elliptic curves with a level 3 structure See [MR09] We wonrsquot need anything


like the full power of the Hopkins-Miller theory here although we do use some ofthe calculations that arise from this point of view See Section 2

We will prove Theorem 01 in three steps

First we prove in Section 3 that there exists a resolution

EhS12 rarr EhG24 rarr EhC6 rarr EhC6 rarr X

where

ElowastX sim= ElowastEhG24

as twisted G2-modules This resolution and the necessary algebraic preliminarieswere announced in [Hen07] and grew out of the work surrounding [GHMR05] Thealgebraic preliminaries are discussed in detail in [Bea15]

Second in section 4 we examine the Adams-Novikov Spectral Sequence

Hlowast(G24 Elowast) =rArr πlowastX

Using a comparison of our resolution with a second resolution due to Henn weshow roughly that certain classes ∆8k+2 isin H0(G24 E192k+48) are permanentcyclesmdashwhich would certainly be necessary if our main result is true The ex-act result is in Corollary 47 These calculations were among the main results inthe first authorrsquos thesis [Bob14] and the key ideas for the entire project can befound there This ratifies a comment of Mark Mahowald that there is a class inπ45LK(2)S

0 which supports non-zero multiplications by η and κ and the only way

this could happen is if ∆2 is a permanent cycle This insight was we think the re-sult of long hours of contemplation of the results of Shimomura and Wang [SW02]and indeed one reason for this entire project is to find some way to catch upwith those amazing calculations We will have more to say about this element ofMahowaldrsquos in Remark 48

Third and finally in section 5 we use a variation of this same comparison argu-ment to produce the equivalence Σ48EhG24 X See Theorem 58 At the very endwe add a remark about the possibility or impossibility of resolutions for LK(2)S

0

itself See Remark 510

There is a number of sections of preliminaries Section 1 provides the usualbackground on the K(n)-local category as well as some more specific informationon mapping spaces Section 2 pulls together what we need about the homotopygroups of various fixed point spectra This draws from many sources and we tryto be complete there

An essential ingredient in our argument is the existence of the other topological

resolution for EhS12 Hennrsquos centralizer resolution from sect34 of [Hen07] We have less

control over the maps in this resolution but as has happened before ([GHM04]see also [GH16]) this resolution provides essential information needed to solveambiguities in the duality resolution It is also much closer to being an Adams-Novikov-style resolution as it is based on relative homological algebra We coversome of this material at the end of section 3


Acknowledgements There is a wonderful group of people working on K(2)-local homotopy theory and this paper is part of an on-going dialog within thatcommunity We are grateful in particular to Agnes Beaudry Charles Rezk andespecially Hans-Werner Henn Of course others have helped as well In particularthe key ideamdashto look for homotopy classes with particularly robust multiplicativepropertiesmdashis due to Mark Mahowald We miss his insight irreplaceable knowl-edge and friendship every day Finally we would like to give a heartfelt thanks tothe referee whose thorough and thoughtful report made this a better paper

Contents

1 Recollections on the K(n)-local category 5

2 The homotopy groups of homotopy fixed point spectra 17

3 Algebraic and topological resolutions 27

4 Constructing elements in π192k+48X 34

5 The mapping space argument 38

References 43

1 Recollections on the K(n)-local category

We begin with the standard material on the K(n)-local category the Moravastabilizer group and Morava E-theory also known as Lubin-Tate theory We thenget specific at n = 2 and p = 2 discussing the role of formal groups arising fromsupersingular elliptic curves We review some material on the homotopy type ofthe spectrum of maps between various fixed point spectra derived from Morava E-theory and then spell out some details of the K(n)-local Adams-Novikov SpectralSequence Finally we discuss the role of the Galois group

11 The K(n)-local category Fix a prime p and let n ge 1 Let Γn be a formalgroup of height n over the finite field Fp of p elements Then for any finite extensioni Fp sube Fq of Fp we form the group Aut(ΓnFq) of the automorphisms of ilowastΓnover Fq We fix a choice of Γn with the property that any extension Fpn sube Fq givesan isomorphism

(11) Aut(ΓnFpn)sim=minusrarr Aut(ΓnFq)

The usual Honda formal group satisfies these criteria this has a formal group lawwhich is p-typical and with p-series [p](x) = xp

n

However if n = 2 then the formalgroup of a supersingular elliptic curve defined over Fp will also do and this will beour preferred choice at p = 2 Define

(12) Sn = Aut(ΓnFpn)


If we choose a coordinate for Γn then any element of Sn defines a power seriesφ(x) isin xFpn [[x]] invertible under composition and the assignment φ(x) 7rarr φprime(0)defines a map

Snminusrarr Ftimespn This is a split surjection and we define Sn to be the kernel of this map this isthe p-Sylow subgroup of the profinite group Sn We then get an isomorphismSn o Ftimespn sim= Sn

Define the (big) Morava stabilizer group Gn as the automorphism group of thepair (Fpn Γn) Since Γn is defined over Fp there is an isomorphism

(13) Gn sim= Aut(ΓFpn) o Gal(FpnFp) = Sn o Gal(FpnFp)

We will often write Gal = Gal(FpnFp) when the field extension is understood

We next must define Morava K-theory There are many variants all of whichhave the same Bousfield class and define the same localization we will choose avariant which works well with Morava E-theory Let K(n) = K(Fpn Γn) be the2-periodic ring spectrum with homotopy groups

K(n)lowast = Fpn [uplusmn1]

and with associated formal group Γn Here the class u is in degree minus2 The groupF0 = Ftimespn o Gal(FpnFp) acts on K(n) and

Fp[vplusmn1n ] sim= (K(n)hF0)lowast = K(n)F0

lowast

where vn = uminus(pnminus1) The spectrum K(n)hF0 is thus a more classical version ofMorava K-theory

We will spend a great deal of time working in the K(n)-local category and whendoing so all our spectra will implicitly be localized In particular we emphasizethat we often write X and Y for LK(n)(X and Y ) as this is the smash product internalto the K(n)-local category

We now define the Morava spectrum E = En = E(Fpn Γn) (We will suppressthe n on En whenever possible to help ease the notation) This is the complexoriented Landweber exact 2-periodic Einfin-ring spectrum with

(14) Elowast = (En)lowast sim= W[[u1 middot middot middot unminus1]][uplusmn1]

with ui in degree 0 and u in degree minus2 Here W = W (Fpn) is the ring of Wittvectors on Fpn Note that E0 is a complete local ring with residue field Fpn theformal group over E0 is a choice of universal deformation of the formal group Γnover Fpn (We will be specific about this choice at n = p = 2 below in subsection12) The group Gn = Aut(Γn) o Gal(FpnFp) acts on E = En by the Hopkins-Miller theorem [GH04] and we have (see Section 15) a spectral sequence for anyclosed subgroup F sube Gn

(15) Hs(FEt) =rArr πtminussEhF

We will collectively call these by the name Adams-Novikov Spectral Sequence Seealso Lemma 129 below If F = Gn itself then EhGn LK(n)S

0 and we arecomputing the homotopy groups of the K(n)-local sphere


Various subgroups of Gn will play a role in this paper especially at n = 2and p = 2 The right action of Aut(Γn) on End(Γn) defines a determinant mapdet Sn = Aut(ΓnFpn)rarr Ztimesp which extends to a determinant map

(16) Gn sim= Sn o Gal(FpnFp)dettimes1 Ztimesp timesGal(FpnFp)

p1 Ztimesp

Define the reduced determinant (or reduced norm) N to be the composition

(17) GnN

22det Ztimesp Ztimesp C sim= Zp

where C sube Ztimesp is the maximal finite subgroup For example C = plusmn1 if p = 2

There are isomorphisms Ztimesp C sim= Zp and we choose one Write G1n for the kernel of

N S1n = Sn capG1

n and S1n = Sn capG1

n The map N Sn rarr Zp is split surjective andwe have semi-direct product decompositions for each of the groups Gn Sn and Snfor example there is an isomorphism

S1n o Zp sim= Sn

If n is prime to p we can choose a central splitting and the semi-direct product isactually a product but that is not the case of interest here

12 Deformations from elliptic curves Here we spell out what we need fromthe theory of elliptic curves at p = 2 this will give us a preferred formal group anda preferred universal deformation Choose Γ2 to be the formal group obtained fromthe elliptic curve C0 over F2 defined by the Weierstrass equation

(18) y2 + y = x3

This is a standard representative for the unique isomorphism class of supersingularcurves over F2 see [Sil09] Appendix A Because C0 is supersingular Γ2 has height 2as the notation indicates Following [Str] let C be the elliptic curve over W(F4)[[u1]]defined by the Weierstrass equation

(19) y2 + 3u1xy + (u31 minus 1)y = x3

This reduces to C0 modulo the maximal ideal m = (2 u1) the formal group G ofC is a choice of the universal deformation of Γ2

Again turning to [Sil09] Appendix A we have

(110) G24def= Aut(C0F4) sim= Q8 o Ftimes4

where Ftimes4 sim= C3 acts on Q8 as the 3-Sylow subgroup of Aut(Q8) Define

(111) G48def= Aut(F4 C0) sim= Aut(C0F4) o Gal(F4F2)

Since any automorphism of the pair (F4 C0) induces an automorphism of the pair(F4Γ2) we get a map G48 rarr G2 This map is an injection and we identify G48

with its image

Remark 112 Let C0[3] be the subgroup scheme of C0 consisting of points oforder 3 over F4 this becomes abstractly isomorphic to Z3timesZ3 The group G48

acts linearly on C0[3] and choosing a basis for the F4-points of C0[3] determinesan isomorphism G48

sim= GL2(Z3) This restricts to an isomorphism group G24sim=

Sl2(Z3)


Remark 113 The following subgroups will play an important role in this paper

(1) C2 = plusmn1 sube Q8(2) C6 = C2 times Ftimes4 (3) C4 any of the subgroups of order 4 in Q8(4) G24 and G48 themselves

The subgroup C4 is not unique but it is unique up to conjugation in G24 and inG2 In particular the homotopy type of EhC4 is well-defined

Remark 114 We have been discussing G24 as a subgroup of S2 but it can also bethought of as a quotient Inside of S2 there is a normal torsion-free pro-2-subgroupK which has the property that the composition

G24minusrarr S2minusrarr S2K

is an isomorphism Thus we have a decomposition K oG24sim= S2 See [Bea15] for

details The group K is a Poincare duality group of dimension 4

13 The functor ElowastX We define

ElowastX = πlowastLK(n)(E andX)

Despite the notation Elowast(minus) is not a homology theory as it does not take arbitrarywedges to sums but it is our most sensitive algebraic invariant on the K(n)-localcategory

Here are some properties of Elowast(minus) See [HS99] sect8 and Appendix A for moredetails Let m sube E0 be the maximal ideal and L = L0 be the 0th derived functorof completion at m Recall that an E0-module is L-complete if the natural mapM rarr L(M) is an isomorphism If M = L(N) then M is L-complete that isL(N) rarr L2(N) is an isomorphism for all N Thus the full subcategory of L-complete modules is a reflexive sub-category of all continuous E0-modules ByProposition 84 of [HS99] ElowastX is L-complete for all X

If N is an E0-module there is a short exact sequence

(115) 0rarr lim1k TorE0

1 (E0mk N)rarr L(N)rarr limkNm

kN rarr 0

Hence if M is L-complete then M is m-complete but it will be complete andseparated only if the right map is an isomorphism See [GHMR05] sect2 for someprecise assumptions which guarantee that ElowastX is complete and separated All ofthe spectra in this paper will meet these assumptions

Since Gn acts on E it acts on ElowastX in the category of L-complete modules Thisaction is twisted because it is compatible with the action of Gn on the coefficientring Elowast We will call L-complete E0-modules with such a Gn-action either twistedGn-modules or Morava modules

For example let F sube Gn be a closed subgroup Then there is an isomorphismof twisted Gn-modules

(116) ElowastEhF sim= map(GnFElowast)


where map(minusminus) denotes the set of continuous maps On the right hand side ofthis equation Elowast acts on the target and the Gn-action is diagonal This needs abit of care as the proof of this result given in [DH04] is given for the Honda formalgroup and it may not be true in general However in analyzing the proof of thecrucial Theorem 2 of [DH04] we see that the isomorphism of (116) follows from themore basic isomorphism

E0E sim= map(Gn E0)

The standard proof of this isomorphism for the Honda formal group (see Theorem12 of [Str00] for example) requires only that our formal group be defined over Fpand satisfy the stabilization requirement of (11)

Remark 117 We add here that Theorem 89 of [HS99] implies that the functor

X 7rarr ElowastX = πlowastLK(n)(En andX)

detects weak equivalences in the K(n)-local category Given a map f X rarr Y the class of spectra Z so that LK(n)(Z and f) is a weak equivalence is closed under

cofibrations and retracts hence if E is in this class then LK(n)S0 is in this class

14 Mapping spectra We collect here some basics about the mapping spectraF (EhF1 EhF2) for various subgroups F1 and F2 of Gn

Remark 118 Let F sube Gn be a closed subgroup We begin with the equivalence

(119) E and EhF map(GFE)

from the local smash product to the localized spectrum of continuous maps It ishelpful to visualize this map as sending xand y to the function gF 7rarr x(gy) We willcontinue this mnemonic below using point-wise defined functions to indicate mapsof spectra which cannot be defined that way We hope the readers can fill in thedetails themselves if not complete details can be found in [GHMR05] sect2

The action of Gn on E in (119) defines the Morava module structure of ElowastEhF

under the isomorphism of (119) this maps to the diagonal action on the functions

(hφ)(g) = hφ(hminus1g)

Note that(ElowastE

hF )Gn = mapGn(GnFElowast) sim= (Elowast)

F

By [DH04] Theorem 2 this extends to an isomorphism

Hlowast(Gn ElowastEhF ) sim= Hlowast(FElowast)

See also Lemma 129 below

Remark 120 We now recall some results from [GHMR05] If X = lim Xi is aprofinite set let E[[X]] = lim E andX+

i where the + indicates a disjoint basepointThen if F1 is a closed subgroup of Gn we have an equivalence

(121) E[[GnF1]] F (EhF1 E)

defined as follows Let FE denote the function spectrum in E-modules Then

E[[GnF1]] FE(map(GnF1 E) E)

FE(E and EhF1 E)

F (EhF1 E)


Next note that the equivalence of (121) is Gn-equivariant with the following actionsin F (EhF1 E) we act on the target and in E[[GnF1]] we act as follows

h(sum

aggF1) =sum

h(ag)hminus1gF1

We can now make the following deductions First suppose F1 = U is open (andhence closed) so that GnF1 = GnU is finite Let F2 be finite Then we haveequivalences prod

F2GnU

EhFx E[[GnU ]]hF2(122)

F (EhU E)hF2

F (EhU EhF2)

The product in the source is over the double coset space and for a double cosetF2xU

Fx = F2 cap xUxminus1 sube F2

Since it depends on a choice of x the group Fx is defined only up to conjugationbut the fixed point spectrum EhFx is well-defined up to weak equivalence

The first map of (122) sends a isin EhFx to the sumsumgFxisinF2Fx

(gminus1a) gxU

We say a word about the naturality of the equivalence of (122) Suppose U subeV sube Gn is a nested pair of open subgroups Then for each double coset F2xU weget a double coset F2xV a nested pair of subgroups

Fx = F2 cap xUxminus1 sube F2 cap xV xminus1 = Gx

and a transfer map

trx EhFxminusrarr EhGx

associated to this inclusion Then we have a commutative diagram

(123)prodF2GnU

EhFx

tr

E[[GnU ]]hF2

g

prodF2GnV

EhGx E[[GnV ]]hF2

where the map tr is the sum of the transfer maps and the map g induced by thequotient GnU rarr GnV

For a more general closed subgroup F1 write F1 = capiUi where Ui sube Gn is openThen for F2 finite we get a weak equivalence

(124) holimi

prodF2GnUi

EhFx F (EhF1 EhF2)

where the product is as before and

Fx = F2 cap xUixminus1 sube F2


depends on i and as in (123) there may be transfer maps in the transition mapsfor the homotopy limit As F2 is finite and the product in (124) is finite thisimplies that the image ofprod

F2GnUj

πlowastEhFxminusrarr

prodF2GnUi

πlowastEhFx

is independent of j for large j It follows that there will be no lim1 term for thehomotopy groups of the inverse limit and hence there is an isomorphism

(125) limi

prodF2GnUi

πlowastEhFx sim= πlowastF (EhF1 EhF2)

15 The K(n)-local Adams-Novikov Spectral Sequence This is the maintechnical tool of this paper and we give a few details of the construction and someof its properties We begin with some algebra from [HS99] Appendix A

Let m sub E0sim= W[[u1 middot middot middot unminus1]] be the maximal ideal An L-complete E0-

module M is pro-free if any one of the following equivalent conditions is satisfied

(1) M sim= L(N) sim= Nandm for some free E0-module N (2) the sequence (p u1 middot middot middot unminus1) is regular on M

(3) TorE01 (Fpn M) = 0

(4) M is projective in the category of L-complete modules

Proposition A13 of [HS99] implies that a continuous homomorphism M rarr N ofpro-free modules is an isomorphism if and only if MmM rarr NmN is an isomor-phism

If F sube Gn is a closed subgroup then ElowastEhF is pro-free by (116)

By Proposition 84 of [HS99] if X is a spectrum with K(n)lowastX concentrated ineven degrees then ElowastX is pro-free concentrated in even degrees Furthermore ifElowastX is pro-free and concentrated in even degrees then

K(n)lowastX sim= Fpn otimesE0 ElowastXsim= (ElowastX)m

Remark 126 We now come to the Adams-Novikov Spectral Sequence in K(n)-local category As in [Str00] Proposition 15 this spectral sequence is obtained bythe standard cosimplicial cobar complex for E = En in the K(n)-local categoryThus we have

Est2sim= πsπt(E

bull andX) =rArr πtminussLK(n)X

The smash products are in the K(n)-local category It is a consequence of the proofof [Str00] Proposition 15 that this spectral sequence converges strongly and has ahorizontal vanishing line at Einfin

Wersquod now like to rewrite the E2-term as group cohomology at least under somehypotheses For any group G let EG be the standard contractible simplicial G-setwith (EG)s = Gs+1 This is a bar construction If G is a topological group thenEG is a simplicial space


Let M be an L-complete Morava module We define

Hs(GnM) = πsmapGn(EGnM)

where mapGn(minusminus) denotes the group of continuous Gn-maps Since there is an

isomorphism mapGn(Gs+1

n M) sim= map(GsnM) we have that Hlowast(GnM) is the sthcohomology group of a cochain complex

M map(GnM) map(G2nM) middot middot middot

Proposition 127 Suppose X = Y and Z where K(n)lowastY is concentrated in evendegrees and Z is a finite complex Then we have an isomorphism

πsπt(Ebull andX) sim= Hs(Gn EtX)

and the K(n)-local Adams-Novikov Spectral Sequence reads

Hs(Gn EtX) =rArr πtminussLK(n)X

Proof If Z = S0 then ElowastX is pro-free in even degrees The result can be foundin Theorem 43 of [BH16] The authors there work with the version of Morava E-theory obtained from the Honda formal group but they need only the isomorphismE0E sim= map(Gn E0) which holds in our case See the remarks after (116) Thekey idea is that this last isomorphism can be extended to an isomorphism

Elowast(Eands andX)minusrarr map(Gsn ElowastX)

for any spectrum X with EtX pro-free for t For more general Z the isomorphismon E2-terms follows from the five lemma

In some crucial cases it is possible to reduce the E2-term to group cohomologyover a finite group Let F sube Gn be a finite subgroup and N an L-complete twistedF -module Define an L-complete module N uarrGn

F as the set of continuous mapsφ Gn rarr N such that φ(hx) = hφ(x) for h isin F This becomes a Morava modulewith

(gφ)(x) = φ(xg)

with g isin Gn and there is an isomorphism

mapGn(Gs+1

n N uarrGn

F ) sim= mapF (Gs+1n N)

of groups of continuous maps Let Hs(FN) = πsmapF (EFN)

Lemma 128 (Shapiro Lemma) Let F sube Gn be a finite subgroup and sup-pose N is an L-complete twisted F -module with the property that NmN is finitedimensional over Fpn Then there is an isomorphism

Hlowast(Gn N uarrGn

F ) sim= Hlowast(FN)

and under these hypotheses on N there is an isomorphism

Hlowast(FN) sim= limk Hs(FNmkN)

Proof Since N is L-complete and NmN is finite the short exact sequence of (115)implies N sim= Nandm


Choose a nested sequence Ui+1 sube Ui sube Gn of finite index subgroups of Gn withthe property that capUi = e Then for all s ge 0 we have

mapGn(Gs+1

n N uarrGn


sim= limk mapF (Gs+1n NmkN)

sim= limk colimi mapF ((GnUi)s+1 NmkN)

The last isomorphism follows from the fact that NmkN is discrete and finite SinceF is finite we have F cap Ui = e for all i greater than some i0 Then for i gt i0 itfollows that (GnUi)bull+1 is a contractible simplicial free F -set and thus we havean isomorphism

πsmapF ((GnUi)bull+1 NmkN) sim= Hs(FNmkN)

Again since F is finite Hs(FNmkN) is a finite abelian group Both isomorphismsof the lemma now follow

Lemma 129 Let Y be a spectrum equipped with an isomorphism of Morava mod-ules

ElowastY sim= map(GnFElowast) sim= ElowastEhF

where F sube Gn is a finite subgroup Then for all finite spectra Z there is anisomorphism

Hlowast(Gn Elowast(Y and Z)) sim= Hlowast(FElowastZ)

Proof We have an isomorphism of Morava modules

Elowast(Y and Z)sim= map(GnFElowastZ)

where the action of Gn on the target is by conjugation (gφ)(x) = gφ(gminus1x) Thereis an isomorphism of Morava modules

map(GnFElowastZ) sim= (ElowastZ)uarrGn

F

adjoint to evaluation at eF The result follows from the Shapiro Lemma 128

Remark 130 Suppose Y = EhF Then Lemma 129 follows from the variantof Theorem 2 of [DH04] appropriate for our formal group indeed we need onlyrequire that F be closed However the proof relies on the construction of EhF

given in that paper and doesnrsquot a priori apply if we only know ElowastY sim= ElowastEhF ndash as

will be the case in our application

Remark 131 There is a map of Adams-Novikov Spectral Sequences

ExtsBPlowastBP (ΣtBPlowast BPlowast) +3

Z(2) otimes πtminussS0

Hs(G2 Et) +3 πtminussLK(2)S

0

but it takes a little care to define Let G(x y) isin E0[[x y]] be the formal group lawof the supersingular curve of (19) since G is the formal group of a Weierstrasscurve it has a preferred coordinate Let Glowast(x y) = uG(uminus1x uminus1y) isin Elowast[[x y]]Then Glowast is a formal group over Elowast with coordinate in cohomological degree 2 and


therefore is classified by a map Z(2) otimes MUlowast rarr Elowast Since Glowast is not evidently2-typical it need not be classified by a map BPlowast rarr Elowast However over a Z(2)-algebra the Cartier idempotent gives an equivalence between the groupoid of allformal group laws and the groupoid of 2-typical formal group laws hence we havea diagram of spectral sequences as needed

ExtsBPlowastBP (ΣtBPlowast BPlowast) +3 Z(2) otimes πtminussS0

Z(2) otimes ExtsMUlowastMU (ΣtMUlowastMUlowast) +3

sim=

OO


=

OO


0

We will use this below in the section on the cohomology of G48

16 The action of the Galois group We now turn to analyzing EhS2 as an equi-variant spectrum over the Galois group As above we will write Gal = Gal(FpnFp)so that Gn sim= Sn o Gal

We begin with the following elementary fact the map Zp rarr W is Galois withGalois group Gal thus it is faithfully flat etale and the shearing map

WotimesZpWrarr map(GalW)

sending a otimes b to the function g 7rarr ag(b) is an isomorphism In fact this shearingmap is certainly an isomorphism modulo p then the statement for W follows fromNakayamarsquos Lemma Faithfully flat descent now implies that the category of Zp-modules is equivalent to the category of twisted W[Gal]-modules under the functorM 7rarrWotimesZp

M the inverse to this functor sends N to NGal

This extends to the following result

Lemma 132 Let K sube Gn be a closed subgroup and let K0 = K cap Sn Supposethe canonical map

KK0minusrarr GnSn sim= Gal

is an isomorphism Then for any twisted Gn-module M we have isomorphisms

Hlowast(KM) sim= Hlowast(K0M)Gal

Hlowast(K0M) sim= WotimesZp Hlowast(KM)

Proof The subgroup Sn acts on E0 through W-algebra homomorphisms hence itacts on M through W-module homomorphisms It follows that we can write thefunctor (minus)K of invariants as a composite functor

twisted Gn-modules(minus)K0

twisted W[Gal]-modules(minus)Gal

Zp-modules

As we just remarked the second of these two functors is an equivalence of categoriesand in particular it is an exact functor The first equation follows The secondequation follows from the first and the fact that the inverse to (minus)Gal is the functorM 7rarrWotimesZp

M


We now give a fact seemingly known to everyone but hard to find in print DrewHeard was the first to point out an error in our original argument others followedquickly We learned the following replacement from Mike Hopkins Agnes Beaudryand the referee We extend our thanks to everyone

Lemma 133 For all p and all n ge 1 we have isomorphisms

H0(Gn E0) sim= ZpH0(Sn E0) sim= W = W (Fpn)

Furthermore H0(Gn Et) = H0(Sn Et) = 0 if t 6= 0

Proof By Lemma 132 we need only do the case of Sn

It is also sufficient to prove this when Γn is the Honda formal group over Fpn Any other height n formal group becomes isomorphic to the Honda formal groupover the algebraic closure of Fp and the general result could then be deduced fromGalois descent

Write φ for lift of Frobenius to Witt vectors W For the Honda formal groupSn is the group of units in the endomorphism ring

End(Γn) sim= W〈S〉(Sn minus p)where W〈S〉 is non-commutative polynomial ring over W on a variable S withSa = φ(a)S when a isinW Thus we have an inclusion Wtimes sub Sn Every k isin Ztimesp sube Sncorresponds to an automorphism Γn rarr [k]Γn

of Γn in particular Ztimesp acts triviallyon E0 and if u isin Eminus2 is the invertible generator we have klowastu = ku If follows thatH0(Ztimesp Et) = 0 if t 6= 0

This leaves the case t = 0 Write K = W[pminus1] Since all elements of Sn fix theconstants W sube E0 there is an inclusion

W sim= H0(WtimesW)rarr H0(Wtimes E0)

Furthermore since E0 is torsion-free it is sufficient to show H0(Wtimes E0[pminus1]) sim=K By [DH95] Lemma 43 there are power series wi = wi(u1 u2 unminus1) isinK[[u1 middot middot middot unminus1]] and an inclusion

W[[w1 middot middot middot wnminus1]]minusrarr K[[u1 middot middot middot unminus1]]

onto an Gn-equivariant sub-algebra which becomes an isomorphism after invertingp in the source Thus it is sufficient to show that

H0(Wtimes J) = 0

where J = W[[w1 middot middot middot wnminus1]]W

By [DH95] Proposition 33 the action of Wtimes on W[[w1 middot middot middot wnminus1]] is diagonalif a isinWtimes then

alowastwi = φi(a)aminus1wi

Let ω isin W be a primitive (pn minus 1)st root of unity and set a = 1 + pω Thenφ(1 + pω) = 1 + pωp If some ij 6= 0 it follows that

(1 + pω)lowastwi11 w

inminus1

nminus1 6= wi11 winminus1

nminus1

as needed


Remark 134 Notice that the proof of Lemma 133 actually shows we need onlyrelatively small subgroups of Gn to get the full invariants Specifically we have

Zp sim= H0(Wtimes o Gal E0)

and if t 6= 0 then H0(Ztimesp Et) = 0 We wonrsquot need this stronger result

Remark 135 In the proof of Lemma 136 below we will use the following ob-servation Suppose we have a spectral sequence Estr that is multiplicative thatis Elowastlowastr is a bigraded ring which is commutative up to sign and dr satisfies theLeibniz rule again up to sign Further suppose R sube E00

r is a commutative subringof dr-cycles and R sube S is an etale extension in E00

r Then every element of S is adr-cycle To see this note that dr restricted to S is a derivation over R and anysuch derivation must vanish indeed depending on your foundations the vanishingof such derivations may even be part of your definition of etale

Lemma 136 For all p and all n ge 1 there is a Gal-equivariant equivalence

Gal+ and LK(n)S0 rarr EhSn

Proof We first prove we have injection W rarr π0EhSn We begin with the isomor-

phism W sim= H0(Sn E0) of Lemma 133 Since Zp sube W is an etale extension andthe Adams-Novikov Spectral Sequence for Sn is a spectral sequence of rings all ofW survives to Einfin and the edge homomorphism provides a surjection π0E

hSn rarrWof rings The kernel of this map nilpotent as an ideal because the spectral sequencehas a horizontal vanishing line at Einfin See Remark 126 We then have a diagram

Zp

sube

π0EhSn

W =

W

Again since Zp sube W is etale the dashed arrow can be completed uniquely Thisyields the injection we need

Now define ω S0 rarr EhSn to be a representative of the homotopy class definedby a primitive (pnminus 1)st root of unity in W We can extend ω to a Gal-equivariantmap f Gal+ and S0 rarr EhS2 inducing the splitting W rarr π0E

hS2 here we use thatthe map Zp[Gal]rarrW sum

agg 7minusrarrsum

agg(ω)

is an isomorphism The map f extends to an isomorphism of twisted Gn-modules

Elowastf Elowast(Gal+ and S0) sim= map(Gal Elowast) sim= map(GnSn Elowast) sim= ElowastEhSn

thus completing the argument

The following result is a topological analog of Lemma 132



is an isomorphism Then there is a Gal-equivariant equivalence

Gal+ and EhK rarr EhK0


Proof This follows from Lemma 136 Define a map EhSn and EhK rarr EhK0 by thecomposition

(138) EhSn and EhK rarr EhK0 and EhK0 rarr EhK0

where the first map is given by the inclusion and the last map is multiplicationWe have

Elowast(EhSn and EhK) sim= map(GnSn Elowast)otimesElowast map(GnKElowast)

sim= map(GnSn timesGnKElowast)

In Elowast-homology the map of (138) then becomes the map

map(GnSn Elowast)otimesElowast map(GnKElowast) sim= map(GnSn timesGnKElowast)minusrarr map(GnK0 Elowast)

induced by the maps on cosets

GnK0 rarr GnK0 timesGnK0 rarr GnSn timesGnK

where the first map is the diagonal and the second map is projection Since KK0sim=

GnSn this map on cosets is an isomorphism therefore the map of (138) is anElowast-isomorphism By Remark 117 this map is a weak equivalence

Remark 139 Combining Lemma 132 and Lemma 137 yields an isomorphism ofspectral sequences

WotimesHlowast(KElowast) +3

sim=

Wotimes πlowastEhK

sim=

Hlowast(K0 Elowast) +3 πlowastEhK0

where the differentials on the top line are the W-linear differentials extended fromthe spectral sequence for K

Remark 140 At n = 2 and p = 2 Lemma 137 applies to the case of K = G48then K0 = G24 This implies that any of the spectra

X(i j)def= Σ24iEhG48 or Σ24jEhG48

has the property that ElowastX(i j) sim= ElowastEhG24 But X(i j) = Σ24iEhG24 if and only

if i equiv j mod 8 This means that in some of the arguments we give to prove ourmain result we will have to produce two homotopy classes rather than one SeeTheorem 58

2 The homotopy groups of homotopy fixed point spectra

Here we collect what we will need about the homotopy groups of EhF where Fruns through the finite subgroups of G1

2 of Remark 113 We will be working entirelyat n = p = 2 and using the formal group from the supersingular curve of (18)Much of whatrsquos needed is in the literature and wersquoll do our best to give referencesHowever much of what is written is for calculations over Hopf algebroids whichis not quite what wersquore doing and the results need translation In addition manyof the results as written include some variant of the phrase ldquowe neglect the bo-patternsrdquo We make this thought precise with the following ad hoc definition


Definition 21 Let F sube G2 be any finite subgroup containing C2 = plusmn1 Thenwe define the bo-patterns L1(πlowastE

hF ) of πlowastEhF to be the image of the map in

homotopy

πlowastEhFminusrarr πlowastLK(1)E

hF

We also define the pure K(2)-classes M2(πlowastEhF ) to be the kernel of the same

map

Thus we have a short exact sequence

0rarrM2(πlowastEhF )rarr πlowastE

hF rarr L1(πlowastEhF )rarr 0

Notice that the bo-patterns are defined as a quotient In most cases this sequenceis not split as modules over the homotopy groups of spheres

Remark 22 The name bo-patterns is something of a misnomer as KO-patternswould be more accurate Here KO is 8-periodic 2-complete real K-theory In allour examples we will have an isomorphism

R(F )otimesZ2 KOlowastsim= πlowastLK(1)E

hF

for some Z2-algebra R(F ) in degree zero While R(F ) otimesZ2KOlowast is 8-periodic

L1(πlowastEhF ) will typically have 8k-periodicity for some k gt 1 As a warning we

mention that this isomorphism is simply as rings we are not claiming LK(1)EhF is

a KO-algebra

Remark 23 Here is more detail to explain our thinking

Let us write S2n for the Z2n-Moore spectrum Then there is a weak equiva-lence

LK(1)X holim vminus11 (X and S2n)

and ifX isK(2)-local a corresponding localized Adams-Novikov Spectral Sequence

(24) lim vminus11 Hlowast(G2 (ElowastX)2n) =rArr πlowastLK(1)X

This spectral sequence doesnrsquot obviously converge

Now suppose C2 = plusmn1 sube F sub G48 Then the spectral sequence (24) forX = EhF becomes

(25) lim vminus11 Hlowast(F (Elowast)2

n) =rArr πlowastLK(1)EhF

Using Stricklandrsquos formulas [Str] it is possible to show that

lim vminus11 Hlowast(F (Elowast)2

n) sim= lim vminus11 Hlowast(C2 (Elowast)2

n)FC2

and that

vminus11 Hlowast(C2 (Elowast)2

n) sim= W((u1))[uplusmn2 η]

where W((u1)) = lim(W2n)[[uplusmn11 ]] and η isin H1(C2 E2) detects the class of the

same name in π1S0 (See (26) below) From this it follows that the spectral

sequence (25) is completely determined by the standard differential d3(v21) = εη3

where v21 = u2

1uminus2 and ε isin F4((u1))times is a unit We can conclude that the spectral

sequence converges and

πlowastLK(1)EhF sim= W((u1))F otimesZ2

KOlowast


and in particular that πlowastLK(1)EhF and L1(πlowastE

hF ) are both concentrated in de-grees congruent to 0 1 2 and 4 modulo 8 It then remains to analyze the pureK(2)-local classes

Now not much of what we just wrote is explicitly in print and it would takequite a few pages to prove in detail But we will put together what we can from theexisting literature to cover the main points case-by-case below See Propositions27 211 and 217

21 The homotopy groups of EhC2 and EhC6 The standard source here isMahowald-Rezk [MR09] which uses the Hopf algebroid approach but also uses thesame elliptic curve we have chosen So the translation is straightforward Here is asummary

The central C2 sube G2 acts trivially on E0 and by multiplication by minus1 on uhence

(26) Hlowast(C2 Elowast) sim= W[[u1]][uplusmn2 α](2α)

where α isin H1(C2 E2) is the image of the generator of H1(C2Z〈sgn〉) under themap which sends the generator of the sign representation to uminus1 Since

v1 = u1uminus1 isin H0(C2 E22)

and the class u1α isin H1(C2 E2) is the image of v1 under the integral Bocksteinthe class η isin π1S

0 is detected by u1α We will also write η = u1α

Proposition 27 The class bdef= u2

1uminus2 reduces to v2

1 in vminus11 Hlowast(C2 Elowast2) There

is an isomorphism

W((u1))[bplusmn1 η](2η) sim= lim vminus11 Hlowast(C2 Elowast2

n)

The standard differential d3(v21) = η3 (see Lemma 221 below) forces a differential

d3(uminus2) = εu1α3

where ε isin F2[[u1]]times Using the Mahowald-Rezk transfer argument [MR09 Prop35] we have ν isin π3S

0 is non-zero in π3EhC2 and detected by α3 this in turn forces

a differentiald7(uminus4) = α7 = αν2

The spectral sequence collapses at E7 and we have the following result

Proposition 28 The homotopy ring πlowastEhC2 is periodic of period 16 with peri-

odicity generator e16 detected by uminus8 The bo-patterns L1(EhC2) are concentratedin degrees congruent to 0 1 2 and 4 modulo 8 and the group of pure K(2)-localclasses M2(EhC2) is generated by the classes

αiek16 k isin Z i = 3 4 5 6

To get the homotopy of EhC6 with C6 = C2 times Ftimes4 we need to know the actionof Ftimes4 We can use Stricklandrsquos calculations [Str] or interpret the Mahowald-Rezkresults Let ω isin Ftimes4 be a primitive cube root of unity Then ωlowastu = ωu andωlowastu1 = ωu1 it follows that ωlowastα = ωminus1α The next result can be deduced fromthese formulas and the fact that

πlowastEhC6 sim= (πlowastE

hC2)Ftimes4



odicity generator e316 detected by uminus24 The bo-patterns L1(EhC6) are concentrated

in degrees congruent to 0 1 2 and 4 modulo 8 and the group of pure K(2)-localclasses M2(EhC6) is generated by the classes

e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5

of degrees 48k + 3 48k + 6 48k + 20 and 48k + 37 respectively Furthermore thehomotopy class ν isin π3S

0 is detected by the class α3 and the class κ isin π20S0 is

detected by e16α4

22 The homotopy groups of EhC4 The standard reference for this calculationis Behrens-Ormsby [BO16] sect21 especially Theorem 213 and Perspective 2 afterRemark 219 See also their Figure 2 Again they use Hopf algebroids Here we hitanother small problem the supersingular elliptic curve they use is different fromthe one we have chosen and thus they have a different action of C4 on a differentversion of Morava E-theory There are two possible solutions One is to do thecalculations over again using Stricklandrsquos formulas The other is to notice that thetwo supersingular curves become isomorphic over the algebraically closed field F2

and to use descent to make the calculations Using either method we obtain thefollowing result Let i isin C4 be a generator and let

z = u1 + ilowastu1 isin H0(C4 E0)

There are further cohomology classes

b2 isin H0(C4 E4) δ isin H0(C4 E8)

and

γ isin H1(C4 E6) ξ isin H2(C4 E8)

Let η isin H1(C4 E2) and ν isin H1(C4 E4) be the images of the like-named classesfrom the BP -based Adams-Novikov Spectral Sequence (see Remark 131)

Proposition 210 There is an isomorphism

Hlowast(C4 Elowast) sim= W[[z]][b2 δplusmn1 η ν γ ξ]R

where R is the ideal of relations given by

2η = 2γ = 4ξ = 0

and

b22 equiv z2δ mod 2

and

δη2 = b2ξ = γ2 b2γ = zδη

and

b2η = zγ γη = zξ

and the final relations involving ν

ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0

Proof This can be obtained from Perspective 2 (after Remark 219) of [BO16] bya three-step process First set γ = γ j = z minus 2 and β = δminus1ξ Second invert δFinally complete at the maximal ideal of H0(C4 E0)


This result is displayed in Figure 1 below presented in the standard Adamsformat the x-axis is tminuss the y-axis is s In this chart the square box representsa copy of W[[z]] the circle a copy of F4[[z]] and the crossed circle otimes a copy ofW[[z]](4 2z) generated by a class of the form ξjδi A solid dot is a copy of F4

annihilated by z it is generated by a class of the form ξjν The solid lines aremultiplication by η or ν as needed and a dashed line indicates that xη = zywhere x and y are generators in the appropriate bidegree

0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ

Figure 1 The cohomology of C4

Notice that Hlowast(C4 Elowast) is 8-periodic with the periodicity class δ and that

timesξ Hs(C4 Elowast)rarr Hs+2(C4 Elowast+8)

is onto for s ge 0 and an isomorphism for s gt 0 In fact δminus1ξ isin H2(C4 E0) isup to multiplication by a unit the image of the periodicity class for the groupcohomology of C4 under the inclusion of trivial coefficients

Z4 sim= H4(C4Z2) sim= H2(C4W)rarr Hs(C4 E0)

Proposition 211 Modulo 2 we have an equivalence b2 equiv v21 and then an isomor-

phism

W((z))[bplusmn12 η](2η) sim= lim vminus1

1 Hlowast(C4 Elowast2n)

The differentials and extensions in this spectral sequence go exactly as in Behrensand Ormsby [BO16] Theorem 2312 We end with the following result

Proposition 212 The homotopy ring πlowastEhC4 is periodic of period 32 with pe-

riodicity generator e32 detected by δ4 The bo-patterns L1(EhC4) are concentratedin degrees congruent to 0 1 2 and 4 modulo 8 and the group of pure K(2)-localclasses M2(EhC4) is generated by the classes ek32x where x is from the followingtable


Class Degree Order E2-namea 1 2 δminus1ξνaη 2 2 δminus2ξ2ν2

ν 3 4 νaν 4 2 δminus1ξν2

ν2 6 2 ν2

ε 8 2 δminus2ξ4

ν3 = ηε 9 2 κ2δminus5ξνκ 14 2 δν2

b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2

c 27 2 δ2γξcη 28 2 δminus1ξ6

The pure K(2) classes in πlowastEhC4 are presented in the Figure 2 the horizontal

bar is multiplication by ν the diagonal bar is multiplication by η Note thatκη2 = 2bν The horizontal scale is the degree of the element but the vertical scalehas no meaning Many of the additive and multiplicative relations are given byexotic extensions in this spectral sequence and the meaning of the original Adams-Novikov filtration becomes attenuated as a result see Behrens-Ormsby [BO16]especially figure 9 for details

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ

Figure 2 The pure K(2) classes in πlowastEhC4

23 The homotopy groups of EhG24 and EhG48 Remarks 139 and 140 yieldan isomorphism of spectral sequences

WotimesZ2Hlowast(G48 Elowast) +3

sim=

Wotimes πlowastEhG48

sim=

Hlowast(G24 Elowast) +3 πlowastEhG24

Therefore we focus on the case of G48 Here the standard sources are [Bau08][DFHH14] and [HM98] although it requires some translation in each case to get theresults we want


The ring H0(G48 Elowast) is isomorphic to the ring of modular forms for supersin-gular elliptic curves at the prime 2 Then there are elements

c4 isin H0(G48 E8) c6 isin H0(G48 E12) ∆ isin H0(G48 E24)

obtained from the modular forms of the same name for our supersingular curveSince this curve is smooth ∆ is invertible and the j-invariant of our curves j =c34∆ isin H0(G48 E0) is defined Then we get an isomorphism

Z2[[j]][c4 c6∆plusmn1](c34 minus c26 = (12)3∆∆j = c34) sim= H0(G48 Elowast)

Modulo 2 we get a slightly simpler answer

F2[[j]][v1∆plusmn1](j∆ = v12

1 ) sim= H0(G48 Elowast2)

Modulo 2 we have congruences

(213) c4 equiv v41 and c6 equiv v6

1

To describe the higher cohomology we make a table of multiplicative generatorsFor each x the bidegree of x is (s t) if x isin Hs(G48 Et) All but micro detect theelements of the same name in πlowastS

0 Furthermore all elements but κ are in theimage of the map (see Remark 131)

ExtlowastlowastBPlowastBP (BPlowast BPlowast)rarr Hlowast(G2 Elowast)rarr Hlowast(G48 Elowast)

Hence we also give the name (the ldquoMRWrdquo is for Miller-Ravenel-Wilson) of a preim-age The Greek letter notation is that of [MRW77]

Class Bidegree Order MRWη (1 2) 2 α1

ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus

The class κ isin π20S0 is detected by the image of β4 in H2(G2 Elowast) The class micro has

a special role which we discuss below in Lemma 221 but we would like to noteright away that

(214) v21η equiv micro modulo 2

The following result is actually much easier to visualize than to write down Seethe Figure 3 below

Theorem 215 There is an isomorphism

H0(G48 Elowast)[η ν micro ε κ κ]R sim= Hlowast(G48 Elowast)

where R is the ideal defined by

(1) the order of the elements of positive cohomological degree

2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0

(2) the relations for ν

ην = 2ν2 = ν4 = microν = 0


(3) the relations for ε

ηε = ν3 νε = ε2 = microε = 0

(4) the relations for κ

ν2κ = 4κ η2κ = εκ = κ2 = microκ = 0

(5) the elements annihilated by modular forms

c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0

(6) the relations between κ and modular forms

c4κ = ∆η4 c6κ = ∆η3micro

(7) and the relations indicated by the congruences of (213) and (214)

micro2 = c4η2 c4micro = c6η c6micro = c24η

This result is presented graphically in Figure 3 below We present it as the E2

page of the Adams-Novikov Spectral Sequence The cohomology is 24-periodic on∆ and the spectral sequence fills the entire upper-half plane In Figure 3 thesquare box represents a copy of Z2[[j]] the circle a copy of F2[[j]] and thecrossed circle otimes a copy of Z2[[j]](8 2j) generated by a class of the form ∆iκj Thesolid bullet represents a class of order 2 annihilated by j and the doubled bullet aclass of order 4 annihilated by j these last classes are always of the form ∆iκjνThe solid lines are multiplication by η or ν as needed and a dashed line indicatesthat xη = jy where x and y are generators in the appropriate bidegree

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ

Figure 3 The cohomology of G48

Remark 216 (1) Many of the later relations can be rephrased as relations formultiplication by j = c34∆ For example Theorem 215 (4) implies

jν = jε = jκ = 0


and (5) impliesjκ = c24η

4

and (6) impliesjmicro = c24c6∆minus1η

These last two equations explain the dashed lines in Figure 4

(2) Multiplication by κ Hs(G48 Et) rarr Hs+4(G48 Et+24) is surjective and anisomorphism if s gt 0 In fact up to a unit ∆minus1κ isin H4(G48 E0) is the imageof the periodicity class in group cohomology for Q8 under the inclusion of trivialcoefficients

Z8 sim= H4(Q8W)G48Q8 sim= H4(G48W)rarr H4(G48 E0)

The congruence (213) and the relations of Theorem 215 now give the followingresult Note that the class c4 becomes invertible in lim vminus1

1 Hlowast(G48 Elowast2n) and we

may define b2 = c6c4 (Warning This class b2 is related to but not quite thesame as the class b2 of Proposition 210 Both uses of b2 appear in the literature)

Proposition 217 The class b2 reduces to v21 in vminus1

1 Hlowast(G48 Elowast2) There areisomorphisms

Z2((j))[bplusmn12 η](2η) sim= lim vminus1

1 Hlowast(G48 Elowast2n)

andF2((j))[vplusmn1

1 η] sim= vminus11 Hlowast(G48 Elowast2)

Under the reduction map Hlowast(G48 Elowast)rarr Hlowast(G48 Elowast2) we have

c4 7rarr v41

c6 7rarr v61

c4κ 7rarr v41κ = ∆η4

micro 7rarr v21η

Under the localization map Hlowast(G48 Elowast)rarr vminus11 Hlowast(G48 Elowast2) we have

∆ 7rarr v121 j

κ 7rarr v81η

4j

and that ν ε and κ map to zero

We have the following see [HM98] [Bau08] or [DFHH14]

Proposition 218 The homotopy ring πlowastEhG48 is periodic of period 192 with

periodicity generator detected by ∆8 The bo-patterns L1(EhG48) are concentratedin degrees congruent to 0 1 2 and 4 modulo 8

Remark 219 We will not try to enumerate the pure K(2)-classes of M2(EhG48)this information is known (by the same references as for Proposition 218) but wewonrsquot need that information in its entirety and it is rather complicated to writedown What we will need can be read off of Figure 4 which is adapted from thecharts created by Tilman Bauer [Bau08] Section 8


This chart shows a section of the Einfin-page of the Adams-Novikov Spectral Se-quence

Hs(G48 Et) =rArr πtminussEhG48

It is in the standard Adams bigrading (tminus s s)

40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2

1Figure 4 The homotopy groups πiE

hG48 for 40 le i le 70

Some additive and multiplicative extensions are displayed as well The non-zeropermanent cycles are in black some other elements mostly built from patternsaround elements of the form ∆jκi have been left in gray for orientation eventhough they do not last to the Einfin-page Bullets with circles are elements of order4 bullets with two circles are elements of order 8 Vertical lines are extensionsby multiplication by 2 lines raising homotopy degree by 1 are η-extensions linesraising homotopy degree by 3 are ν-extensions

The lines 0 le s le 2 display the bo-patterns the adorned boxes and circles allrepresent ideals of either Z2[[j]] or F2[[j]]

sim= Z2[[j]]

sim= (2) sube Z2[[j]]

sim= (4 j) sube Z2[[j]]

sim= F2[[j]]

sim= (j) sube F2[[j]]

Elements not falling into one of these patterns are annihilated by j The η-extensionfrom (t minus s s) = (65 3) entry is ambiguous We mark it as non-zero because wemay choose as Bauer does the two generators of the group of pure K(2)-classes inπ65E

hG48 to be

e[45 5]κ and e[51 1]κ

where e[45 5] isin π45EhG48 and e[51 1] isin π51E

hG48 are generators detected by ∆κηand ∆2ν respectively The class e[51 1]κ is detected on the s = 3 line by ∆2κν ande[51 1]κη 6= 0

We now record from Figure 4 some data about our crucial homotopy classes


Lemma 220 There is an isomorphism

Z2 sim= π45EhG48

The generator is detected by the class

∆κη isin H5(G48 E50)

The class ∆κ2η2 isin H10(G48 E76) is a non-zero permanent cycle detecting a gen-erator of the subgroup π66E

hG48 of the pure K(2)-classes of that degree

We close with some remarks on the role of micro in the d3 differentials

Lemma 221 Let micro isin H1(G2 E6) be the image of the class

α3 isin Ext1BPlowastBP (Σ6BPlowast BPlowast)

Then in any of the Adams-Novikov Spectral Sequences

Hlowast(FElowast) =rArr πtminussEhF

and for any x isin Hlowast(FElowast) we have

d3(xmicro) = d3(x)micro+ xη4

In the spectral sequences

(222) Hlowast(FElowast2) =rArr πtminuss(EhF and S2)

we have

d3(v21x) = d3(x)v2

1 + xη3 + y

where yη = 0 Finally in the spectral sequence (222) we have

d3(v41x) = v4

1d3(x)

Proof In the Adams-Novikov spectral sequence

ExtsBPlowastBP (ΣtBPlowast BPlowast) =rArr Z(2) otimes πtminussS0

we have d3(α3) = η4 (In fact by [MRW77] Corollary 423 η4 6= 0 at E2 and

E162sim= Z2 generated by α3 The differential is then forced) Since the fixed point

spectral sequence is a module over this standard Adams-Novikov Spectral Sequencethe first formula follows The second formula follows because v2

1η = α3 in

ExtsBPlowastBP (ΣtBPlowast BPlowast2)

The third formula follows from the fact that S2 has a v41-self map

3 Algebraic and topological resolutions

In this section we review the centralizer resolution constructed by Hans-WernerHenn [Hen07] sect34 and then begin the construction of the topological duality reso-lution The details of the algebraic duality resolution can be found in [Bea15]

The two resolutions have complementary features While we will not try to makethis thought completely precise the duality resolution reflects in an essential waythe fact that the group S1

2 is a virtual Poincare duality group of dimension 3 Thecentralizer resolution on the other hand is much closer to being an Adams-Novikov


tower as there is an underlying relative homological algebra in the spirit of Miller[Mil81] See Remark 324 below

31 The centralizer resolution Hennrsquos centralizer resolutions grew out of hispaper [Hen98] which used the centralizers of elementary abelian subgroups of Snto detect elements in the cohomology of Sn At the prime 2 this approach needsa slight modification as the maximal finite 2-group in S2 is Q8 which is not ele-mentary abelian

Remark 31 In (110) we defined G24 sube S12 as the image of a group of auto-

morphisms of a supersingular elliptic curve The group S12 fits into a short exact

sequence

1minusrarr S12

S2N Z2minusrarr 1

where N is the reduced determinant map of (17) Let

π = 1 + 2ω

be an element of S2 where ω isinWtimes is a cube root of unity Notice that π is not anelement of S1

2 because N(π) = 3 Then we define Gprime24 = πG24πminus1 sube S1

2 This is asubgroup isomorphic to G24 but not conjugate to G24 in S1

2

Note that multiplication by π defines an equivalence EhG24 EhGprime24 For com-

plete details on this and more see [Bea15]

We now have the following result from sect34 of [Hen07] This is the algebraiccentralizer resolution

Theorem 32 There is an exact sequence of continuous S12-modules

0rarr Z2[[S12C6]]rarr Z2[[S1

2C2]]rarrZ2[[S12C6]]oplus Z2[[S1

2C4]]

rarr Z2[[S12G24]]oplus Z2[[S1

2Gprime24]]

εminusrarr Z2 rarr 0 (33)

The map ε is the sum of the augmentation maps

We will call this a resolution even though the terms are not projective as Z2[[S12]]-

modules It is an F-projective resolution an idea we explore below in Remark 324

Remark 34 Suppose we write

P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

Then for any profinite S12-module M we get a spectral sequence

Epq1sim= ExtqZ2[[S12]]

(PpM) =rArr Hp+q(S12M)

The E1-terms can all be written as group cohomology groups for example

E0q1sim= Hq(G24M)timesHq(Gprime24M)


We will call this the algebraic centralizer resolution spectral sequence In manyapplications the distinction between the groups G24 and Gprime24 disappears For ex-ample if M is a G2-module (such as EnX for some spectrum X) then multiplicationby π induces an isomorphism Hq(G24M) sim= Hq(Gprime24M)

Remark 35 We can induce the resolution (33) of S12-modules up to a resolution

of G2-modules and obtain an exact sequence

0rarr Z2[[G2C6]]rarr Z2[[G2C2]]rarr Z2[[G2C6]]oplus Z2[[G2C4]]

rarr Z2[[G2G24]]oplus Z2[[G2G24]]rarr Z2[[G2S12]]rarr 0

Since Gprime24 is conjugate to G24 in G2 we have Z2[[G2G24]] sim= Z2[[G2Gprime24]] as G2-

modules and we have made that substitution If F is any closed subgroup of G2then the equivalence of (119) gives us an isomorphism of twisted G2-modules

HomZ2[[G2]](Z2[[G2F ]] Elowast) sim= ElowastEhF

Combining these observations we a get an exact sequence of twisted G2-modules

(36) 0rarr ElowastEhS12 rarr

ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4

rarr ElowastEhC2 rarr ElowastE

hC6 rarr 0

We then have the following result this is the topological centralizer resolution ofTheorem 12 of [Hen07]

Theorem 37 The algebraic resolution of 36 can be realized by a sequence ofspectra

EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6

All compositions and all Toda brackets are zero modulo indeterminacy

Remark 38 The vanishing of the Toda brackets in this result has several impli-cations To explain these and for future reference we write echoing the notation ofRemark 34

F0 = EhG24 times EhG24

F1 = EhC6 times EhC4(39)

F2 = EhC2

F3 = EhC6

Then the resolution of Theorem 37 can be refined to a tower of fibrations underEhS

12

(310) EhS12 Y2

Y1 EhG24 times EhG24 = F0

Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO


Alternatively we could refine the resolution into a tower over EhS12 Let us write

X

Zoo

Y

for a cofiber sequence (ie a triangle) X rarr Y rarr Z rarr ΣX Then we have adiagram of cofiber sequences

(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3

where each of the compositions Fiminus1 rarr Ci rarr Fi is the map Fiminus1 rarr Fi in theresolution

The towers (310) and (311) determine each other This is because there is adiagram with rows and columns cofibration sequences

(312) Σminus(s+1)Cs+1

EhS12

Ys

ΣminussCs

EhS12

Ysminus1

ΣminussFs lowast Σminuss+1Fs

Using the tower over EhS12 of (311) we get a number of spectral sequences for

example if Y is any spectrum we get a spectral sequence for the function spectrum

F (YEhS12)

(313) Est1 = πtF (Y Fs) =rArr πtminussF (YEhS12)

Up to isomorphism this spectral sequence can be obtained from the tower of (310)this follows from (312)

Remark 314 It is direct to calculate ElowastCs and ElowastYs for the layers of the twotowers If we define Ks sube ElowastFs to be the image of ElowastFsminus1 rarr ElowastFs then ElowastCs sim=Ks and more if we apply Elowast to (311) we get a collection of short exact sequences

ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3

Notice that this implies that each of the dotted arrows of (311) has Adams-Novikov

filtration one Finally the cofibration sequence EhS12 rarr Ys rarr ΣminussCs+1 induces a

short exact sequence

0rarr ElowastEhS12 rarr ElowastYs rarr ΣminussKs+1 rarr 0


32 The duality resolution first steps We have the algebraic duality resolu-tion from [Bea15] The groups G24 and Gprime24 are defined in Remark 31


0rarr Z2[[S12G

prime24]]rarr Z2[[S1

2C6]]rarr Z2[[S12C6]]rarr Z2[[S1

2G24]]εrarrZ2 rarr 0(316)

where ε is the augmentation

Remark 317 As in Remark 34 we get a spectral sequence Suppose we write

Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]

Then for any profinite S12-module M such as ElowastX = (E2)lowastX for some finite spec-

trum X we get a spectral sequence


(QpM) =rArr Hp+q(S12M)

which we will call the algebraic duality resolution spectral sequence

As in Remark 35 and (36) we immediately have the following consequence

Corollary 318 There is an exact sequence of twisted G2-modules

0rarr ElowastEhS12 rarr ElowastE

hG24 rarr ElowastEhC6 rarr ElowastE

hC6 rarr ElowastEhG24 rarr 0

The first of these maps is induced by the map on homotopy fixed point spectra

EhS12 rarr EhG24 induced by the subgroup inclusion G24 sube S1

2

Wersquod now like to prove the following result paralleling Theorem 37 it alsoappears in [Hen07] The main work of the next two sections and indeed the maintheorem of this paper is to identify X

Proposition 319 The algebraic resolution of 318 can be realized by a sequenceof spectra

EhS12

qminusrarrEhG24 rarr EhC6 rarr EhC6 rarr X

with ElowastX sim= ElowastEhG24 as a twisted G2-module All compositions and all Toda

brackets are zero modulo indeterminacy

Remark 320 A consequence of the last sentence of this result is that this reso-

lution can be refined to a tower of fibrations under EhS12

(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO

or to a tower over EhS12

(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X


As in Remark 314 the dotted arrows have Adams-Novikov filtration 1 Examiningthis last diagram we see that X can be defined as the cofiber of D2 rarr EhC6 andit will follow as in Remark 314 that ElowastX sim= ElowastE

hG24 Thus Proposition 319 isequivalent to the following result See also [Hen07] or [Bob14]

Lemma 323 The truncated resolution



hC6

can be realized by a sequence of spectra

EhS12minusrarr EhG24 rarr EhC6 rarr EhC6

such that all compositions are zero and the one Toda bracket is zero modulo inde-terminacy

Proof The map EhS12qminusrarr EhG24 is induced by the inclusion map on homotopy fixed

point spectra induced by the subgroup inclusion G24 sube S12 To realize the other

maps and to show the compositions are zero we prove that Hurewicz map

π0F (EhF EhC6)minusrarr HomMor(E0EhF E0E

hC6)

to the category of Morava modules is an isomorphism for F = S12 or F = G24 To

see this first note there is an isomorphism

π0F (EhF EhC6) = π0E[[G2F ]]hC6 sim= H0(C6 E0[[G2F ]])

This follows from (125) and the fact that π0EhK = H0(KE0) whenever K sube C6

see sect21 Then we can finish the argument by using (118) and (121) to show

H0(C6 E0[[G2F ]]) sim= HomE0[[G2]](E0[[G2C6]] E0[[G2F ]])

sim= HomMor(E0EhF E0E

hC6)

This leaves the Toda bracket To see that it is zero modulo indeterminacy weshow that the indeterminacy is the entire group To be specific we show that the

inclusion EhS12 rarr EhG24 induces a surjection

πlowastF (EhG24 EhC6)minusrarr πlowastF (EhS12 EhC6)

Using (121) we can rewrite this map as

πlowastE[[G2G24]]hC6minusrarr πlowastE[[G2S12]]hC6

Again using (121) the inclusion EhS12 rarr EhG24 induces a map of C6-spectra

E[[G2G24]] F (EhG24 E)rarr F (EhS12 E) E[[G2S1

2]]

It is thus sufficient to show that the quotient map on cosets

G2G24minusrarr G2S12

has a C6-splitting Since G2sim= S2 o Gal(F4F2) every coset in G2S1

2 has a rep-resentative of form πiφεS1

2 where π is as in Remark 31 φ isin Gal(F4F2) is theFrobenius i isin Z2 and ε = 0 or 1 The splitting is then given by

πiφεS12 7minusrarr πiφεG24


33 Comparing the two resolutions There is a map from the centralizer towerto the duality tower we will not prove that here In the end we will only need asmall part of the data given by such a map and what we need is in Remark 327

Remark 324 The underlying algebra for the centralizer resolutions fits well therelative homological algebra usually deployed in building an Adams-Novikov towerthis goes back to Miller in [Mil81] among other sources

Here is more detail Let F be the set of conjugacy classes of finite subgroups ofS1

2 A continuous S12-module P is F-projective if the natural mapoplus

FisinFZ2[[S1

2]]otimesZ2[[F ]] Pminusrarr P

is split surjective where F runs over representatives for the classes in F The classof F-projectives is the smallest class of continuous S1

2-modules closed under finitesums retracts and containing all induced modules Z2[[S1

2]]otimesZ2[[F ]] M where M isa continuous F -module

The class of F-projectives defines a class of F-exact morphisms there are enoughF-projectives there are F-projective resolutions and so on All of this and moreis discussed in sect35 of [Hen07]

Comparing the two towers now begins with the following result see Remark (d)after Proposition 17 in [Hen07]

Proposition 325 The centralizer resolution (33) is an F-projective resolutionof the trivial S1

2-module Z2

Thus if we write Pbull rarr Z2 for the centralizer resolution (33) and Qbull rarr Z2 forthe duality resolution (316) then standard homological algebra gives us a map ofresolutions unique up to chain homotopy

Qbull

gbull

Z2

=

Pbull Z2

The map g0 Q0 rarr P0 can be chosen to be the inclusion onto the first factor

(326) Q0 = Z2[[S12G24]]

i1 Z2[[S12G24]]oplus Z2[[S1

2Gprime24]] = P0

Remark 327 This immediately gives a map from the centralizer resolution spec-tral sequence of Remark 34 to the duality resolution spectral sequence of Remark317 This map is independent of the choice of gbull at the E2-page This can belifted to a map from the centralizer tower to the duality tower although we donrsquotneed that here and wonrsquot prove it We note that (326) implies there is a commu-tative diagram where the horizontal maps are the edge homomorphisms of the two


spectral sequences

Hlowast(S12 Elowast)

plowast

=

Hlowast(G24 Elowast)timesHlowast(G24 Elowast)

(g0)lowast

Hlowast(S1

2 Elowast) qlowast Hlowast(G24 Elowast)

and (g0)lowast is projection onto the first factor This can be realized by a diagram ofspectra where the map g0 is again projection onto the first factor

EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24

4 Constructing elements in π192k+48X

We now turn to the analysis of the homotopy groups of X where Σminus3X is thetop fiber in the duality tower see Proposition 319 We have an isomorphism ofMorava modules ElowastX sim= ElowastE

hG24 and hence by Proposition 127 and Lemma 129a spectral sequence


See also Remark 130 The cohomology if G24 is discussed in Theorem 215 In thissection we show roughly that ∆8k+2 isin H0(G24 E192k+48) is a permanent cyclemdashwhich would certainly be necessary if our main result is true The exact result isbelow in Corollary 47 In the next section we will use this and a mapping spaceargument to finish the identification of the homotopy type X

The statements and the arguments in this section have a rather fussy naturebecause the spectrum X has no a priori ring or module structure and in particularthe W-algebra structure on Hlowast(G24 Elowast) does not immediately extend to a W-module structure on πlowastX

The results of this section were among the main results in the first authorrsquos thesis[Bob14] and the key ideas for the entire project can be found there

We begin by combining Remark 139 and Lemma 220 to obtain the followingresult Note that 45 equiv 5 modulo 8 so there is no contributions from the bo-patternsin that degree In all degrees the bo-patterns lie in Adams-Novikov filtration at most2 The following is an immediate consequence of Lemma 137 and Lemma 220


F4sim= π45E

hG24

We can chose an F4 generator detected by the class



The class ∆κ2η2 isin H10(G24 E76) is a non-zero permanent cycle detecting an F4

generator of the subgroup of π66EhG24 consisting of the elements of Adams-Novikov

filtration greater than 2

Let p EhS12 rarr EhG24timesEhG24 be the augmentation in the topological centralizer

resolution of Theorem 37 This is the same map as from the top to the bottom ofthe centralizer resolution tower (310)

Lemma 42 Let k isin Z The map

plowast π192k+45EhS12minusrarr π192k+45(EhG24 times EhG24)

is surjective If x isin π192k+45EhS12 has the property that plowast(x) 6= 0 then x has

Adams-Novikov filtration at most 5 xκη 6= 0 and xκη is detected by a class ofAdams-Novikov filtration at most 10

Proof For the first statement we examine the homotopy spectral sequence of thecentralizer tower (313) In this case this spectral sequence reads

πtFs =rArr πtminussEhS12

The fibers Fs are spelled out in (39) We are asking that the edge homomorphism

plowast π192k+45EhS12minusrarr π192k+45F0

be surjective The crucial input is that

πkEhC6 sube πkEhC2 = 0

for k = 45 46 and 47 and that π45EhC4 = 0 See Proposition 29 and Figure 2

The final statement follows from Lemma 41

Now let q EhS12 rarr EhG24 be the augmentation in the topological duality reso-

lution of Proposition 319 This is also the projection from the top to the bottom

of the duality resolution tower (321) Let i Σminus3X rarr EhS12 be the map from the

top fiber of the duality tower Consider the commutative diagram

(43) π192k+45Σminus3Xilowast

r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4

π192k+45(EhG24 times EhG24)(g0)lowast

π192k+45EhG24

The bottom row is short exact and induced by the map between the resolutionsSee Remark 327 The map r is defined by this diagram and the fact that thecomposition

πlowastΣminus3Xminusrarr πlowastE

hS12 qlowastminusrarr πlowastEhG24

is zero

Proposition 44 The map

r π192k+45Σminus3X rarr F4


is surjective If y isin π192k+45Σminus3X is any class so that r(y) 6= 0 then y is detectedby a class

fdef= f(j)∆8k+2 isin H0(G24 E192k+48) sim= W [[j]]∆8k+2

Furthermore

fκη isin H5(G24 E192k+73)

is a non-zero permanent cycle in the spectral sequence for πlowastX

Proof In the diagram (43) the map plowast is onto and any element x in the kernelof (g0)lowast must have filtration at least 1 in the homotopy spectral sequence of theduality tower Since πkE

hC6 = 0 for k = 46 and k = 47 by Proposition 212 anysuch element must be the image of class y isin πlowastΣminus3X This shows r is surjective

The map Σminus3X rarr EhS12 raises Adams-Novikov filtration by 3 see the diagram

of (322) and the remarks thereafter If r(y) = plowastilowast(y) 6= 0 then by Lemma 42 ymust have Adams-Novikov filtration at most 2 however by Theorem 215 and thechart of Figure 3 we have that

H1(G24 E192k+49) = 0 = H2(G24 E192k+50)

Thus y must have filtration 0 Similarly 0 6= yκη must have filtration at leastfive and at most 7 Again we examine the chart of Figure 3 to find it must havefiltration 5 and be detected in group cohomology as claimed

Recall that we are writing S2 for the mod 2 Moore spectrum

Proposition 45 Let y isin π192k+48X be detected by

f = f(j)∆8k+2 isin H0(G24 Elowast)

If r(y) 6= 0 then f and fκη are non-zero permanent cycles in the spectral sequence

Hlowast(G24 Elowast2) =rArr πlowast(X and S2)

Proof This follows from Proposition 44 and the fact that

H5(G24 E192k+73)rarr H5(G24 E192k+732)

is injective This last statement can be deduced from the long exact sequence incohomology induced by the short exact sequence 0 rarr Elowast rarr Elowast rarr Elowast2 rarr 0 andthe fact that

H6(G24 E192k+73) = 0

See Figure 3

The crucial theorem then becomes

Theorem 46 Let y isin π192k+48X and let


be the image of y under the edge homomorphism

πlowastXminusrarr H0(G24 Elowast)

If f(j) equiv 0 modulo (2 j) then r(y) = 0


Proof We will show that if f(0) equiv 0 modulo 2 then fκ = 0 in Elowastlowast4 (X and S2) Wecan then apply Proposition 45

Under the assumption f(0) equiv 0 modulo 2 we have

f = jg(j)∆8k+2 isin H0(G24 Elowast2)

We will show that in the Adams-Novikov Spectral Sequence for LK(2)(X andS2) wehave

d3(v81g(j)∆8k+2micro) = fκ

The result will follow

We appeal to Theorem 215 Remark 216 and the chart of Figure 3 We have

jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4

Since fκ is a d3-cycle d3 is η-linear and

η4 E192k+4803 2 sim= H0(G24 (E2)192k+48)rarr H4(G24 E192k+52) sim= E192k+48+48

3

is injective we have that

d3(v81g(j)∆8k+2) = 0

It now follows from Lemma 221 that

d3(v81g(j)∆8k+2micro) = v8

1g(j)∆8k+2η4 = fκ

This is what we promised

The next result has a slightly complicated statement because we donrsquot know yetthat πlowastX is a W-module

Corollary 47 There is a commutative diagram

π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4

where the bottom map is some possibly non-trivial isomorphism of groups and

ε(f(j)∆8k+2) = f(0) mod (2 j)

There are homotopy classes xki isin π192k+48X i = 1 2 detected by classes

fi(j)∆8k+2 isin H0(G24 E192k+48)

so that f1(0) and f2(0) span F4 as an F2 vector space

Proof This is an immediate consequence of Theorem 46


Remark 48 Lemma 44 produces classes f isin π45Σminus3X for which fκη 6= 0 The

image of any such f in π45EhS12 is non-zero and has Adams-Novikov filtration at

least 3 It is natural to ask what we know about these classes In particular doesone of these classes come from the homotopy groups of sphere itself under the unit

map πlowastS0 rarr πlowastE

hS12

There are classes x isin π45S0 detected in the Adams Spectral Sequence by h3

4 =h2

3h5 Note any such class has filtration 3 There is a choice for x that seem tosupport non-zero multiplications by many of the basic elements in πlowastS

0 includingη and κ It would be easy to guess that this class is mapped to the class we haveconstructed but wersquove not yet settled this one way or another

For more about this class in π45S0 see the chart ldquoThe Einfin-page of the classical

Adams Spectral Sequencerdquo in [Isa14] The fact that some class detected by h23h5

can have a non-zero κ multiplication can be found in Lemma 4114 of [Isa] Seealso Table 33 of that paper Note that Isaksen is careful to label this lemma astentative as this is in the range where the homotopy groups of spheres still needexhaustive study In the table A33 of [Rav86] a related class is posited to bedetected in the Adams-Novikov Spectral Sequence by the class γ4 also of filtration3 but note the question mark there

5 The mapping space argument

We would like to extend the results of the Section 4 in the following way Letι S0 rarr EhG48 be the unit and r the composition

π192k+48X sim= π192k+45(Σminus3X)rarr π192k+45EhG24 sim= F4

defined in (43)

Theorem 51 The composite

π192k+48F (EhG48 X)ιlowast π192k+48X

r F4

is surjective

We can use this result to build maps out of EhG48 as follows Recall from (121)that if F sube G2 is a closed subgroup then there is an isomorphism Elowast[[G2F ]] sim=ElowastEhF of Elowast[[G2]]-modules Also Proposition 319 and the Universal CoefficientTheorem give an isomorphism ElowastX sim= Elowast[[G2G24]]


Consider the following diagram Note we are using that Eminus48 = E48

(52) π48(EhG48 X)ilowast

H

π48X

H

HomG2(E0[[G2G24]] Eminus48[[G2G48]])

ιlowast HomG2(E0[[G2G24]] Eminus48)

sim=

W[[j]]∆2 sim= (E48)G24

ε

F4

where the maps labelled H are the Hurewicz maps for Elowast(minus) and the map ε reducesmod (2 j) By Corollary 47 the vertical composition on the right is r up to someautomorphism of F4 Proposition 44 and Corollary 47 then yield the followingcorollary to Theorem 51 using the case when k = 0

Corollary 53 Let f(j)∆2 isin H0(G24 E48) Then there is a map

φ Σ48EhG48 rarr X

so that ιlowast(φ) equiv f(0) modulo 2

We will use this result to show that there is an equivalence Σ48EhG24 rarr X SeeTheorem 58 below

We now begin the proof of Theorem 51 Let p EhS12 rarr EhG24 times EhG24 be the

projection from the top to the bottom of the centralizer resolution tower


plowast π192k+45F (EhG48 EhS12)rarr π192k+45(F (EhG48 EhG24)times F (EhG48 EhG24))

is surjective

Proof We apply F (EhG48 minus) to the centralizer tower and examine the resultingspectral sequence in homotopy See (313) The spectral sequence reads

πtF (EhG48 Fs) =rArr πtminussF (EhG48 EhS12)

and the fibers Fs are described in (39) Thus we need to know

0 = π192k+45F (EhG48 EhC6 or EhC4)

= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)

We can use (124) Note that C2 is central so all of the subgroups Fx contain C2Therefore the crucial input is as before


for k = 45 46 and 47 and that π45EhC4 = 0 See Propositions 28 29 and 212

See also Figure 2


Let q EhS12 rarr EhG24 be the projection from the top to the bottom of the duality

resolution tower Using Remark 327 we now can produce a commutative diagramwhere we have abbreviated F (EhG48 Y ) as F (Y ) and wersquore writing n = 192k+ 45

(55) πnF (Σminus3X)

r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast

πn(F (EhG24)times F (EhG24))(g0)lowast

ιlowast

πnF (EhG24)

ιlowast

πnE

hG24 πn(EhG24 times EhG24)(g0)lowast

πnEhG24

The maps labelled (g0)lowast are induced from the map g0 EhG24 times EhG24 rarr EhG24

discussed in Remark 327 It is projection onto the first factor The lower two rowsare split short exact the maps labelled ιlowast are all onto and the maps plowast and qlowast areonto The map r is then defined by the requirement that the upper right squarecommute It is the analog of the map r of (43) and in fact we have a commutativediagram

π192k+45F (EhG48 Σminus3X)ιlowast

r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)

ιlowast π192k+45EhG24 sim= F4

Since EhG24 is an EhG48-module spectrum the evaluation map

ι F (EhG48 EhG24)rarr EhG24

is split Hence Theorem 51 follows from the next result

Lemma 56 The map

r π192k+45F (EhG48 Σminus3X)rarr π192k+45F (EhG48 EhG24)

is onto

Proof The proof is an exact copy of the first part of the argument for Proposition44 generalized to mappings spaces that is we examine the homotopy spectral

sequence built from the duality tower for F (EhG48 EhS12) In the diagram (55) the

map qlowast is onto and any element x in the kernel of (g0)lowast must have filtration at least 1in the homotopy spectral sequence of the duality tower Since πkF (EhG48 EhC6) =0 for k = 46 and k = 47 using (124) and Propositions 28 and 29 any suchelement must be the image of class from πlowastF (EhG48 Σminus3X)

Remark 57 Note that all of these arguments would work with replacing EhG48

with EhG24

The next result is our main theorem

Theorem 58 There is an equivalence

Σ48EhG24minusrarr X


realizing the given isomorphism of Morava modules

ElowastEhG24

sim=minusrarr ElowastX

Proof We actually use the given isomorphism of Morava modules to produce a(non-equivariant) equivalence

C+2 and Σ48EhG48 Σ48(EhG48 or EhG48)rarr X

Here C2 = Gal = Gal(F4F2) Then we will apply Lemma 137

We begin with some algebra Recall from Remark 114 that the 2-Sylow subgroupS2 sube S2 can be decomposed as KoQ8 Since G24 = Q8 oFtimes4 and G48 = G24 oGalwe have that Elowast[[G2G24]] and Elowast[[G2G48]] are free Elowast[[K]] modules of rank 2 andrank 1 respectively Since K is a finitely generated pro-2-group the ring E0[[K]]is a complete local ring with maximal ideal mK given by the kernel of the reducedaugmentation

E0[[K]]minusrarr E0minusrarr F4

We will produce a map

f Σ48(EhG48 or EhG48)rarr X

so that the map of E0[[K]]-modules

Elowastf E48Xminusrarr E48(Σ48(EhG48 or EhG48))

is an isomorphism modulo mK Then by the appropriate variant of NakayamarsquosLemma (see Lemma 43 of [GHMR05]) it will be an isomorphism of E0[[K]]-modules and hence of E0[[G2]]-modules as required

Since G24sim= Q8 oFtimes4 and G2

sim= ((KoQ8)oFtimes4 )oGal we have an isomorphismof K-sets G2G24

sim= KtKφ where φ is the Frobenius in the Galois group Then wehave the following commutative diagram It is an expansion of the diagram (52)


ιlowast

HomK(E0[[K]]oplus E0[[Kφ]] Eminus48[[K]])

ιlowast

HomG2(E0[[G2G24]] Eminus48)

sim=

HomK(E0[[K]]oplus E0[[Kφ]] Eminus48)

sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24

The top two horizontal maps are forgetful maps remembering only the K actionThe third horizontal map is the composition

W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2

and the bottom map is x 7rarr (x φ(x)) The top vertical maps are induced by themap Elowast(EhG48) rarr ElowastS0 given by the unit the middle vertical map evaluates a


homomorphism at 1 isin E0[[G2G24]] note we are again using that Eminus48 sim= E48The final map is reduction modulo the maximal ideals in both cases

By Corollary 53 we can produce two maps

fi Σ48EhG48 rarr X i = 1 2

so that (fi)lowast(ι) equiv ωi∆2 modulo (2 j) where ω isin F4 is the primitive cube root ofunity We now examine the fate of

E0(fi) E0[[G2G24]]minusrarr Eminus48[[G2G48]]

as we work from the upper left to the bottom right of this diagram Using theformulas of the previous paragraph we have

E0(f1) 7rarr (ω ω2) and E0(f2) 7rarr (ω2 ω)

Finally letf = f1 or f2 Σ48(EhG48 or EhG48)rarr X

If we apply Elowast(minus) to this map we get a map

Elowastf E0[[G2G24]]rarr Eminus48[[G2G48]]times Eminus48[[G2G48]]

which yields a map of K-modules

Elowastf E0[[K]]2minusrarr Eminus48[[K]]2

which modulo the maximal ideal mK gives the map

F24minusrarr F2

4

given by the matrix (ω ω2

ω2 ω

)with determinant ω2 + ω = 1 Thus Elowastf is an isomorphism as needed

We can now complete the proof of Theorem 01 and construct the topologicalduality resolution Proposition 319 and Theorem 58 imply the following Recallfrom Theorem 29 that EhC6 is 48-periodic

Corollary 59 There exists a resolution of EhS12 in the K(2)-local category at the

prime 2


Remark 510 In [GHMR05] working at the prime 3 the authors were able toproduce a topological resolution for LK(2)S

0 = EhG2 itself by the same methods

that produced the resolution for EhG12 The resolution for the sphere was essentially

a double of that for EhG12 Not only do the methods of [GHMR05] not apply to the

case p = 2 itrsquos very unlikely that there is a topological resolution of the sphere

or for EhS2 which could be obtained by doubling the resolution of EhS12 There

are any number of difficulties but the first obstacle is that we have only a semi-direct decomposition S1

2oZ2sim= S2 at p = 2 rather than the product decomposition

G12timesZ3

sim= G2 at the prime 3 This makes the algebra much harder and it only getsworse from there Other short topological resolutions are possible of course andcould be very instructive This is the subject of current research by Agnes Beaudryand Hans-Werner Henn


References

[Bau08] T Bauer Computation of the homotopy of the spectrum tmf In Groups homotopy

and configuration spaces volume 13 of Geom Topol Monogr pages 11ndash40 GeomTopol Publ Coventry 2008

[Bea15] Agnes Beaudry The algebraic duality resolution at p = 2 Algebr Geom Topol

15(6)3653ndash3705 2015[Bea17] Agnes Beaudry The chromatic splitting conjecture at n = p = 2 Geom Topol

21(6)3213ndash3230 2017

[Beh06] Mark Behrens A modular description of the K(2)-local sphere at the prime 3 Topol-ogy 45(2)343ndash402 2006

[Beh12] Mark Behrens The homotopy groups of SE(2) at p ge 5 revisited Adv Math

230(2)458ndash492 2012

[BH16] Tobias Barthel and Drew Heard The E2-term of the K(n)-local En-Adams spectralsequence Topology Appl 206190ndash214 2016

[BO16] Mark Behrens and Kyle Ormsby On the homotopy of Q(3) and Q(5) at the prime 2

Algebr Geom Topol 16(5)2459ndash2534 2016[Bob14] Irina Bobkova Resolutions in the K(2)-local Category at the Prime 2 ProQuest LLC

Ann Arbor MI 2014 Thesis (PhD)ndashNorthwestern University

[Bou79] A K Bousfield The localization of spectra with respect to homology Topology18(4)257ndash281 1979

[DFHH14] CL Douglas J Francis AG Henriques and MA Hill Topological Modular Formsvolume 210 of Mathematical Surveys and Monographs Amer Math Soc 2014

[DH95] Ethan S Devinatz and Michael J Hopkins The action of the Morava stabilizer group

on the Lubin-Tate moduli space of lifts Amer J Math 117(3)669ndash710 1995[DH04] Ethan S Devinatz and Michael J Hopkins Homotopy fixed point spectra for closed

subgroups of the Morava stabilizer groups Topology 43(1)1ndash47 2004

[GH04] P G Goerss and M J Hopkins Moduli spaces of commutative ring spectra InStructured ring spectra volume 315 of London Math Soc Lecture Note Ser pages

151ndash200 Cambridge Univ Press Cambridge 2004

[GH16] Paul G Goerss and Hans-Werner Henn The Brown-Comenetz dual of the K(2)-localsphere at the prime 3 Adv Math 288648ndash678 2016

[GHM04] Paul Goerss Hans-Werner Henn and Mark Mahowald The homotopy of L2V (1) for

the prime 3 In Categorical decomposition techniques in algebraic topology (Isle ofSkye 2001) volume 215 of Progr Math pages 125ndash151 Birkhauser Basel 2004

[GHM14] P Goerss H-W Henn and M Mahowald The rational homotopy of the K(2)-local

sphere and the chromatic splitting conjecture at the prime 3 and level 2 Doc Math191271ndash1290 2014

[GHMR05] P Goerss H-W Henn M Mahowald and C Rezk A resolution of the K(2)-localsphere at the prime 3 Ann of Math (2) 162(2)777ndash822 2005

[GHMR15] P Goerss H-W Henn M Mahowald and C Rezk On Hopkins Picard groups for

the prime 3 and chromatic level 2 J Topol 8(1)267ndash294 2015[Hen98] Hans-Werner Henn Centralizers of elementary abelian p-subgroups and mod-p coho-

mology of profinite groups Duke Math J 91(3)561ndash585 1998[Hen07] H-W Henn On finite resolutions of K(n)-local spheres In Elliptic cohomology vol-

ume 342 of London Math Soc Lecture Note Ser pages 122ndash169 Cambridge Univ

Press Cambridge 2007

[HG94] M J Hopkins and B H Gross The rigid analytic period mapping Lubin-Tate spaceand stable homotopy theory Bull Amer Math Soc (NS) 30(1)76ndash86 1994

[HKM13] H-W Henn N Karamanov and M Mahowald The homotopy of the K(2)-localMoore spectrum at the prime 3 revisited Math Z 275(3-4)953ndash1004 2013

[HM98] M Hopkins and M Mahowald From elliptic curves to homotopy theory 1998

Unpublished Available from httphopfmathpurdueeduHopkins-Mahowald

eo2homotopypdf

[HS99] M Hovey and N P Strickland Morava K-theories and localisation Mem Amer

Math Soc 139(666)viii+100 1999[Isa] Daniel C Isaksen Stable stems Mem Amer Math Soc To Appear


[Isa14] Daniel C Isaksen Classical and motivic Adams charts ArXiv1401 4983[ mathAT]

January 2014

[Lad13] O Lader Une resolution projective pour le second groupe de morava pour p ge 5et applications Available from httpstelarchives-ouvertesfrtel-00875761

document 2013

[Mil81] Haynes R Miller On relations between Adams spectral sequences with an applicationto the stable homotopy of a Moore space J Pure Appl Algebra 20(3)287ndash312 1981

[MR09] M Mahowald and C Rezk Topological modular forms of level 3 Pure Appl Math

Q 5(2 Special Issue In honor of Friedrich Hirzebruch Part 1)853ndash872 2009[MRW77] Haynes R Miller Douglas C Ravenel and W Stephen Wilson Periodic phenomena

in the Adams-Novikov spectral sequence Ann of Math (2) 106(3)469ndash516 1977

[Rav84] D C Ravenel Localization with respect to certain periodic homology theories AmerJ Math 106(2)351ndash414 1984

[Rav86] Douglas C Ravenel Complex cobordism and stable homotopy groups of spheres vol-ume 121 of Pure and Applied Mathematics Academic Press Inc Orlando FL 1986

[Sil09] Joseph H Silverman The arithmetic of elliptic curves volume 106 of Graduate Texts

in Mathematics Springer Dordrecht second edition 2009[Str] N P Strickland Level three structures Unpublished

[Str00] N P Strickland Gross-Hopkins duality Topology 39(5)1021ndash1033 2000

[SW02] Katsumi Shimomura and Xiangjun Wang The Adams-Novikov E2-term for πlowast(L2S0)at the prime 2 Math Z 241(2)271ndash311 2002

[SY95] Katsumi Shimomura and Atsuko Yabe The homotopy groups πlowast(L2S0) Topology

34(2)261ndash289 1995

School of Mathematics Institute for Advanced Study Princeton NJ 08540 USA

E-mail address ibobkovamathiasedu

Department of Mathematics Northwestern University Evanston IL 60208 USA

E-mail address pgoerssmathnorthwesternedu




4 Constructing elements in 192k+48X


References


point topological resolutions become a useful way to organize the contributions ofthe finite subgroups The key to unlocking this idea is the Hopkins-Miller theoremwhich implies that Gn and hence all of its finite subgroups act on En and thatLK(n)S

0 EhGnn See [DH04] for this and more

The prototypical example is at n = 1 and p = 2 Adams and Baird [Bou79] andRavenel [Rav84] showed that here we have a fiber sequence

LK(1)S0 rarr KO

ψ3minus1minusminusminusrarr KO

where KO is 2-complete real K-theory For a suitable choice of a height one formalgroup Γ1 we can take E1 = K where K is 2-complete complex K-theory ThenG1 = Aut(Γ1F2) sim= Ztimes2 is the units in the 2-adic integers and C2 = plusmn1 sube Ztimes2acts through complex conjugation We can then rewrite this fiber sequence as

LK(1)S0 = EhG1

1 minusrarr EhC21

ψ3minus1minusminusminusrarr EhC21

and ψ3 is a topological generator of Ztimes2 plusmn1 Z2

For higher heights the topological resolutions will not be simple fiber sequencesbut finite towers of fibrations with the successive fibers built from EhFi where Firuns over various finite subgroups of Gn

In [GHMR05] the authors generalized the fiber sequence of the K(1)-local caseto the case n = 2 and p = 3 One way to say what they proved is the following Letus write E = E2 to simplify notation First we have a split short exact sequence ofgroups

1 rarr G12

G2N Z3 rarr 1

where N is obtained from a determinant (See (17)) and hence a fiber sequence



12

where ψ is any element of G2 which maps by N to a topological generator of Z3

Then second there exists a resolution of EhG12

EhG12 rarr EhG24 rarr Σ8EhSD16 rarr Σ40EhSD16 rarr Σ48EhG24

Here resolution means each successive composition is null-homotopic and all possibleToda brackets are zero modulo indeterminacy thus the sequence refines to a towerof fibrations

EhG12 X2

X1 EhG24

Σ45EhG24

OO

Σ38EhSD16

OO

Σ7EhSD16

OO

The maximal finite subgroup of G2 of 3-power order is a cyclic group C3 oforder 3 it is unique up to conjugation in G1

2 The group G24 is the maximal finitesubgroup of G2 containing C3 The subgroup SD16 is the semidihedral group oforder 16 Because of the symmetry of this resolution and because G1

2 is a virtualPoincare duality group of dimension 3 this is called a duality resolution Thisresolution and related resolutions were instrumental in exploring the K(2)-localcategory at primes p gt 2 See [GHM04] [HKM13] [Beh06] [GHM14] [GHMR15]








12





12 is innocuous



prime 2













where









0








Contents






References 43






n













lowast












p1 Ztimesp


(17) GnN




N S1n = Sn capG1



S1n o Zp sim= Sn



(18) y2 + y = x3


(19) y2 + 3u1xy + (u31 minus 1)y = x3







with its image




Sl2(Z3)
















kN rarr 0







E0E sim= map(Gn E0)













Note that(ElowastE


F






(121) E[[GnF1]] F (EhF1 E)



FE(E and EhF1 E)

F (EhF1 E)



h(sum

aggF1) =sum

h(ag)hminus1gF1


F2GnU


F (EhU E)hF2

F (EhU EhF2)





(gminus1a) gxU



and a transfer map



(123)prodF2GnU

EhFx

tr

E[[GnU ]]hF2

g

prodF2GnV

EhGx E[[GnV ]]hF2



(124) holimi

prodF2GnUi

EhFx F (EhF1 EhF2)





F2GnUj


prodF2GnUi

πlowastEhFx


(125) limi

prodF2GnUi






(3) TorE01 (Fpn M) = 0







Est2sim= πsπt(E




















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References








12





12 is innocuous



prime 2













where









0








Contents






References 43






n













lowast












p1 Ztimesp


(17) GnN




N S1n = Sn capG1



S1n o Zp sim= Sn



(18) y2 + y = x3


(19) y2 + 3u1xy + (u31 minus 1)y = x3







with its image




Sl2(Z3)
















kN rarr 0







E0E sim= map(Gn E0)













Note that(ElowastE


F






(121) E[[GnF1]] F (EhF1 E)



FE(E and EhF1 E)

F (EhF1 E)



h(sum

aggF1) =sum

h(ag)hminus1gF1


F2GnU


F (EhU E)hF2

F (EhU EhF2)





(gminus1a) gxU



and a transfer map



(123)prodF2GnU

EhFx

tr

E[[GnU ]]hF2

g

prodF2GnV

EhGx E[[GnV ]]hF2



(124) holimi

prodF2GnUi

EhFx F (EhF1 EhF2)





F2GnUj


prodF2GnUi

πlowastEhFx


(125) limi

prodF2GnUi






(3) TorE01 (Fpn M) = 0







Est2sim= πsπt(E




















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References






where









0








Contents






References 43






n













lowast












p1 Ztimesp


(17) GnN




N S1n = Sn capG1



S1n o Zp sim= Sn



(18) y2 + y = x3


(19) y2 + 3u1xy + (u31 minus 1)y = x3







with its image




Sl2(Z3)
















kN rarr 0







E0E sim= map(Gn E0)













Note that(ElowastE


F






(121) E[[GnF1]] F (EhF1 E)



FE(E and EhF1 E)

F (EhF1 E)



h(sum

aggF1) =sum

h(ag)hminus1gF1


F2GnU


F (EhU E)hF2

F (EhU EhF2)





(gminus1a) gxU



and a transfer map



(123)prodF2GnU

EhFx

tr

E[[GnU ]]hF2

g

prodF2GnV

EhGx E[[GnV ]]hF2



(124) holimi

prodF2GnUi

EhFx F (EhF1 EhF2)





F2GnUj


prodF2GnUi

πlowastEhFx


(125) limi

prodF2GnUi






(3) TorE01 (Fpn M) = 0







Est2sim= πsπt(E




















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References



Contents






References 43






n













lowast












p1 Ztimesp


(17) GnN




N S1n = Sn capG1



S1n o Zp sim= Sn



(18) y2 + y = x3


(19) y2 + 3u1xy + (u31 minus 1)y = x3







with its image




Sl2(Z3)
















kN rarr 0







E0E sim= map(Gn E0)













Note that(ElowastE


F






(121) E[[GnF1]] F (EhF1 E)



FE(E and EhF1 E)

F (EhF1 E)



h(sum

aggF1) =sum

h(ag)hminus1gF1


F2GnU


F (EhU E)hF2

F (EhU EhF2)





(gminus1a) gxU



and a transfer map



(123)prodF2GnU

EhFx

tr

E[[GnU ]]hF2

g

prodF2GnV

EhGx E[[GnV ]]hF2



(124) holimi

prodF2GnUi

EhFx F (EhF1 EhF2)





F2GnUj


prodF2GnUi

πlowastEhFx


(125) limi

prodF2GnUi






(3) TorE01 (Fpn M) = 0







Est2sim= πsπt(E




















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References











lowast












p1 Ztimesp


(17) GnN




N S1n = Sn capG1



S1n o Zp sim= Sn



(18) y2 + y = x3


(19) y2 + 3u1xy + (u31 minus 1)y = x3







with its image




Sl2(Z3)
















kN rarr 0







E0E sim= map(Gn E0)













Note that(ElowastE


F






(121) E[[GnF1]] F (EhF1 E)



FE(E and EhF1 E)

F (EhF1 E)



h(sum

aggF1) =sum

h(ag)hminus1gF1


F2GnU


F (EhU E)hF2

F (EhU EhF2)





(gminus1a) gxU



and a transfer map



(123)prodF2GnU

EhFx

tr

E[[GnU ]]hF2

g

prodF2GnV

EhGx E[[GnV ]]hF2



(124) holimi

prodF2GnUi

EhFx F (EhF1 EhF2)





F2GnUj


prodF2GnUi

πlowastEhFx


(125) limi

prodF2GnUi






(3) TorE01 (Fpn M) = 0







Est2sim= πsπt(E




















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References




p1 Ztimesp


(17) GnN




N S1n = Sn capG1



S1n o Zp sim= Sn



(18) y2 + y = x3


(19) y2 + 3u1xy + (u31 minus 1)y = x3







with its image




Sl2(Z3)
















kN rarr 0







E0E sim= map(Gn E0)













Note that(ElowastE


F






(121) E[[GnF1]] F (EhF1 E)



FE(E and EhF1 E)

F (EhF1 E)



h(sum

aggF1) =sum

h(ag)hminus1gF1


F2GnU


F (EhU E)hF2

F (EhU EhF2)





(gminus1a) gxU



and a transfer map



(123)prodF2GnU

EhFx

tr

E[[GnU ]]hF2

g

prodF2GnV

EhGx E[[GnV ]]hF2



(124) holimi

prodF2GnUi

EhFx F (EhF1 EhF2)





F2GnUj


prodF2GnUi

πlowastEhFx


(125) limi

prodF2GnUi






(3) TorE01 (Fpn M) = 0







Est2sim= πsπt(E




















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References
















kN rarr 0







E0E sim= map(Gn E0)













Note that(ElowastE


F






(121) E[[GnF1]] F (EhF1 E)



FE(E and EhF1 E)

F (EhF1 E)



h(sum

aggF1) =sum

h(ag)hminus1gF1


F2GnU


F (EhU E)hF2

F (EhU EhF2)





(gminus1a) gxU



and a transfer map



(123)prodF2GnU

EhFx

tr

E[[GnU ]]hF2

g

prodF2GnV

EhGx E[[GnV ]]hF2



(124) holimi

prodF2GnUi

EhFx F (EhF1 EhF2)





F2GnUj


prodF2GnUi

πlowastEhFx


(125) limi

prodF2GnUi






(3) TorE01 (Fpn M) = 0







Est2sim= πsπt(E




















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References



E0E sim= map(Gn E0)













Note that(ElowastE


F






(121) E[[GnF1]] F (EhF1 E)



FE(E and EhF1 E)

F (EhF1 E)



h(sum

aggF1) =sum

h(ag)hminus1gF1


F2GnU


F (EhU E)hF2

F (EhU EhF2)





(gminus1a) gxU



and a transfer map



(123)prodF2GnU

EhFx

tr

E[[GnU ]]hF2

g

prodF2GnV

EhGx E[[GnV ]]hF2



(124) holimi

prodF2GnUi

EhFx F (EhF1 EhF2)





F2GnUj


prodF2GnUi

πlowastEhFx


(125) limi

prodF2GnUi






(3) TorE01 (Fpn M) = 0







Est2sim= πsπt(E




















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References



h(sum

aggF1) =sum

h(ag)hminus1gF1


F2GnU


F (EhU E)hF2

F (EhU EhF2)





(gminus1a) gxU



and a transfer map



(123)prodF2GnU

EhFx

tr

E[[GnU ]]hF2

g

prodF2GnV

EhGx E[[GnV ]]hF2



(124) holimi

prodF2GnUi

EhFx F (EhF1 EhF2)





F2GnUj


prodF2GnUi

πlowastEhFx


(125) limi

prodF2GnUi






(3) TorE01 (Fpn M) = 0







Est2sim= πsπt(E




















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References



F2GnUj


prodF2GnUi

πlowastEhFx


(125) limi

prodF2GnUi






(3) TorE01 (Fpn M) = 0







Est2sim= πsπt(E




















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References

















(gφ)(x) = φ(xg)


mapGn(Gs+1

n N uarrGn




Hlowast(Gn N uarrGn







mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References



mapGn(Gs+1

n N uarrGn















F









0






sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References





sim=

OO


=

OO


0
















Zp-modules


M
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References
















H0(Wtimes J) = 0






inminus1


nminus1

as needed













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References













Zp

sube

π0EhSn

W =

W




agg 7minusrarrsum

agg(ω)























sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References















sim=

Wotimes πlowastEhK

sim=













homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References




homotopy


hF


map







hF




a KO-algebra













n)FC2

and that






where v21 = u2




KOlowast















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References















is an isomorphism


n)












hC2)Ftimes4





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References





e3k16α

3 e3k16α

6 e3k+116 α4 e3k+2

16 α5



detected by e16α4






and






2η = 2γ = 4ξ = 0

and


and


and



ν2 = 2ξ 2ν = zν = ην = b2ν = γν = 0





0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References




0 2 4 6 8 10 12 14 16 18 20 22 24 260

2

4

6

8

otimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

otimesotimes

κ

δγ

ξ







phism









ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References




ν2 6 2 ν2

ε 8 2 δminus2ξ4


b 19 4 δ2νκ 20 4 δξ2

ηκ 21 2 ηξ3

bν 22 4 δ2ν2




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

a

νε

κ

cb

κ




sim=


sim=









F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References







F2[[j]][v1∆plusmn1](j∆ = v12




1






ν (1 4) 4 α22

micro (1 6) 2 α3

ε (2 10) 2 β2

κ (2 16) 2 β3

κ (4 24) 8 minus









2η = 4ν = 2micro = 2ε = 2κ = 8κ = 0


ην = 2ν2 = ν4 = microν = 0







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References







c4ν = c6ν = c4ε = c6ε = c4κ = c6κ = 0







0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300

2

4

6

8

10

12

14

otimes

micro

otimes

otimesκ



jν = jε = jκ = 0



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References



4












andF2((j))[vplusmn1



c4 7rarr v41

c6 7rarr v61


micro 7rarr v21η


∆ 7rarr v121 j

κ 7rarr v81η

4j










40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References





40 44 48 52 56 60 64 680

4

8

12

∆κη

∆κ2η

2

2

2





sim= Z2[[j]]



sim= F2[[j]]



hG48 to be

e[45 5]κ and e[51 1]κ






Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References



Z2 sim= π45EhG48














we have

d3(v21x) = d3(x)v2

1 + xη3 + y


d3(v41x) = v4

1d3(x)






1η = α3 in












sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References






sequence

1minusrarr S12

S2N Z2minusrarr 1


π = 1 + 2ω




2







2C4]]


2Gprime24]]






P0 = Z2[[S12G24]]times Z2[[S1

2Gprime24]]

P1 = Z2[[S12C6]]times Z2[[S1

2C4]]

P2 = Z2[[S12C2]]

P3 = Z2[[S12C6]]

















ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References












ElowastEhG24

timesElowastE

hG24

rarrElowastE

hC6

timesElowastE

hC4


hC6 rarr 0



EhS12

pminusrarrEhG24

timesEhG24

rarrEhC6

timesEhC4

rarr EhC2 rarr EhC6





F2 = EhC2

F3 = EhC6


12

(310) EhS12 Y2


Σminus3F3

OO

Σminus2F2

OO

Σminus1F1

OO



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References



X

Zoo

Y


(311) EhS12

p

C1oo

C2oo

C3oo

F0

F1

F2

F3




EhS12

Ys

ΣminussCs

EhS12

Ysminus1




F (YEhS12)




ElowastEhS12$$

$$

ElowastC10oo

ElowastC20oo

ElowastC30oo

sim=

ElowastF0

ltlt ltlt

ElowastF1

ltlt ltlt

ElowastF2

ltlt ltlt

ElowastF3








0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References




0rarr Z2[[S12G






Q0 = Z2[[S12G24]]

Q1 = Q2 = Z2[[S12C6]]

Q3 = Z2[[S12G

prime24]]













2



EhS12






(321) EhS12 Z2

Z1 EhG24

Σminus3X

OO

Σminus2EhC6

OO

Σminus1EhC6

OO


(322) EhS12

q $$

D1oo

D2oo

D3oo

EhG24

ltlt

EhC6

ltlt

EhC6

ltlt

X







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References







hC6








hC6)








hC6)








2]]












FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References






FisinFZ2[[S1








2-module Z2


Qbull

gbull

Z2

=

Pbull Z2


(326) Q0 = Z2[[S12G24]]


2Gprime24]] = P0



spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References


spectral sequences


plowast

=


(g0)lowast

Hlowast(S1



EhS12

p

=

EhG24 times EhG24

g0

EhS

12

q EhG24










F4sim= π45E

hG24


























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References
























r

π192k+45EhS12 =

plowast

π192k+45EhS12

qlowast

F4


π192k+45EhG24




is zero






Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References




Furthermore







H1(G24 E192k+49) = 0 = H2(G24 E192k+50)








H5(G24 E192k+73)rarr H5(G24 E192k+732)


H6(G24 E192k+73) = 0

See Figure 3















jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References









jκ = c24η4 = v8

1η4

and hence that

fκ = jg(j)∆8k+2κ = v81g(j)∆8k+2η4



3


d3(v81g(j)∆8k+2) = 0



1g(j)∆8k+2η4 = fκ




π192k+48X

r

H0(G24 E192k+48)

ε

F4 sim=

F4


ε(f(j)∆8k+2) = f(0) mod (2 j)


fi(j)∆8k+2 isin H0(G24 E192k+48)








hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References






hS12


4 =h2









defined in (43)



r F4

is surjective





H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References




H

π48X

H



sim=

W[[j]]∆2 sim= (E48)G24

ε

F4



φ Σ48EhG48 rarr X







is surjective





= π192k+46F (EhG48 EhC2)

= π192k+47F (EhG48 EhC6)




See also Figure 2





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References





r

πnF (EhS12)

=

plowast

πnF (EhS12)

qlowast

πnF (EhG24)

ιlowast


ιlowast

πnF (EhG24)

ιlowast

πnE


πnEhG24




r

π192k+45Σminus3X

r

π192k+45F (EhG48 EhG24)





Lemma 56 The map


is onto





with EhG24






ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References



ElowastEhG24
















ιlowast


ιlowast


sim=


sube

(E48)G24 sim= W[[j]]

E248sim= (W[[u1]])2

F4

F24


W[[j]]sube W[[u1]]

1timesφ (W[[u1]])2
















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References















F24minusrarr F2

4


ω2 ω




prime 2










G12timesZ3



References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References


References





21(6)3213ndash3230 2017



230(2)458ndash492 2012



























eo2homotopypdf





January 2014


document 2013












34(2)261ndash289 1995










References



January 2014


document 2013












34(2)261ndash289 1995










References

topological resolutions in k(2)-local homotopy theory …pgoerss/papers/xtheoremnew.pdf ·...

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