e–one, two, three infinity - university of notre damewgd/drafts/haynesslides.pdfambushes before...
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E–One, Two, Three . . . Infinity†
A tour through part of the mathematical world ofHaynes R. Miller
June 25, 2008, Bonn
† Sorry, George
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Charge to the speaker
Remain unstable!
Charge to the speaker
Remain unstable!
Charge to the speaker
Remain unstable!
Nothing chromatic!
A brief review of spectral sequencesThings to watch out for in what’s coming
Spectral sequence: a machine for calculation, which can. . .
vanishcollapsespawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out
All this drama (and more) takes place in papers by Haynes.
A brief review of spectral sequencesThings to watch out for in what’s coming
Spectral sequence: a machine for calculation, which can. . .
vanish
collapsespawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out
All this drama (and more) takes place in papers by Haynes.
A brief review of spectral sequencesThings to watch out for in what’s coming
Spectral sequence: a machine for calculation, which can. . .
vanishcollapse
spawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out
All this drama (and more) takes place in papers by Haynes.
A brief review of spectral sequencesThings to watch out for in what’s coming
Spectral sequence: a machine for calculation, which can. . .
vanishcollapsespawn an heir
suffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out
All this drama (and more) takes place in papers by Haynes.
A brief review of spectral sequencesThings to watch out for in what’s coming
Spectral sequence: a machine for calculation, which can. . .
vanishcollapsespawn an heirsuffer a localization
impersonate a rivaldivulge a (partial) secretwork absolutely everything out
All this drama (and more) takes place in papers by Haynes.
A brief review of spectral sequencesThings to watch out for in what’s coming
Spectral sequence: a machine for calculation, which can. . .
vanishcollapsespawn an heirsuffer a localizationimpersonate a rival
divulge a (partial) secretwork absolutely everything out
All this drama (and more) takes place in papers by Haynes.
A brief review of spectral sequencesThings to watch out for in what’s coming
Spectral sequence: a machine for calculation, which can. . .
vanishcollapsespawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secret
work absolutely everything out
All this drama (and more) takes place in papers by Haynes.
A brief review of spectral sequencesThings to watch out for in what’s coming
Spectral sequence: a machine for calculation, which can. . .
vanishcollapsespawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out
All this drama (and more) takes place in papers by Haynes.
A brief review of spectral sequencesThings to watch out for in what’s coming
Spectral sequence: a machine for calculation, which can. . .
vanishcollapsespawn an heirsuffer a localizationimpersonate a rivaldivulge a (partial) secretwork absolutely everything out
All this drama (and more) takes place in papers by Haynes.
The unstable Ad Spec Seq for generalized homologyBendersky, Curtis, Miller
X a space; R = MU, BP, connective ring spectrum
CX = Ω∞(R ∧ X ) X // CX//// C2X · · ·oo
E2¬ab(X ) =⇒ π∗X
E2-page: spawned by another SS (R, X nice)
Untori Primj R∗X =⇒ E2¬ab(X ) =⇒ π∗X
Partial secret revealed for R = BP1-line for X = odd sphere
On relations between Adams spectral sequenceswith an application to the stable homotopy of the Moore space
Fancy footwork for Moore spectrum M
E2BP(E2
Z/pM) Mahowald +3
May
E2Z/pM
Adams
E2BPM
Novikov+3 π∗M
Algebra in (May) gives topology in (Adams).E3[1/φ] = E∞[1/φ]
π∗M[1/φ] & LK (1)M
A localization theorem in homological algebraAlgebraic background for last slide
A = S⊗E , S =coalgebra, E =exterior Hopf algebra
CotorE(k , k) = k [q]
CotorA(k , M) = module over k [q]
Compute q−1 CotorA(k , M)
q−1E2M =
Eroded by dMay2 = dAdams
2
Unstable credentials: generalized by Hess & Levi (2007)
C∗Ω(homotopy fibre)
A localization theorem in homological algebraAlgebraic background for last slide
A = S⊗E , S =coalgebra, E =exterior Hopf algebra
CotorE(k , k) = k [q]
CotorA(k , M) = module over k [q]
Compute q−1 CotorA(k , M)
q−1E2M =
Eroded by dMay2 = dAdams
2
Unstable credentials: generalized by Hess & Levi (2007)
C∗Ω(homotopy fibre)
Spec seq for homology of an infinite delooping
X an ∞-loop space
QX = Ω∞Σ∞X X QXoo // Q2X · · ·oooo
E2¬ab(X ) =⇒ H∗B∞X
E2-page: spawned by another SS
Untori Indecj H∗X =⇒ E2¬ab(X ) =⇒ H∗B∞X
Indecj = 0, j 6= 0, 1“Λ-algebra” complex for Untor∗
Collapse and work absolutely everything out X = ZDyer-Lashof =⇒ Steenrod
The Segal conjecture for elementary abelian p-groupsAdams, Gunawardena, Miller
Segal conjecture G finite
S0 ⇐⇒∐
BΣn
Map(BG, S0)?⇐⇒ Map(BG,
∐BΣn)
Map(BG, S0)?∼ ∨O Σ∞B Aut(O)
Reductions1 Enough: G a p-group (May-McClure)2 Enough: G elementary abelian, equivariantly (Carlsson)3 G elem abel: Ext-calculation (AGM) + glue (May-Priddy)
Ext-calculation V = (Z/p)n, S = β(H1)
E2( S−1H∗BV ) ∼= p(p−1)2 E2( Σ−nZ/p )
Unstable!ttiiiiii
BG
The Segal conjecture for elementary abelian p-groupsAdams, Gunawardena, Miller
Segal conjecture G finite
S0 ⇐⇒∐
BΣn
Map(BG, S0)?⇐⇒ Map(BG,
∐BΣn)
Map(BG, S0)?∼ ∨O Σ∞B Aut(O)
Reductions1 Enough: G a p-group (May-McClure)2 Enough: G elementary abelian, equivariantly (Carlsson)3 G elem abel: Ext-calculation (AGM) + glue (May-Priddy)
Ext-calculation V = (Z/p)n, S = β(H1)
E2( S−1H∗BV ) ∼= p(p−1)2 E2( Σ−nZ/p )
Unstable!ttiiiiii
BG
The Sullivan conj. on maps from classifying spaces
Sullivan conjecture G finite, X fin. dim.
Map(BG, X )?∼ X
Enough: G = Z/p, π1X = ∗
X 1-connected, V = Z/p
CX = Ω∞(HZ/p ∧ X ) X // CX//// C2X · · ·oo
E2¬ab(X ) =⇒ π∗ Map(BV , X )
E2-page: spawned by another SS H = H∗BV
Unexti (ΣjH, Primk H∗X ) =⇒ E2¬abX =⇒ π∗ Map
Sullivan conjecture: proof
Unexti (ΣjH, Primk H∗X ) =⇒ E2¬abX =⇒ π∗ Map
X fin. dim’l =⇒ Primk H∗X bounded =⇒ Unext0 vanishes
Dualize, use homotopy theory of simpl. commutative algebrassuspensionreverse unstable Adams Spectral Sequence
No maps (finite dim’l)→ H∗BZ/p
Unexti vanishes for i > 0
Dualize, prove H∗BZ/p is injective as an unstable moduleretract of limit of dual Brown-Gitler modules
Sullivan conjecture: proof
Unexti (ΣjH, Primk H∗X ) =⇒ E2¬abX =⇒ π∗ Map
X fin. dim’l =⇒ Primk H∗X bounded =⇒ Unext0 vanishesDualize, use homotopy theory of simpl. commutative algebras
suspensionreverse unstable Adams Spectral Sequence
No maps (finite dim’l)→ H∗BZ/p
Unexti vanishes for i > 0
Dualize, prove H∗BZ/p is injective as an unstable moduleretract of limit of dual Brown-Gitler modules
Sullivan conjecture: proof
Unexti (ΣjH, Primk H∗X ) =⇒ E2¬abX =⇒ π∗ Map
X fin. dim’l =⇒ Primk H∗X bounded =⇒ Unext0 vanishes
Dualize, use homotopy theory of simpl. commutative algebrassuspensionreverse unstable Adams Spectral Sequence
No maps (finite dim’l)→ H∗BZ/p
Unexti vanishes for i > 0Dualize, prove H∗BZ/p is injective as an unstable module
retract of limit of dual Brown-Gitler modules
Sullivan conjecture: proof
Unexti (ΣjH, Primk H∗X ) =⇒ E2¬abX =⇒ π∗ Map
X fin. dim’l =⇒ Primk H∗X bounded =⇒ Unext0 vanishesDualize, use homotopy theory of simpl. commutative algebras
suspensionreverse unstable Adams Spectral Sequence
No maps (finite dim’l)→ H∗BZ/p
Unexti vanishes for i > 0Dualize, prove H∗BZ/p is injective as an unstable module
retract of limit of dual Brown-Gitler modules
Generalized Sullivan conjecture
Homotopy fixed points G acts on X
X G = MapG( ∗ , X )
X hG = MapG(EG, X ) = EG ×G X // BGvv R_l
Sullivan fixed point conjecture X finite, G = Z/p
X hG ?∼p X G
Trivial action ⇐⇒ Sullivan conjecture
Theorem: OK mod π1 subtlety
E2 over Ap⊗H∗B =⇒ π∗( E // Bww Q_m
)
SS for π∗X hG impersonates SS for π∗X G
And more, and more, and more
An Incomplete List
1 Vanishing lines (Miller-Wilkerson)2 Equivariant function spaces (Mann-Miller-Miller)3 Homotopical uniqueness (D-Miller-Wilkerson)4 Double suspension (Harper-Miller)5 Stiefel manifolds
And more, and more, and more
An Incomplete List
1 Vanishing lines (Miller-Wilkerson)
2 Equivariant function spaces (Mann-Miller-Miller)3 Homotopical uniqueness (D-Miller-Wilkerson)4 Double suspension (Harper-Miller)5 Stiefel manifolds
And more, and more, and more
An Incomplete List
1 Vanishing lines (Miller-Wilkerson)2 Equivariant function spaces (Mann-Miller-Miller)
3 Homotopical uniqueness (D-Miller-Wilkerson)4 Double suspension (Harper-Miller)5 Stiefel manifolds
And more, and more, and more
An Incomplete List
1 Vanishing lines (Miller-Wilkerson)2 Equivariant function spaces (Mann-Miller-Miller)3 Homotopical uniqueness (D-Miller-Wilkerson)
4 Double suspension (Harper-Miller)5 Stiefel manifolds
And more, and more, and more
An Incomplete List
1 Vanishing lines (Miller-Wilkerson)2 Equivariant function spaces (Mann-Miller-Miller)3 Homotopical uniqueness (D-Miller-Wilkerson)4 Double suspension (Harper-Miller)
5 Stiefel manifolds
And more, and more, and more
An Incomplete List
1 Vanishing lines (Miller-Wilkerson)2 Equivariant function spaces (Mann-Miller-Miller)3 Homotopical uniqueness (D-Miller-Wilkerson)4 Double suspension (Harper-Miller)5 Stiefel manifolds
Quote I for the dayfrom Massey-Peterson towers and maps from classifying spaces
. . .the Massey-Peterson theory should beregarded as a piece of light artillery, with whichone can move quickly and execute smallambushes before wheeling in 〈heavier guns〉.
Quote II for the dayfrom The Sullivan conjecture on maps from classifying spaces
In summary, the proof proceeds by constructinga long chain of spectral sequences and thenshowing that, miraculously, the initial term of theinitial spectral sequence is trivial.
Just another example of. . .
Just another example of. . .