cobordism category of manifolds with baas-sullivan singularities, part 2

7/28/2019 Cobordism Category of Manifolds With Baas-Sullivan Singularities, Part 2

1/64

COBORDISM CATEGORY OF MANIFOLDS

WITH BAAS-SULLIVAN SINGULARITIES, PART 2

NATHAN PERLMUTTER

Abstract. For a given collection of closed manifolds k = (P1, . . . , P k), we construct a

cobordism category Cobkd

of embedded manifolds with Baas-Sullivan singularities of type

k . Our main results identify the homotopy type of the classifying spaces BCobk

dof these

cobordism categories with that of the infinite loop-space of certain spectrum MT(d)k . Thecase of single singularity has been covered by the authors paper [14].

1. Introduction and Statement of Main Results

1.1. Manifolds with Baas-Sullivan singularities. This paper continues the study westarted in [14] where we have determined the homotopy type of the classifying space BCobdfor a cobordism category of smooth manifolds of dimension d 2 with Baas-Sullivan singu-larity = (P). Here we extend those constructions and results for manifolds with multipleBaas-Sullivan singularities. To simplify our presentation we assume for most of the paperthat all manifolds are smooth and unoriented.

We fix a sequence = (P1, . . . , P k, . . . ) of smooth closed manifolds and denote pi := dim Pi .Then for a positive integer k , we denote by k the truncated list k := (P1, . . . , P k).

We recall briefly the definition of a manifold with Baas-Sullivan Singularities of type k =(P1, . . . , P k), postponing details to Section 2. It is convenient to denote by P0 the single point.

We say that a compact manifold with corners W, dim W = d, i s a k -manifold if theboundary W is given a decomposition

W = 0W 1W kW

with the property that for each subset I {0, 1, . . . , k} , the intersection

IW :=iI

iW

is a (d |I|)-dimensional manifold with corners, where |I| is the size of I. Furthermore, we

suppose that for each I {0, 1, . . . , k}, the manifold IW is given a factorization,

IWI= IW P

I where PI =iI

Pi ,

for some (d |I|

iIpi)-dimensional manifold with corners IW.1

arXiv:1306.40

45v1

[math.AT]18Jun2013


2/64

2 NATHAN PERLMUTTER

By definition, the manifold IW is a k -manifold as well for each subset I {0, 1, . . . , k} .The submanifold 0W is referred to as the boundary of the k -manifold W. Closed k -manifolds are the ones for which 0W is empty.

To get actual Baas-Sullivan singularities, we define an equivalence relation on W by declaring

two points, x, y W to be equivalent if there exists a subset I {0, 1, . . . , k} such that xand y both belong to IW and

prI(x) = p rI(y) where prI : (IW = IW PI) IW is the projection.

Quotienting-out with respect to this equivalence relation, we obtain a manifold Wk withBaas-Sullivan singularities of the type k .

Remark 1.1. It is more convenient to work with k -manifolds. We take special care toensure that all of our constructions factor through the above equivalence relation.

We denote by the graded bordism group of unoriented manifolds, and by k a graded

bordism group of manifolds with Baas-Sullivan Singularities of type k . We denote for now,

0 := . For varying integer k , the groups k are related to each other by the well-knownBockstein-Sullivan exact couple:

(1) k1

ik ''

k1

Pkoo

k

k

77

The map Pk is given by multiplication by the manifold Pk and is of degree pk , ik is themap given by inclusion, and k is the degree (pk + 1) map given by sending a k -manifoldW to the k1 -manifold kW, see definitions in Section 2.

The classifying spectra for these cobordism theories, which we will denote by MOk , can beconstructed inductively as cofibres of maps between spectra starting with MO := MO0 . Thehomomorphism P1 : of degree p1 = dim P1 defines a map of spectra

P1 : p1MO MO.

We then setMO1 := Cofibre(P1 :

p1MO MO).

Then, assuming that MOk1 is defined, we obtain a map Pk : pkMOk1 MOk1 given

by the homomorphism Pk : k1

k1 of degree pk = dim Pk . We define by induction:

MOk := Cofibre(Pk : pkMOk1 MOk1).

The exact couple (1) is induced by the resulting cobfre sequence,pkMOk1 MOk1 MOk

pk+1MOk1

For constructions and calculations relating to this exact couple, see [4]. An important exampleof the above is the case when k = 1 and P1 is equal to the set of n discrete points. In thiscase 1 -manifolds are sometimes referred to as Z/n-manifolds.


3/64

COBORDISM CATEGORY FOR MANIFOLDS WITH SINGULARITIES 3

1.2. Cobordism categories. Motivated by the ideas in [2], we will construct a cobordismcategory of manifolds with Baas-Sullivan singularities and determine the homotopy-type ofits classifying space.

First, we recall that in [6], the authors construct a cobordism category Cobd whose morphisms

are d-dimensional submanifolds W [a, b] Rd that intersect the walls {a, b} Rdorthogonally in W. This category is topologically enriched in such a way so that there arehomotopy equivalences,

Ob(Cobd) M

B Diff(M), Mor(Cobd) =W

B Diff(W; in, out)

where M varies over diffeomorphism classes of (d 1)-dimensional closed manifolds and Wvaries over diffeomorphism classes of d-dimensional cobordisms. Here Diff(M) is the groupof diffeomorphisms of M and Diff(W; in, out) is the group of diffeomorphims of W thatrestrict to diffeomorphisms of the incoming and outgoing boundaries. In [6], the authorsdetermine the homotopy type of the classifying space of Cobd . In particular, they prove a

weak homotopy equivalence(2) BCobd

1MT(d),

where MT(d) is a certain Thom-spectrum. This deep result, could be interpreted as aparametrized version of the classical Thom-Pontryagin construction.

Following this work on cobordism categories, we construct a cobordism category of k -manifolds. Roughly, the construction goes as follows (for more details, see Section 2).

We denote Rk+ := [0, )k and Rk+I := {(t1, . . . , tk) R

k+ | ti = 0 if i / I} for I a subset of

{1, . . . , k}. We then fix once and for all smooth embeddings

(3) i : Pi Rpi+mi for each Pi from with mi > pi = dim Pi ,

for each k 0. Now we are ready to construct a topologically enriched category Cobkd . Themorphisms are given by embedded d-dimensional k -cobordisms,

W

[a, b] Rk+ R

d1+

1ik

Rpi+mi

such that for each subset I {1, . . . , k},

W

[a, b] Rk+Ic R

d1+

1ik

Rpi+mi

= IW

where Ic is the compliment of I in {1, . . . , k} . It is required that the submanifold IW has

factorization,IW = IW P

I

where,

IW

[a, b] Rk+Ic R

d1+ iIc

Rpi+mi


4/64

4 NATHAN PERLMUTTER

and the submanifold

PI iI

Rpi+mi

is given by a product of the embeddings from (3). Furthermore, we have

0W

{a, b} Rk+ R

d1+

1ik

Rpi+mi

.

All intersections of W with the boundary of [a, b] Rk+ Rd1+

1ik

Rpi+mi are required

to be orthogonal. This category is topologized in such a way similar to as in [6] so that thereare homotopy equivalences,

Ob(Cobkd ) MB Diff(M)k , Mor(Cobkd ) W

B Diff(W)k

where M varies over diffeomorphism classes of (d 1)-dimensional closed (0M = ) k -manifolds and W varies over diffeomorphism classes of d-dimensional k -manifolds withboundary. The diffeomorphism groups Diff (W)k are defined to be the space of all smoothmaps g : W W that are diffeomorphisms of W as a manifold with corners, i.e. g(IW) =IW for all I, with the additional property that for each subset I {0, . . . , k}, the restrictionof g to IW has the factorization

(4) g |IW = gI IdPI

where gI : IW IW is a diffeomorphism of a manifold with corners. Such diffeomor-

phisms descend to unique homeomorphisms of the singular manifold Wk obtained whenquotienting-out by the equivalence relation described earlier. The construction of this cat-egory depends on our choice of embeddings from (3). However, we will latter see that forany two choices of collections of embeddings of the manifolds from the list , the resultingcategories will be isomorphic, provided the dimension of the ambient space is large enough soas to make the embeddings isotopic.

Remark 1.1. If k = 0, then the category Cob0d is the same cobordism category of smoothmanifolds studied in [6]. The case of a single singularity type, i.e. k = 1, is covered in [14].

1.3. First results. To state our first results, we need more definitions. We will need to com-

pare the categories Cobkd to other similar cobordism categories of manifolds with singularitysets that differ slightly from . For a non negative integer k , we denote by k the legnth

k list (P1, . . . , P k, , . . . , ) obtained by swapping the last entries of k with single points.

Using these singularity sets we define categories Cobk

d in the same way as before. With

this definition in place, for any pair k the category Cob1k

d can be identified with the


5/64


pull-back of the diagram,

Cobk

d

k+1

Cob1k1

dpk+11

Pk+1 // Cob1k1

d1

where k+1 is the functor defined by sending a morphism W to k+1W, which in thiscase equals k+1W. The map Pk+1 is the functor defined by sending a morphism Wto W Pk+1 . Taking the pull-back of this diagram has the effect of adding one singularity

type, in this case singularities of type Pk+1 , to the morphisms and objects in Cobk

d .

From this identification, we see that the category Cobkd can be defined inductively starting

with Cobkk

d , by iterating this pull-back construction for each from k down to 0. Applyingthe classifying space functor, we obtain a cartesian diagram

(5) BCob1k

d//

Bk+1

BCobk

d

Bk+1

BCob1k1

dpk+11

B(Pk+1) // BCob1k1

d1 .

This brings us to the statement of our first results:

Theorem 1.1. The above diagram (5) is homotopy-cartesian for all and k with 0 < k .

We then with some work identify the homotopy fibre of the vertical maps of the above

homotopy-cartesian square with the space BCob1k1d . For the case that = 1 this yields:

Theorem 1.2. For all k there is a homotopy fibre-sequence,

BCobk1d

// BCobkd

Bk // BCobk1d1pk

.

The fact that (5) is homotopy cartesian implies that the homotopy type of the classifying

space BCobkd can be determined inductively starting with BCobkk

d , by iterating the thehomotopy-pull-back construction with respect to the maps B(k+1) and B(Pk+1) for

each from k down to 0. For the case = k , the category Cobkk

d is the cobordism category

of manifolds with corners. This category has been studied in [7]. There, the author identifiesthe homotopy type of the classifying space of this category with that of the infinite loop-spaceof a homotopy-colimit of a certain diagram comprised of the spectra MT(n). We will identify

the homotopy type of BCobk

d with the infinite loop-space of a similar homotopy colimit.Below we give a rough description of the construction.


6/64

6 NATHAN PERLMUTTER

1.4. Cubes of spaces and spectra. To state our results which determines the weak ho-motopy type of the classifying spaces BCobkd , we have to consider k -dimensional cubicaldiagrams of spaces and spectra. We denote by Top or Spec the corresponding categories. Ak -dimensional cubical diagram of spaces or spectra (which we will sometimes refer to as k -

cubes of spaces) is a contravarient functor from the lattice of subsets of a finite set {1, . . . , k} ,which we denote by 2k , to the categories Top or Spec. Let

X : 2k Top,

be such a functor. For instance, for k = 1, 2, 3, the functor X produces the correspondingdiagrams:

(6) X X{1}

X{2} X{1,2}

X X{1}

oo

oo

X{2,3} X{1,2,3}

X{3} X{1,3}

X{2} X{1,2}

X X{1}

zz

oo

zz

oo

zz

oo

zzoo

Now, given a functor X : 2k Top, we associate a space called the total homotopy cofibre

of X , which we denote by tCofibre X . The total homotopy cofibre is defined inductively asfollows. First if k = 1, we set

tCofibre X := Cofibre X.

This makes sense because in this case Cofibre X = Cofibre(X{1} X).

Now assume that it is defined for all (k 1)-cubes. We denote by X,k the (k 1)-cube obtained by restricting X to the sub-lattice of all subsets containing the element k {1, . . . , k}. We denote by X,k the (k 1)-cube obtained by restricting to the sub-lattice ofall subsets disjoint from k .

For instance, if k = 3, the top square gives X,3 , and the bottom one gives X,3 , see (6).Clearly, there is a map of (k 1)-cubes X,k X,k which induces a map

tCofibre X,k tCofibre X,k.

We use this map and the induction hypothesis to define

tCofibre X := Cofibre(tCofibre X,k tCofibre X,k).

1.5. Weak homotopy type of BCobkd . Next, we would like to construct a k -cube ofspectra MT(d)k . First, recall that the spectrum MT(d) from the main theorem in [6]is defined as follows. The (n + d)-th space of this spectrum is given by the Thom spaceMT(d)n+d := Th(U

d,n) where U

d,n G(d, n) is the orthogonal compliment to the canonical


7/64


bundle Ud,n G(d, n) over the Grassmanian manifold G(d, n) consisting of d-dimensionalvector subspaces of the Euclidian space Rd+n .

The bundle Ud,n+1 G(d, n + 1) restricts over G(d, n) to the direct sum Ud,n

1 , where 1

is a trivial line bundle. Then the maps

S1 Th(Ud,n) Th(Ud,n+1)

are the structure maps 1MT(d)n+d MT(d)n+1+d of the spectrum MT(d).

Recall that our manifolds Pi come together with embeddings i : Pi Rpi+mi , see (??). For

a subset I {1, . . . , k}, we denote by pI and mI the sums

iI pi and

iI mi , and by PI

the product of manifolds Pi , i I. We notice that the embeddings i induce the productembedding

I : PI RpI+mI.

Let I J {1, . . . , k}. Then we have the corresponding embedding

I\J : PJ\I RpJ\I+mJ\I.

We denote by NJ\I the normal bundle of the embedding I\J which comes together with aGauss map

NJ\I UpJ\I,mJ\I

of the embedding I\J. We obtain a corresponding Thom-Pontryagin map

(7) J\I : SpJ\I+mJ\I Th(UpJ\I,mJ\I).

Now we define a k -cube of spectra, MT(d)k defined by setting

MTJ(d)k :=

|J|MT

(d pJ |J|) for each J {1, . . . , k} .Thus the spectrum MTJ(d)k is the vertex in our cubical diagram corresponding to J.

2MT(dp{2,3} 2)

uu

3MT(dp{1,2,3} 3)oo

tt1MT(dp1 1)

2MT(dp{1,3} 2)oo

1MT(dp2 1)

uu

2MT(dp{1,2} 2)oo

ttMT(d) 1MT(dp1 1)

oo

Figure 1. The cubical diagram of spectra MT(d)k for the case that k = 3.


8/64

8 NATHAN PERLMUTTER

Then for I J {1, . . . , k}, we will denote the edge in the cubical diagram connecting thevertices associated to I and J by

J,I : |J|

MT(d pJ |J|) |I|MT(d pI |I|).

The construction of J,I goes as follows. There is a natural map of the Grassmanians

I,J : G(d pJ |J|, n mJ) G(pJ\I, mJ\I) G(d pI |J|, n mI)

which sends a pair of vector subspaces

(, ) G(d pJ |J|, n mJ) G(pJ\I, mJ\I)

to their product RdpJ|J|+nmJ RpJ\I+mJ\I.

The map I,J gives the standard pairing map of the Thom spaces:

J,I : Th(UdpJ|J|,nmJ

) Th(UpJ\I,mJ\I) Th(UdpI|J|,nmI

)

Then the maps I,J and I,J yield a map I,J given by the composition:

Th(UdpJ|J|,nmJ)SpJ\I+mJ\ISpI+mI Th(UdpJ|J|,nmJ)Th(U

pJ\I,mJ\I

)SpI+mI

Th(UdpI|J|,nmI) SpI+mI

//IdJ\IId

))

I,J

J\IId

which in turn induces a map of spectra:

(8) J,I : |J|

MT(d pJ |J|) |J|

MT(d pI |J|).Now let

Th(UdpI|J|,nmI) Th(UdpI|I|,nmI

)

be the map induced by the natural embedding

G(d pI |J|, n mI) G(d pI |I|, n mI),

which is given by sending a (d pI |J|)-dimensional subspace to the product R|J||I| .This induces a map of spectra

(9) jJ,I : |J|

MT(d pI |J|) |I|MT(d pI |I|).

We then set J,I := jJ,I J,I to be the edge in our cubical diagram of spectra connecting

the J-th vertex to the I-th vertex. With some care, this construction can be carried out sothat the cubical diagram obtained by putting together the maps J,I commutes on the nose(though it is relatively easy prove that the diagram commutes up to homotopy). Thus ourfunctor MT(d)k : 2

k Spec is well defined. We denote,

MT(d)k := tCofibreMT(d)k .


9/64


This leads to us to the statement of our main theorem:

Theorem 1.3. There is a weak homotopy equivalence,

BCobkd

1MT(d)k .

The above equivalence is constructed using a relative-parametric version of the classical Thom-Pontryagin construction. This theorem for the case where k = 1 was proven in [14].

Now, recall the Thom-Pontryagin maps

PJ\I : SpJ\I+mJ\I Th(UpJ\I,mJ\I)

used in the construction of the cube MT(d)k . By standard Thom-Pontryagin theory, thehomotopy classes of these maps depend only on the cobordism classes of the manifold PJ\I.From this observation we have:

Corollary 1.4. Let

be a sequence of manifolds obtained by replacing each Pi from the

sequence by a manifold Pi cobordant to Pi . Then the classifying space BCob kd of theresulting cobordism category Cob

kd , is weakly homotopy equivalent to BCob

kd , for all k 0.

1.6. Plan of the paper. The paper is structured as follows. Section 2 is devoted to carefullyconstructing the category BCobkd . Along the way we define the appropriate spaces of em-beddings, diffeomorphism groups, and moduli-spaces of k -manifolds. In Sections 3 and 4 weconstruct sheaves Ck,d and D

kd , defined on the category of smooth manifolds, which reduce

to the sheaves Cd and Dd from [6] in the case k = 0. The representing spaces of these sheaveswill be seen to weakly homotopy equivalent to Cobkd and BCob

kd respectively. In Section

5 we prove Theorems 1.1 and 1.2. Section 6 is devoted to basic constructions and resultsregarding cubical diagrams. In section 7 we give a construction of the spectrum MT(d)k and

in section 8 we give a proof of Theorem 1.3.

1.7. Acknowledgements. The author would like to thank Boris Botvinnik for suggestingthis particular problem and for numerous helpful discussions on the subject of this paper.The Author would also like to thank the Rose Hills Foundation for their financial support.The author is also grateful to Nils Baas for his encouraging remarks and to Oscar RandalWilliams for very helpful critical comments on the first part of this project.

2. The Cobordism Category

We start with the same sequence of closed smooth manifolds = ( P1, . . . , P n, . . . ) specified

in the introduction. We denote pi := dim(Pi). We fix once and for all a positive integer k .We denote by k the truncated list of manifolds k = (P1, P2, . . . , P k). We will have to workwith contracted or augmented lists kj or k+j , but in order to keep all of our constructionsconsistent, the integer k will be of the same value throughout the whole paper.

For 0 k we denote by k the list (P1, . . . , P k, , . . . , ) of the length k .


10/64

10 NATHAN PERLMUTTER

As it was outlined in the introduction we construct a cobordism category of embedded k -manifolds. That is, for each positive integer n and 0 k , we construct a cobordism

category Cobk

d,n with morphisms given by d-dimensional k -cobordisms embedded in (d+n)-

dimensional Euclidean space. Objects will be given by closed (d1)-dimensional k -manifolds

embedded in (d + n 1)-dimensional Euclidean space.

2.1. The Objects. In this section we construct the space of objects of our cobordism cate-

gories Coblk

d,n .

For each positive integer i , we fix once and for all integers mi with mi > pi = dim(Pi), andsmooth embeddings

(10) i : Pi Rpi+mi .

For each subset I {1, . . . , k}, we denote the product embedding,

I :=iI i : PI iIRpi+mi .We denote by the collection of embeddings {1, . . . , k}. We now introduce some notationthat will be used throughout the paper.

Notational Convention 2.1. Denote by k the set {1, . . . , k} . We define,

p = p1 + + pk, m = m1 + + mk,

and for any subset I k,

pI =iI

pi, mI =iI

mi.

For any integer n with n k + p + m, we denoten := n k + p + m.These conventions will become important when we define the space of embeddings of a k -manifold.

For each subset I k we define,

Rp+mI := {(x1, . . . , xk) R

p1+m1 Rpk+mk | xi = 0 if i / I}.

We will also use the spaces,

RkI := {(t1, . . . , tk) Rk | ti = 0 if i I },

Rk+I := {(t1, . . . , tk) RkI | ti 0 for all 1 i k }.

For a subset I k we set

[0, 1)kI := {(t1, . . . , tk) [0, 1)k | ti = 0 ifi / I}.


11/64


For a list of positive constants = (1, . . . , k), we will denote,

[0, )kI := {(t1, . . . , tk) [0, 1)kI | ti < i for each i}.

If I k , we denote by Ic the compliment of I in k . We will frequently use the following

identifications:[0, 1)kJ\I [0, 1)

kI = [0, 1)

kJ, R

p+mI R

p+mJ\I = R

p+mJ , and R

k+I R

k+J\I = R

k+J.

for any subsets I J k. We will be using these notational conventions throughout thepaper.

We now give a definition of a closed k -manifold.

Definition 2.1. Let M be a compact (d 1)-dimensional manifold with k -order corners.We say that M is a closed-k -manifold if it satisfies the following conditions:

i. The boundary M is given a decomposition

M = 1M kMof the boundary M into a union of (d 2)-dimensional manifolds with (k 1)-ordercorners such that for all subsets I k , the intersection

IM :=iI

iM

is a manifold with (k |I|)-order corners with its boundary given by

(IM) =

jk

(jM IM).

(ii) The faces IM are given compatible collar embeddings, i.e., for all I k, thereexist compatible collar embeddings

hI : IM [0, 1)kI M.

Compatibility here means the following. Let I J k be subsets. We require thatthe embedding hJ : JM [0, 1)

kJ M maps the subspace

JM [0, 1)kJ\I JM [0, 1)

kJ

into IM [0, 1)kI . We denote this restriction by hJ|J\I, and require that hJ factorsthrough the composition:

JM [0, 1)kJ

=

hJ

--JM [0, 1)kJ\I [0, 1)kI

hJ|J\IId // IM [0, 1)kIhI // M

where we are identifying JM [0, 1)kJ\I [0, 1)kI with JM [0, 1)

kJ , see (??).


12/64


(iii) For each subset I {1, . . . , k l} there are diffeomorphisms,

I : IM= // IM PI

where IM is a (d 1 |I| pI)-dimensional manifold with corners which satisfies

the conditions (i) and (ii), and PI =iI Pi is as in the above notational convention.Furthermore these diffeomorphisms must be compatible in the following sense: if I Jand J,I : JM IM is the corresponding inclusion, then the map

I J,I 1J : JM P

J IM PI

is the identity on the direct factor of the PI.

It follows directly from the above definition that for i {1, . . . , k } iM is a closed k1 -manifold of dimension d 2 pi and that for j {k + 1, . . . , k}, jW is a closed

1k1

manifold of dimension d 2. We note also that a closed k -manifold is automatically a jk

manifold for all j .

We now define the space of diffeomorphisms of a closed k manifold.

Definition 2.2. Let M a (d 1)-dimensional closed k -manifold as above. We again denoteby hI and I the collar embeddings and product structures given in the previous definition.Let i for i k, be positive constants and denote = (1, . . . , k). We define

Diff(M)k

to be the space of diffeomorphisms g : M M, of a manifold with corners, subject to thefollowing conditions:

i. For each subset I k , we have g(IM) = IM.

ii. The map g respects collars about each face iM of widths i in the following way.For each I k , we require that

g hI(w, t1, . . . , tk) = hI(g|IM(w), t1, . . . , tk)

for w IM, where (t1, . . . , tk) [0, )kI .

iii. For each I {1, . . . , k } , the restrictions g |IM have the factorizations,

g |IM= 1I (gIM IdPI) I

with gIM : IM IM a diffeomorphism of IM satisfying the conditions i. andii.

The space Diff(M)

k

is topologized using the C

-Whitney topology.To eliminate dependence on , we take the direct limit,

(11) Diff(M)k := colim

0Diff(M)

k .

We now proceed to define a space of embeddings of k -manifolds.


13/64


Recall that we have smooth embeddings i : Pi Rpi+mi . for each i. We denote by the

collection of embeddings {1, . . . , k}, and

I :=

iIi : P

I

iIRpi+mi = Rp+mI .

Below we use the notation introduced in Notational Convention 2.1. For what follows let Ma d 1-dimensional closed k -manifold.

Definition 2.3. For each i k let i be a positive constant. Denote = (1, . . . , k). We

define Emb(M,Rk+ Rd1+n Rp+m)

k,

to be the space of smooth embeddings of manifoldswith corners,

f : M Rk+ Rd1+n Rp+m,

subject to the following conditions:

i. For each I k ,

f(IM) Rk+Ic Rd1+n Rp+m.

ii. The map f respects collars of widths i in the following way. Let

iI : [0, )kI R

k+I

be the standard inclusion. Then for all I k , the following diagram commutes

IM [0, )kI

proj

fhI // Rk+I (Rk+Ic R

d1+n Rp+m)

proj

[0, )kIiI // Rk+I

where in the upper-right corner we are using the identification,

Rk+I Rk+Ic R

d1+n Rp+m = Rk+Ic Rd1+n Rp+m.

iii. For each I {1, . . . , k } there is a factorization:

f |IM = fIM PI

where the PI are the embeddings specified in (10) and

fIM : IM [a, b] Rk+Ic R

d1+n Rp+mIc

is an embedding which satisfies conditions i. and ii. given above.

The space Emb(M,Rk+ Rd1+n Rp+m)

k,

is topologized using the C -Whitney topology.

To eliminate dependence on , we take the direct limit,

(12) Emb(M,Rk+ Rd1+n Rp+m)

k, := colim

0Emb(M,Rk+ R

d1+n Rp+m)k, .


14/64


Let M be a 1k -manifold. Since M is then a k -manifold as well, the space

Emb(M,Rk+ Rd1+n Rp+m)

k,

is defined. For what follows, to save space we temporarily denote

Hnk, := Rk+{k+1}c R

d1+nRp+m{k+1}c.

The space Emb(M,Rk+ Rd1+n Rp+m)

1k

, can be identified with the pull-back of thediagram,(13)


k,

k+1

Emb(k+1M, Hnk,)

1k1,

k+1// Emb(k+1M,Rk+{k+1}c R

d1+n Rp+m)1k1,

where the right vertical map sends an embedding

g : M Rk+ Rd1+n Rp+m

to its restriction g |k+1M and the bottom horizontal map sends an embedding

f : kl+1M Rk+{k+1}c R

d1+n Rp+m{k+1}c

to the product embedding

f Pk+1 : k+1M Pk+1 (Rk+{k+1}c R

d1+n Rp+m{k+1}c) Rp+m{k+1}.

Remark 2.1. The spaces of diffeomorphisms Diff(M)+1k can be realized as the pull-back of

Diff(M)k

k+1

Diff(k+1M)1k1

IdPk+1 // Diff(k+1M)1k1

in a similar way.

The following lemma will prove to be quite useful.

Lemma 2.1. For each all k and l , the map

k+1 : Emb(M,Rk+R

d1+nRp+m)k, Emb(k+1M,R

k+{k+1}cR

d1+nRp+m)1k1,

is a Serre-fibration.

Proof. The proof follows from Theorem A.1 given in Appendix A.


15/64


This lemma implies that the space Emb(M, Rk+ Rd1+n Rp+m)

1k , is actually the

homotopy pull-back of the diagram (13).

In the case that = k , the space Emb(M, Rk+ Rd1+n Rp+m)

kk, is simply the space of

embeddings of a manifold with corners, the product structure on faces IM are forgotten.

We take the direct limit to define,

(14) Emb(M,Rk+ Rd1+ Rp+m)

k, = colim

nEmb(M,Rk+ R

d1+n Rp+m)k,.

This brings us to the following result:

Theorem 2.2. Let M be a jk -manifold. Then Emb(M,Rk+ R

d1+ Rp+m)k, is weakly

contractible for all j . In particular, the space

Emb(M,Rk+ Rd1+ Rp+m)k,,

which corresponds to the case where j = 0, is weakly contractible for any k -manifold M.

Proof. We prove this result by induction on the difference k . Now, by Theorem 2.7 of [7],for any k 0 the space Emb(M,Rk+ R

d1+ Rp+m)kk, is weakly contractible. This space

is the space of neat embeddings of a manifold with corners. Now suppose the that theoremholds for all sr manifolds where r s k . We will show that this implies that

Emb(M,Rk+ Rd1+ Rp+m)

1k ,

is weakly contractible. Lemma 2.1 together with (13) implies that the diagram,(15)


1k ,

k+1

i // Emb(M,Rk+ Rd1+ Rp+m)

k,

k+1

Emb(k+1M, Hk,)

1k1,

k+1// Emb(k+1M,Rk+{k+1}cR

d+Rp+m)1k1 ,

is homotopy-cartesian where Hk, in the diagram above has the same meaning as in (13).By the induction hypothesis, the lower-left, lower-right, and upper-right spaces are weaklycontractible. This together with the fact that that the diagram is homotopy-cartesian impliesthat the space


1k

is weakly contractible as well. This proves the theorem.

The proof of the above theorem demonstrates the utility of the pull-back construction ( 13).This method of proof will be used throughout this paper.

The above construction depends on the choice of embeddings i : Pi Rpi+mi . However, the

homeomorphism type of the spaces Emb(M,Rk+ Rd1+n Rp+m)

1k

, do not.


16/64


Theorem 2.3. For each , the homeomorphism type of the space


k,

does not depend on our choice of embeddings .

Proof. Proof is given in Appendix A

For each choice of and positive integer n, the group Diff(M)k acts freely (and smoothly)

on the space Emb(M,Rk+ Rd1+n Rp+m)

k, , by pre-composition,

(g, f) f g

for g a diffeomorphism and f an embedding. We denote by

Bn(M)lk :=


k,

Diff(M)k

the quotient space induced by the group action. Notice that the underlying set of Bn(M)k isthe set of all k -submanifolds ofR

k+ R

d1+n Rp+m diffeomorphic to M. Indeed the action

of Diff(M)k identifies any two embeddings with the same image. We have the following:

Theorem 2.4. For each k and , the quotient map

q : Emb(M,Rk+ Rd1+n Rp+m)

k, Bn(M)

k

is a locally trivial principal fibre-bundle with structure group Diff(M)k .

Proof. Proof of this theorem is essentially the same as Lemma A.1 from [ 14]. We refer thereader there.

The local-triviality of q along with the smooth structures on


k, and Diff(M)

k

make the spaces Bn(M)k into (infinite dimensional) manifolds. The inclusion maps


k, Emb(M,Rk+ R

d1+n+1 Rp+m)k,

are compatible with the action of Diff(M)1k and so we can take the direct limit to define

(16) B(M)k = colim

nBn(M)

k .

Theorem 2.4 combined with Theorem 2.2 implies that the projection map

q : Emb(M,Rk+ Rd1+ Rp+m)

k, B(M)

k

is the universal Diff(M)k -principal bundle and that it is homotopy equivalent to the classi-

fying space B Diff(M)k .


17/64


Using the well-known Borel construction we define,

(17) En(M)k := Emb(M,Rk+ R

d1+n Rp+m)k,

Diff(M)k

M.

We obtain a fibre-bundle:

En(M)k Bn(M)

k

with fibre M and structure group Diff(M)1k . This fibre bundle comes with a natural

embedding

En(M)k (Bn(M)

k Rk+ R

d1+n Rp+m).

2.2. Morphisms. We now must define a lk -bordism between two closed lk manifolds. Let

Ma and Mb be closed lk -manifolds of dimension d 1. For = a, b, let h

I and

I be the

collar embeddings and product structure maps associated to Ma and Mb from Definition 2.1.

Definition 2.4. A d-dimensional manifold with k+1 -order corners is said to be a lk -bordism

from Ma to Mb if it satisfies the following conditions:i. The boundary W is given the decomposition,

W = (Ma Mb) 1W kW

such that for each subset I k, the intersection IW := iIWi is a d |I|-dimensional manifold with k |I| +1-order corners just as in condition i. of definition2.1. Furthermore, for each I we have,

IW M = IM for = a,b.

ii. There are collar embeddings

hI : IW [0, 1)kI W

which satisfy the same compatibility conditions given in condition ii. of Definition 2.1.Furthermore, for each I there are embeddings,

jI : IM [0, 1) IW

which make the following diagram commute,

IM [0, 1)kI [0, 1)= (IM [0, 1)) [0, 1)kI

jIId //

hIId

IW [0, 1)kI

M [0, 1)

j

// Wwhere above it is understood that M = M and j

:= j .iii. For each subset I {1, . . . , k l} , there are diffeomorphisms,

I : IW= // IW PI


18/64


which satisfy the same compatibility conditions as given in condition iii. of Definition2.1. Furthermore, for each I, the composition,

IM PI(I)

1

// IM = IW MI|IWM // IW PI

is factors as a product iI,IdPI : IMPI IWPI, where iI, is an embedding.

In the above definition we will refer to Ma Mb as the boundary of the k -bordism W. Wewill sometimes denote this boundary by W or by 0W as we did in the introduction.

Remark 2.2. It follows directly from the definition that for i {1, . . . , k l}, iW is alk1 -bordism from iMa to iMb and that for j {k l + 1, . . . , k}, jW is a

l1k1 -bordism

from jMa to jMb .

W

Ma Mb

1W

1W

1Ma

1Ma

1Ma

1Ma

Figure 2. Above is a lk

- bordism in the case that k = l = 1.

We now proceed to define the spaces of diffeomorphisms and embeddings of a lk -bordism.The definitions will be similar to the spaces of diffeomorphisms and embeddings of closedlk -manifolds.

Let Ma and Mb closed d 1-dimensional lk manifolds with collar embeddings and productstructures given by hI and

I for = a, b. Let W be a d-dimensional

lk -bordism from

Ma to Mb with collar embeddings hI, jI and product structures I given as in the previous

definition.

Definition 2.5. Let 0, 1, . . . , k be positive constants and denote := (1, . . . , k) (notice

that we exclude 0 from this list). We define Diff(W; )k,0 to be the space of diffeomorphisms

g : W W, of a manifold with corners, subject to the following conditions:

i. For each subset I k we have, g(IW) = IW. Furthermore, it is required that

the restrictions of g to Ma and Mb are both elements of the spaces Diff(Ma)k

and

Diff(Mb)k

.


19/64


ii. The map g respects collars about each face iW of widths i for i k and collarsabout M of width 0 for = a, b in the following way. For each I k, it is requiredthat

g hI(w, t1, . . . , tk) = hI(g|IW(w), t1, . . . , tk)

for w IW, where (t1, . . . , tk) [0, )kI , and ti < i for all i = 1, . . . , k . Further-

more, it is required that

g jI(m, t) = jI((g|IM (m), t)

for m M, t [0, 0).iii. For each I {1, . . . , k } , the restrictions g |IW have the factorizations,

g |IW= 1I (gIW IdPI) I

with gIW : IW IW a diffeomorphism of IW satisfying the conditions i. andii.

The space Diff(W; )k

is topologized using the C -Whitney topology.

As before we eliminate the dependence on by taking the direct limit,

(18) Diff(W; )k = colim

0Diff(W; )

k

,0 .

Now let the same collection of embeddings i : Pi Rpi+mi used in the previous section.

Definition 2.6. Let 0, 1, . . . , k be positive constants and set := (1, . . . , k). We define

Emb(W, [0, 1] Rk+ Rd1+n Rp+m)

k,

,0

to be the space of smooth embeddings of a manifold with corners

f : W [0, 1] Rk+ Rd1+n Rp+m,


i. For each subset I k it is required that,

f(IW) [0, 1] Rk+Ic R

d1+n Rp+m,

f(Ma) {0} Rk+ R

d1+n Rp+m and f(Mb) {1} Rk+ R

d1+n Rp+m.

Furthermore the restriction of f to M, for = a, b, is an element of the space

Emb(M, Rk+ R

d1+n Rp+m)k

.

ii. For each i k the map f respects collars of widths i about the faces iW in thesame way as in Definition 2.3. Furthermore, the restriction of fja to Ma [0, 0) is

given by the formula,

(m, t) (t, f(m)) [0, 1] (Rk+ Rd1+n Rp+m) for m Ma and t [0, 0)

and the restriction of f jb to Mb [0, 0) is given by the formula,

(m, t) (1 t, f(m)) [0, 1] (Rk+ Rd1+n Rp+m) for m Mb and t [0, 0) .


20/64


iii. For each I {1, . . . , k } there is a factorization:

f |IM = fIM PI

just as in condition iii. of Definition 2.3.

We eliminate dependence on and 0 by taking the direct limit,(19)


k, := colim

0Emb(W, [0, 1] Rk+ R

d1+n Rp+m)k,

,0 .

We also may take the direct limit as n to define,(20)

Emb(W, [0, 1] Rk+ Rd1+ Rp+m)

k, = colim

nEmb(W, [0, 1] Rk+ R

d1+n Rp+m)k,.

Now, we obtain direct analogues of Lemma 2.1 and Theorems 2.2, and 2.4 from the previoussection. Specifically the topology of the spaces Emb(W, [0, 1] Rk+ R

d1+n Rp+m)k, is

independent of our choice of embeddings and the space

Emb(W, [0, 1] Rk+ Rd1+ Rp+m)

k,

is weakly contractible for all 0 k . The proofs go through in exactly the same way asbefore. Furthermore, the space Emb(W, [0, 1] Rk+ R

d1+n Rp+m)1k

, can be identifiedwith the pull-back of the diagram,(21)

Emb(W,[0, 1] Rk+ Rd1+n Rp+m)

k,

k+1

Emb(k+1W, [0, 1] Hnk,)

1k1,

k+1// Emb(k+1W, [0, 1] Rk+{k+1}c R

d1+n Rp+m)1k1,

where as before, Hnk, := Rk+{k+1}c R

d1+n Rp+m{k+1}c .

For W a k -bordism from Ma to Mb , the space Diff(W; )k acts freely and smoothly on the

space Emb(W, [0, 1] Rk+ Rd1+n Rp+m)

k, and so we may for each n define the spaces

Bn(W; )k in exactly the same way as in the previous section. Furthermore, the quotient

map


k, Bn(W; )

k

is a principle Diff(W; )k - fibre-bundle. We take the direct limit as n to define,

(22) B(W; )k = colim

n

Bn(W; )k .

The space B(W; )k is a model for the classifying space B Diff(W; )

k .

Using the well-known Borel construction we define,

(23) En(W; )k := Emb(W, [0, 1] Rk+ R

d1+n Rp+m)k,

Diff(W;)k

W.


21/64


We obtain a fibre-bundle:En(W; )

k Bn(W; )

k

with fibre M and structure group Diff(W; )1k . This fibre bundle comes with a natural

embedding

En(W; )k (Bn(W; )

k [0, 1] Rk+ Rd1+

n Rp+m).

Remark 2.3. We emphasize the similarity with the corresponding construction from [6]. This

embedding makes the bundle En(W; )k Bn(W; )

k universal in the following sense. If

f : X Bn(W; )k is a smooth map from a smooth manifold X of dimension j , then the

pull-back

f(Bn(W; )k) = {(x, v) X [0, 1] Rk+ R

d1+n Rp+m | (f(x), v) Bn(E)k}

is a smooth (j + d)-dimensional k -submanifold

E X [0, 1] Rk+ Rd1+n Rp+m

such that the projection onto X is a fibre-bundle with fibre W and structure group Diff(W; )k

.Furthermore, E is a k -manifold and it can be easily verified that for each x X, the inclu-sion map of the fibre Ex over x into {x} [0, 1] Rk+ R

d1+n Rp+m satisfies the conditionsof Definition 2.3. Any such embedded fibre-bundle

E X [0, 1] Rk+ Rd1+n Rp+m

with the property that for each x X the inclusion of the fibre

Ex {x} [0, 1] Rk+ R

d1+n Rp+m

satisfies all conditions of Definition 2.3, is induced by a unique smooth map

f : X Bn(W; )k .

Similar remarks apply to the bundle

En(M)k Bn(M)

k

for a closed k -manifold M.

2.3. The Category Cobk

d,n . We are now ready to define the category Cobk

d,n . An object of

Cobk

d,n is a pair (M, a) with a R and M Rk+ R

d1+nRp+m a closed (d1)-dimensional

k -submanifold with 0M = such that the inclusion map of M into Rk+ R

d1+n Rp+m

is an element of the space Emb(M,Rk+ Rd1+n Rp+m)

k,.

A non-identity morphism of Cob

k

d,n from (Ma, a) to (Mb, b) is a triple (W,a,b) with a < bandW [a, b] Rk+ R

d1+n Rp+m

a d-dimensional k submanifold such that the inclusion map of W is an element of the space

Emb(W, [0, 1] Rk+ Rd1+n Rp+m)k,,,


22/64


after a linear rescaling in the first coordinate. Two morphisms (W,a,b) and (V , c, d) can becomposed if b = c and the submanifolds

W ({b} Rk+ Rd1+n Rp+m) and (V {c} Rk+ R

d1+n Rp+m)

are equal. In this case, the composition of (W,a,b, ) of (V,c,d) is given by (W V,a,d).Condition iii. of Definition 2.3 (the condition requiring embeddings of k -manifolds to respectcollars) ensures that the union WV is a smooth manifold with corners and so the compositionis well defined. It is easy to check that this composition rule is associative.

We want to make Cobkd,n into a topological category. Observe that as sets we have isomor-

phisms,

(24)

Ob(Cobkd,n)

=M

Bn(M)k R,

Mor(Cobkd,n)

=

WBn(W; )

k {(a, b) R2 | a < b } Ob(Cob

kd,n),

where M varies over diffeomorphism classes of closed d 1-dimensional k manifolds andW varies over diffeomorphism classes of d-dimensional k -bordisms. These isomorphisms as

sets follow from the fact that for each W, as a set Bn(W; )k , is precisely equal to all d-

dimensional embedded k -bordisms of the space [0, 1]Rk+ R

d1+nRp+m, diffeomorphic to

W as a k -bordism. We then take (24) to be the definition of the category Cobk

d,n . Definedin this way we see that composition, and the target and source maps are all continuous.

We take the direct limit as n to define,

(25) Cobk

d = colimn

Cobk

d,n.

Now, in a way similar to (13), we can identify Cob1kd,n with the pull-back of the diagram

(26) Cobkd,n

k+1

Cob1k1

dpk+11,nmk+1

Pk+1 // Cob1k1

d1,n.

The bottom-horizontal map is the functor defined by sending a morphism

W [a, b] Rk+{k+1}c Rd1+n Rp+m{k+1}c,

to the product

W Pk+1 [a, b] Rk+{k+1}c R

d1+n Rp+m{k+1}c Rp+m{k+1}.

The right-vertical map is the functor which sends a morphism

W [a, b] Rk+ Rd1+n Rp+m


23/64


to

k+1(W) [a, b] Rk+{k+1}c R

d1+n Rp+m.

Notice that the definition of the functors Pkl+1 depend on our choice of the embeddings

from (10).The category Cob

kk

d,n is the cobordism category of manifolds with corners studied in [7].The category of interest, the one appearing in the statement of the main theorem from the

introduction is Cob0k

d,n , which we will often denote by Cobkd,n . Using (26), we can realize

Cobkd,n inductively starting with Cob

kk

d,n , by iterating the above pull-back construction from = k down to 0. Since geometric realization of simplicial spaces preserves finite limits,applying B( ), the classifying space functor to diagram (26) yields a cartesian square,

(27) BCob1k

d,n//

BCobkd,n

k+1

BCob1k1

dpk+11,nmk+1

Pk+1 // BCob1k1

d1,n.

Theorem 1.1 states that this cartesian square is in fact homotopy-cartesian.

Remark 2.4. The construction of this category does depend on our choice of collection ofembeddings . In order for all of our contructions relating to this category to be well definedand consistant, we will need to stick with this choice throughout the paper. However, it followsfrom Theorem A.2 and (24) that the isomorphism type of this category, as a topologicallyenriched category, does not depend on .

3. A Sheaf Model for Cobkd,n

In order to determine the homotopy type of the classifying space of the topological category

Cobk

d,n , we will need to study certain sheaves defined on the category of smooth manifolds

(without boundary) that are modeled on these spaces Cobk

d,n and BCobk

d,n .

3.1. A Recollection of Sheaves. Let X denote the category with objects given by smoothmanifolds without boundary with morphisms given by smooth maps.

Definition 3.1. By a sheaf (set valued) on X we mean a contravariant functor F from X toSets which satisfies the following condition. For any good open covering {Ui | i } of someX Ob(X), and every collection si F(Ui) satisfying si |UiUj= sj |UiUj for all i, j ,there is a unique s F(X) such that s |Ui= si for all i .

This is the same definition used in [13].


24/64


Definition 3.2. Let F be a sheaf on X. Two elements s0 and s1 of F(X) are said to beconcordant if there exists s F(X R) that agrees with pr(s0) in an open neighborhoodof X (, 0] and agrees with pr(s1) in an open neighborhood of X [1, ), where

pr : XR X is the projection on the first factor.

We denote the set of concordance classes of F(X) by F[X]. The correspondence X F[X]is clearly functorial in X.

Definition 3.3. For a sheaf F we define the representing space, denoted by |F| , to be thegeometric realization of the simplicial set given by the formula k F(ke) where

ke := {(x0, x1, . . . , xn) Rn+1 |

xi = 1}

is the standard extended k -simplex.

From this definition it is easy to see that any map of sheaves F G induces a map between

the representing spaces |F| |G| .

Definition 3.4. Let F be a sheaf on X. Assume A X is a closed subset, and let s be agerm near A, i.e., s colimU F(U) with U ranging over all open sets containing A in X.Then we define F(X, A; s) to be the set of all t F(X) whose germ near A coincides withs. Then two elements t0 and t1 are concordant relative to A and s if they are related bya concordance whose germ near A is the constant concordance equal to s. The set of suchrelative concordance classes is denoted by F[X, A; s].

Now, any element z F() determines a point in |F| which we also denote by z. Also, forany X Ob(X), such an element z determines an element, which we give the same name,

z F(X) by pulling back by the constant map. In [13, 2.4.3] it is proven that there is anatural bijection of sets

(28) [(X, A), (|F|, z)] = F[X, A; z].

Here the set on the left hand side is the set of homotopy classes of maps of pairs. Thenon-relative case of this isomorphism with A the empty set holds as well.

Using these observations we define the homotopy groups of a sheaf by setting

(29) n(F, z) := F[Sn, ; z].

By (28) we get n(F, z) = n(|F|, z) for any choice of z F().

Definition 3.5. A map of sheaves F G is said to be a weak equivalence if it induces ahomotopy equivalence |F| |G| of representing spaces.

The following result will give us a useful way to determine when a map of sheaves is a weakequivalence.


25/64


Proposition 3.1 (Relative surjectivity criterion). Let : F G be a map of sheaves.Suppose that induces a surjective map

F[X, A; s] G[X, A; (s)]

for every X X with a closed subset A X and any germ s colimU F(U) where U rangesover the neighborhoods of A in X. Then is a weak equivalence.

Proof. See [13].

In addition to set-valued sheafs, we will also have to consider sheaves on X which take valuesin the category of small categories, which we denote by CAT. A CAT-valued sheaf on Xis a contravariant functor from X to CAT satisfying the same sheaf condition with respectto good open covers given in Definition 3.1. Notice that for a CAT-valued sheaf F, for eachpositive integer k , one has set-valued sheaves NkF defined by sending X Ob(X) to thek -th nerve set of the category F(X).

In the case that F is a CAT-valued sheaf, the rule,

k F(ke)

defines a simplicial category. One can define a topological category |F| by setting

Ob(|F|) := |N0F|, Mor(|F|) := |N1F|.

One can construct the classifying space of the topological category |F| by taking the geometric-realization of the diagonal bi-simplicial space given by

n NnF(ne ).

We have

(30) B|F| = |k NkF(ke)|.

For details see [13, 4.1]. A map of CAT-valued sheaves : F G induces a map of spaces

B|| : B|F| B|G|.

This induced map B|| is a weak homotopy equivalence if for each non-negative integer k ,the induced maps

Nk() : NkF NkG,

are weak equivalences of set-valued sheaves.

3.2. The Sheaf Model. We define a CAT-valued sheaf on X whose representing space ishomotopy equivalent to Cobkd .

Remark 3.1. So far in this paper we have defined closed-lk -manifolds and lk -bordisms. In

this section and throughout the rest of this paper we will encounter manifolds with cornerswhich satisfy all conditions of definition 2.1, the definition closed lk -manifold, with the one


26/64


27/64


ii. For each I {1, . . . , k } , the Ith face IW has the factorization IW = IW PI

where

IW X (a 0, b + 0) Rk+Ic R

d1+n Rp+mIc

is a (dim(X) + d pI |I|)-dimensional lk -submanifold and PI Rp+m

I

is the closed

submanifold given in (10).iii. For each I k, the map and its restrictions |IW are submersions such that

the restriction of to the collar IW [0, ]I has the factorization,

IW [0, ]Iproj // IW

|IW // X.

Furthermore, if I {1, . . . , k } , the restriction |IW has the further factorization,

IW = IW PIproj // IW

IW // X

where the map IW is a submersion as well.

iv. The map (, f) is a proper map. For each I k, the restriction of f to the collarIW [0, ]I has the factorization,

IW [0, ]Iproj // IW

f|IW // R .

If I {1, . . . , k } the restriction f | IW has the further factorization,

IW = IW PIproj // IW

fIW // R

where (IW, fIW) is a proper map as well.v. For each I k , the restrictions of ( |IW, f |IW) to the pre-images

(IW, fIW)1(X ( 0, + 0)) for = a, b

are both submersions.

Remark 3.2. The factorization from condition iii. of the above definition implies that thefibres 1(x) are all k -manifolds of dimension d. Condition iv. implies that for any sub-manifold Y X (a 0, b + 0) that is transverse to the restriction (, f) |IW for allI k, the inverse image (, f)1(Y) is a k manifold as well. For details on the proof ofthese statements, see [7]. There the author proves similar statements regarding transversalityfor manifolds with corners. The situation for k -manifolds is similar. These points will beused in proofs to come.

Condition v. of the above definition implies that for (W,,f) Ck,

d,n (X; a,b,0, ), therestrictions f |1(x) are transverse to both a(x) and b(x) for all x X. This implies thatthe pre-image (, f)1(X [a, b]) is a lk -bordism from

Ma := (, f)1(X {a}) to Mb := (, f)

1(X {b}).


28/64


This condition also implies that the pre-images (, f)1(X ( 0, + 0)) are elements of

the sets Ck,

d,n (X; ,, , 0) for = a, b. From these observations we see that for each X, theset

ab Ck,

d,n (X; a,b, , 0)

has the structure of a category. Morphisms,

(W, W, fW) Ck,

d,n (X; a,b, , 0) and (V, V, fV) Ck,

d,n (X; c,d, , 0)

can be composed if b = c and if the pre-images (W, fW)1(X {b}) and (V, fV)

1(X {b})agree. We now eliminate dependence on 0 and by setting

(31) Ck,

d,n (X; a, b) = colim000

Ck,

d,n (X; a,b,0, ).

Definition 3.7. For X Ob(X), we set,

C

k,

d,n (X) = ab

C

k,

d,n (X; a, b)

with union ranging over all smooth functions a, b : X R with a b and

{x X | a(x) = b(x)}

an open subset of X.

The assignment X Ck,

d,n (X) for X X is a contravriant functor. Indeed, if g : X Y

is a smooth map then for any embedded lk -manifold

(, f , j) : W Y (a 0, b + 0) (Rk+ R

d1+n Rp+m)

representing an element of Ck,

d,n (Y), it follows from condition iii. of definition 3.6 that thespace g(W) defined by the pull-back,

g(W)

g // W

X

g // Y

is a k -manifold, see remark 3.2. Furthermore it has natural embedding

(, f ,j) : g(W) X (a 0, b + 0) (R

k+ R

d1+n Rp+m)

induced by (, f , j), which satisfies all conditions of definition 3.6, thus it yields an element

of Ck,

d,n (X). One can check also that Ck,

d,n satisfies the sheaf condition on X. Furthermore

for each X Ob(X), by the above discussion, Ck,

d,n (X) has the structure of a category, it

follows that Ck,

d,n is a CAT-valued sheaf on X.


29/64


Using maps induced by inclusion, Ck,

d,n Ck,

d,n+1, we set

Ck,

d = colim*

nC

k,

d,n ,

where by colim

*

we mean colimit in the category of sheaves which is taken by sheafifying thecolimit taken in the category of presheaves. There is a homotopy equivalence

(32) |Ck,

d | colimn

|Ck,

d,n |,

see [14, 5.2].

Definition 3.8. Let

Ckd,n(X; a,b,0, ) C

k,

d,n (X; a,b,0, )

be the subset satisfying the further condition:

vi. For x X let Ja(x) be the interval ((a 0)(x), (a + 0)(x)) R and let

Va = (, f)1({x} Ja(x)) {x} Ja(x) Rk+ R

d1+

n

Rp

+ m

.Then

Va = {x} Ja M {x} Ja(x) Rk+ R

d1+n Rp+m

for some (d 1)-dimensional k -submanifold M. The same condition must hold forthe function b.

We then proceed to define

Clkd,n(X; a, b) := colim

000

Clkd,n(X; a,b,0, ),

andC

lkd,n(X) :=

ab

Clkd,n(X; a, b).

As with Clk,

d,n , the contravariant functor X Clkd,n(X) is a sheaf on X. We also set

Clk

d,n = colim*

nC

lk

d,n

to get rid of dependence on n.

This added condition from definition 3.8 implies that for (W,,f; a, b) Clkd,n(X; a,b,0, )(X),

for all x X, the inclusion map of the pre-mage (, f)1({x} [a(x), b(x)]) into

[a(x), b(x)] Rk+ Rd1+n Rp+m

is an element of the space Emb(W, [a(x), b(x)] Rk+ Rd1+n Rp+m)

k, . This implies that

the fibre-bundle : W X is classified by a smooth map g : X Mor(Cobk

d,n), see(24). From this observation we have:


30/64


Proposition 3.2. For all n, the sheaf Clk

d,n is isomorphic to the category valued sheaf givenby

X C(X, Mor(Cobk

d,n)).

The category structure on C(X, Mor(Cobk

d,n)) is given by point-wise multiplication.

Proof. Proof follows from the above discussion.

Proposition 3.3. For each n there are weak homotopy equivalences

B|Ck

d,n| BCobk

d,n.

Proof. Proof is the same as Proposition 2.9 of [6].

For each n there is a map of sheaves in : C

k

d,n Ck,

d,n induced by inclusion.Proposition 3.4. For each n, the above map in induces a weak homotopy equivalence,

B|Ckd,n| B|C

k,

d,n |.

Proof. The proof is almost the same as proof of Proposition 4.4 of [ 6]. The only potentialcomplication arrises in dealing with transversality in the context of lk -manifolds. We providea sketch of the proof.

We show that in induces a weak equivalence of set-valued sheaves

NkCk

d,n

NkC,kd,n

for each k = 0, 1, 2, . . . by the relative surjectivity criterion.

Let X Ob(X). Fix a smooth function : R [0, 1] which is 0 near (, 13 ] and is 1

near [ 23 , ), such that

0 everywhere and

> 0 on 1(0, 1). Given smooth functionsa b : X R with (a b)1(0) X an open subset, we define : X R X R bythe formulas

(x, u) = (x, x(u)),

x(u) =

a(x) (b(x) a(x)) ( ua(x)

b(x)a(x) ) if a(x) < b(x)

a(x) if a(x) = b(x)

Suppose that W C,kd,n (X; a, b) with a b. Condition v. of Definition 3.6 implies that the

maps (, f) and are transverse, i.e. the product map

(, f) : W XR (XR) (XR)


31/64


is transverse to the diagonal . Furthermore, condition v. of Definition 3.6 implies that forall I k , (, f) |IW and are transverse and if I {1, . . . , k l}, then (IW, fIW) and are transverse. These transversality conditions imply that the space

W := W = {(x,u,z) | (z) = x, f(z) = x(u)}

is a lk -submanifold of X R W. Using the embedding

(, f , j) : W XRRk+ Rd1+n Rp+m,

we can rewrite W as

W = {(x,u,r)|(x, x(u), r) W} XRRk+ R

d1+n Rp+m.

By inspection of the formula for , it follows that

W (X (, a + ) Rk+ R

d1+n Rp+m) = Ma (, a + )

W (X (b , ) Rk

+ Rd1+n

Rp+m

) = Mb (b , )for closed lk -manifolds Ma and Mb where = 1 on (b a)

1(0) and = 13

(b a) otherwise.

Thus W determines an element ofCk

d,n(X; a, b). We now show that W is actually concordant

to W in C,kd,n (X; a, b). Define

s(u) = (s)(u) + (1 (s))u

with any smooth function from R to [0, 1] for which = 0 near (, 0] and = 1 near[1, ). Define : X RR XR as (x,s,u) = (x, x(s, u)) where

x(s, u) = a(x) + (b(x) a(x))s( ua(x)b(x)a(x) if a(x) < b(x),(s) a(x) + (1 (s))u if a(x) = b(x).Like before, is transverse to ( |IW, f |IW) for all I k and to (IW, fIW) for allI {1, . . . , k } and so

W = {((x, s), u , r)|(x, x(s, u), r) W} (X R) RRk+ R

d1+n Rp+m

is a k -submanifold and defines the required concordance in C,kd,n (X R) from W to W .

This proves that in induces surjections

N0Ck

d,n[X] N0C,kd,n [X] and N1C

k

d,n[X] N1C,kd,n [X].

The argument needed to prove relative surjectivity is similar. This proves that

NkCk

d,n

NkC

,kd,n

is a weak equivalence for k = 0, 1. The case for general k is similar.


32/64


4. The Sheaf Dk

d

4.1. The Sheaf. In this section we define sheaves Dk

d,n whos representing spaces are weakly

homotopy equivalent to the classifying spacesBCob

k

d,n .Definition 4.1. Let X Ob(X), and := (1, . . . , k) : X (0, )k be a smooth function.

We define Dkd,n,(X) to be the set of embedded, (dim(X) + d)-dimensional

kl -submanifolds

(, f , j) : W XR (Rk+ Rd1+n Rp+m)


i. For each I k , the face IW, is embedded with a collar parametrized by X. Bythis we mean that there is equality,

W (XR [0, ]I Rk+Ic R

d1+n Rp+m) = IW [0, ]I.

Here, we are identifying W with its image under (, f , j).ii. For each I {1, . . . , k } , the Ith face IW has the factorization IW = IW P

I

whereIW XRR

k+Ic R

d1+n Rp+mIc

is a (dim(X) + d pI |I|)-dimensional lk -submanifold and PI Rp+mI is the closed

submanifold given in (10).iii. The map and the restrictions |IW are submersions. Furthermore they satisfy

exactly the same factorization conditions as in condition iii. of Definition 3.6.iv. The map (, f) is a proper map. Furthermore (, f) and the restrictions

( |IW, f |IW) satisfy exactly the same factorization conditions as in condition iv. of

Definition 3.6.Remark 4.1. The above definition looks quite similar to the definition of C

k,

d,n (X). However

condition v. of the definition of Ck,

d,n implies that the map : W X is a proper-

submersion and hence a fibre-bundle for an element (W,,f) Ck,

d,n (X; a, b). Since the abovedefinition does not include this condition we cannot assume that the submersion : W Xa fibre bundle for (W,,f) D

kd,n,(X). The map will not be proper in general and the

diffeomorphism types of the fibres 1(x) may change with x.

We now take the direct limit over all functions : X (0, )k to define

(33) Dkd,n(X) := colim0 D

kd,n,(X).

The assignment X Dk

d,n(X) is a contravariant functor in the same way as described for

Ck

d,n in the previous section. It can be verified easily that Dk

d,n satisfies the sheaf conditionwith respect to locally finite open covers on elements of Ob(X).


33/64


Taking the direct limit over the direct system given by the maps Dk

d,n Dk

d,n+1 induced byinclusion, we define,

(34) Dk

d = colim*

nD

k

d,n

where as before, the colim* is the direct limit in the category of sheaves. As in [14] we have

a homotopy equivalence |Dk

d | colimn

|Dk

d,n| .

The definition of Dk

d,n depends on the choice embeddings from (10). It will latter be seen

that the homotopy type of |Dk

d,n| does not depend on these choices.

4.2. A Fibre Sequence. We now specialize to the case where = 0. We denote,

Dkd,n := D

0k

d,n.

Consider the element (W,,f) Dkd,n(X) wtih X arbitrary. Note that kW is a k1 -

submanifold of

XRRk+{k}c Rd1+n Rp+m{k}c

and so (kW,,f) is naturally an element of Dk1d1pk,nmk

(X). As a result, there is a mapof sheaves,

k : Dkd,n D

k1dpk1,nmk

given by sending an element W Dkd,n(X) to kW Dk1dpk1,nmk

(X).

Theorem 4.1. For all n > 0, there is a homotopy fibre sequence

|Dk1d,n | // |D

kd,n|

|k|// |D

k1d1pk,nmk | .

In order to prove this theorem we will need to use some results regarding the concordancetheory of sheaves on X. We recall now from [13, 4.1.5] the definition of a certain property forsheaves analogous to the homotopy lifting property for spaces.

Definition 4.2. A map of sheaves

: F G

is said to have the Concordance Lifting Property if for any X X, given s F(X) andh G(XR) such that there exists > 0 with

pr

((s)) |X(,) = h |X(,),where pr : XR X is the projection, then there exists h F(X R) such thath |X(,) = pr((s)) |X(,) and (h) = hwhere is some positive real number, possibly different than .


34/64


Any element z G() gives rise to an element (which we give the same name) z G(X) forX Ob(X) by pulling back over the constant map.

Definition 4.3. Let : F G and z G() be as above. The fibre of the map over zis the sheaf Fz defined by

Fz (X) = {s F(X) | (s) = z}.

In [13, A.2.4] the authors prove the following theorem:

Proposition 4.2. Suppose given sheaves E, F, G on X and maps of sheaves

u : E G and v : F G.

Let E G F be the fibred product of u and v . If u has the concordance lifting property, thenthe projection E G F F has the concordance lifting property and the following square ishomotopy cartesian:

|E G F| //

|F||v|

|E|

|u|// |G|.

For this we get an immediate corollary as a special case:

Corollary 4.3. If v : F G has the concordance lifting property then there is a homotopyfibre sequence,

|Fvz| |F| |G|

for any choice of z G().

We now apply these results to the map k : Dkd,n D

k1dpk1,nmk

.

For 0 < k , there are maps,

k+1 : Dk

d,n D1k1

d1,n,

k+1 : D1k

d,n D1k1

dpk+11,nmk+1,

given by,

W k+1W and W k+1W.

There is also an inclusion map

i : D1k

d,n Dk

d,n

given by simply forgetting the product structure k+1W = k+1WPk+1 on the k+1 -th face of the boundary of W. Now in a way similar to the the situation with the spaces of


35/64


embeddings of k -manifolds from (13), the following diagram is cartesian

(35) D1k

d,n

i //

k+1

Dkd,n

k+1

D1k1

dpk+11,nmk+1

Pk+1 // D1k1

d1,n.

We will need the following technical lemma:

Lemma 4.4. For any k and > 0, the map

k+1 : Dk

d,n D1k1

d1,n

has the concordance lifting property.

Proof. Fix an element X Ob(X) and let W D1k1

d1,n(X R) be a concordance. Let

V Dk

d,n(X) be an element which satisfies the following condition: there exists > 0 suchthat

(36) pr(k+1V) |X(,) = W |X(,)

where pr : XR X is the projection. We will construct a concordance V Dkd,n(XR)such that

k+1

V = W and

V |X(,) = pr

(V) |X(,)for some > 0, possibly different from . This will prove the lemma.

First we make the observation that equation (36) implies that the restriction W |X(,) iscylindrical. This means that

(37) W |X(,) = 1(X {t}) (, ) for any t <

where : W X R is the projection that comes with W from the definition as an

element of D1k1

d,n (XR). This is a fact which we will exploit in out our construction of the

concordance V .Recall that by definition of D

k

d,n(X), the face k+1V is embedded into the space

X RRk+ Rd1+n Rp+m

with a collar whose width is parametrized by X. Specifically, there exists a smooth function

k+1 : X (0, ),

such that

V (X RRk+{k+1}c [0, k+1) Rd1+n Rp+m) = k+1V [0, k+1).


36/64


In the above equation k+1V [0, k+1) is given by the set

{(v, s) k+1V [0, ) | 0 s < k+1((v)) },

where is the projection onto X. The space

XRRk+{k+1}c [0, k+1) R

d1+n

Rp+m

is given by the set of all tuples (x,r, (t1, . . . , tk), y) such that

(x, r) XR, 0 tk+1 < k+1(x), and y Rd1+n Rp+m.

Let > 0 be as in (36). Let : R [0, 1) R [0, 1) be a smooth function which satisfiesthe following conditions:

i. The image of is contained in the subspace R [0, 23 ) [0,23 ) [0, 1).

ii. When restricted to the subspace R [0, 13

) [0, 3

) [0, 1), is equal to the identityfunction. Specifically, (t, s) = (t, s) if t 3 or if s

13

.

ii. For (t, s) R

[

2

3 , 1), (t, s) = (

2 , s).

(0, 0)3

23

13

23

Id

Id Id

2

Figure 3. Above is a schematic for the function .

Let : R [0, 1) R be the projection (t, s) t , for (t, s) R [0, 1). Using these twofunctions and , we define, : XR [0, k+1) XRby the formula

(38) (x, t, s) (x, (t,

s

k+1(x) )).

It follows directly from the definition of and that for all x X,

(a) if s k+1(x)3 or t 3 , then (x,t,s) = (x, t),

(b) if s 2k+1(x)3 , then (x,t,s) = (x, 2 ) for all x and t.


37/64


We now form the pull-back, (W)

// W

XR [0, k+1) // XR

where : W X R is the projection onto X R. By definition of D1k1

d1,n(X R), is a submersion and for each I k , the restriction |IW is a submersion as well. Itfollows from the factorization in condition iii. of Definition 4.1 that (W) is a k -manifold,see remark 3.2. Furthermore, the induced map is a submersion as well. The space (W)is by definition equal to the space

(39) {((x,t,s), w) XR [0, k+1) W | (x,t,s) = (w)}and so there is a natural embedding

i : (W) (X R) RRk{k+1}c [0, k+1) Rd1+n Rp+minduced by the inclusion of W into (X R) RRk{k+1}c R

d1+n Rp+m . From nowon we refer to (W) as the submanifold determined by the above embedding.It can be verified directly using condition (b) and (39) that the intersections

(W) (XR) RRk{k+1}c ( 2k+13 , k+1) Rd1+n Rp+m,prV

(XR) RRk{k+1}c (

2k+13 , k+1) R

d1+n Rp+m

,

are equal. Denoting by V the second of the above intersections, this implies that the unionV (W)is k -submanifold of (XR) RR

k+ R

d1+n Rp+m . We setV := V (W).We claim that V is the desired lift of the concordance W that we seek. It is easy to checkthat the projection onto (X R) is a submersion and the projection onto (X R) R isproper. Furthermore, it follows from condition (a) that the intersection

V

(XR) RRk{k+1}c [0,

k+13

) Rd1+n Rp+m

,

is equal to the product k+1W[0, k+13 ) and so k+1W is embedded with a parametrized

collar. Thus the manifold V does determine an element of Dkd,n(X R). We have from (a),that k+1V , which is given by the intersectionV [(XR) RRk{k+1}c Rd1+n Rp+m],


38/64


is equal to k+1W. One also sees that the intersection

V

(X (, 3 )) RR

k+ R

d1+n Rp+m

,

is equal to prV |X(, 3 ) . This implies that the element of D

kd,n(X R) determined by V

is the desired lift of W which extends prV |X(, 3

) . This proves the lemma.

Applying this lemma together to the pull-back diagram (35), Proposition 4.2 implies thefollowing theorem:

Theorem 4.5. The cartesian diagram

(40) |D1k

d,n ||i|

//

|k+1|

|Dk

d,n|

|+1|

|D1k1

dpk+11,nmk+1|

|P+1| // |D1k1

d1,n|

obtained from (35) is homotopy cartesian.

Proof. Follows from the previous lemma.

We now identify the homotopy-fibres of the vertical maps in the above diagram. Since thediagram is homotopy-cartesian, the homotopy fibres of both columns will be the same.

The element D1k1

d1,n( ), given by the empty set, determines an element

D1

k1d1,n(X) for all X Ob(X). Denote by Fk+1 the fibre-sheaf over of the map

k+1 : Dk

d,n D1k1

d1,n.

This is defined by

Fk+1 (X) = {W D

kd,n(X) | k+1(W) = }.

Lemma 4.6. The fibre sheaf Fk+1 is isomorphic to the sheaf D

1k1

d,n

Proof. Elements of D1k1

d,n (X) are given by 1k1 -submanifolds of

XRRk1+ (, ) Rd1+n Rp+m.

For X Ob(X), an element of Fk+1 (X) is given by a

k -submanifold

W XRRk+ Rd1+n Rp+m


39/64


satisfying all conditions of the definition of Dk

d,n(X) with the added property that

W (XRRk+{k}c Rd1+n Rp+m) = k+1W = .

SoF

k+1

(X) can be characterized as the subset ofD

1k1

d,n (X) consisting of all manifolds thatlie in

X RRk1+ (0, ) Rd1+n Rp+m.

Choosing a diffeomorphism,

: XRRk1+ (, ) Rd1+n Rp+m XRRk1+ (0, ) R

d1+n Rp+m

that is identical on the factors of X, Rk1+ , Rd1+n , and Rp+m , we can define a map

D1k1

d,n Fk+1

by sending W D1k1

d,n (X) to (W), which is an element of Fk+1 (X). This map is clearly

natural in X since the map is identical on the X-factor. Since is a diffeomorphism, thismap of sheaves is invertible, and thus is a natural isomorphism.

The previous lemmas implies that for all 0 < k there is a homotopy-fibre sequence

(41) |D1k1

d,n |// |D

kd,n|

|k+1| // |D1k1

d1,n|.

This combined with the fact that diagram (40) is homotopy cartesian implies that for all0 < k there is a homotopy fibre sequence,

|D1k1

d,n |// |D

1k

d,n ||k+1| // |D

1k1

dpk+11,nmk+1|.

Applying this theorem to the case when = 1, the homotopy-fibre sequence (41) implies thatthere is a homotopy fibre sequence,

|Dk1d,n | |D

kd,n| |D

k1d1pk,nmk

|.

This proves Theorem 4.1.

5. The Classifying Space of Cobk

d

In this section we show that for each n there is a weak homotopy equivalence

BCob

k

d,n |D

k

d,n|.Combining this with the results of Section 8 will prove Theorem 1.3 stated in the introduction.

We define a category-valued sheaf Dk,

d,n which we can map to both Dkd,n and C

k,

d,n . In thissection we will need a bit more technical machinery relating to CAT-valued sheaves. We willneed to consider Co-cycle Sheaves.


40/64


5.1. Cocycle Sheaves. Here we will review the definition of Co-cycle sheaves and state themain results that we need to use. For more details on Co-cycle sheaves, we refer the readerto [13].

Definition 5.1. Let F be any CAT-valued sheaf on X. The is an associated set valued

sheaf F on X. Choose once and for all an uncountable set J. An element of F(X) is apair (U, ) where U = {Uj | j J} is a locally finite open cover of X, indexed by J, and is a collection of morphisms, RS N1F(US) indexed by pairs R S of non-empty finitesubsets of J and US = iSUi , subject to the conditions:

i. RR = IdcR for an object cR N0(UR),ii. For each non-empty finite R S, RS is a morphism from cS to cR |US (where by

condition i. IdcS = SS and IdcR = RR ),iii. For all triples R S T of finite non-empty subsets of J, we have

RT = (RS |UT) ST.

Theorem 5.1. There is a weak homotopy equivalence

|F| B|F|.

Proof. See Theorem 4.1.2 of [13].

Remark 5.1. The above homotopy equivalence is natural in the following sense. The assign-ments

(42) F B|F| and F |F|

are functors from the category of CAT-valued sheafs on X to the category of TopologicalSpaces. In [13, A.3], the authors connect these two functors (42) through a zig-zag of natural

transformations of functors sending CAT-valued sheafs to Spaces. The authors then showthat these natural transformations induce equivalences upon passing to the homotopy category.

Remark 5.2. A set valued sheaf F can be considered a category-valued sheaf by definingF(X) to be the object set of a category with only identity morphisms. In this way it makessense to define F. In this case with F a set valued sheaf, there is a forgetful projectionF F which is is a weak equivalence. For proof of this see Section 4.2 of [13].

5.2. The Classifying Space. We now define a Poset-valued sheaf Dk,

d,n which we can map

to both Ck,

d,n and Dk

d,n .

Definition 5.2. For X X, we define Dk,

d,n (X) to be the set of quadruples (W; , f , a)subject to the following conditions:

i. (W; , f) Dk

d,n(X),ii. a : X R is a smooth function,


41/64


iii. the function f : W R and all restrictions f |IW are fibrewise transverse to awith respect to the submersion .

By fibrewise transverse we mean that for each x X the restriction of f (and f |IW for each

I) to 1

(x) is transverse to the point a(x). For each X X, the set Dk,

d,n (X) actuallyhas the structure of a category, namely a partially ordered set. The objects are the elements(W; , f , a) as described above. A morphism is given by a quintuple (W; ,f,a,b) where

(W; , f , a) and (W; , f , b) are elements of Dk,

d,n (X) and a(x) b(x) for all x X. In this

way Dk,

d,n (X) has the structure of a partially ordered set. Notice that there are no morphisms

connecting elements (W; , f , a) and (V; , f

, b) if W = V .

For each positive integer n, there is a map of sheaves

n : Dk,

d,n Ck,

d,n

defined as follows. Let (W; , f , a) be an object of D

k,

d,n (X). There exists a smooth function : X (0, ) such that f is fibre-wise transverse to all other smooth functions g : X Rsuch that a g a + . The properness of (, f) then implies that the restriction of(, f) to the open subset

W := (, f)1(X (a , a + )),

is a proper submersion W X (a , a + ). Thus the class [W], as 0, is a well

defined element of Ck,

d,n (X; a, a) hence an element of Ob(Ck,

d,n (X)). We define,

n(W; , f , a) := ([W], , f , a , a).

This defines n on the level of objects, it is defined similarly on the level of morphisms thus

giving a map of CAT-valued sheaves.We now state:

Proposition 5.2. The map : Dk,

d,n Ck,

d,n defined above induces a weak homotopyequivalence

B|Dk,

d,n | // B|C

k,

d,n |.

Proof. This proposition is proven by showing that n induces homotopy equivalences

|Nk(Dk,

d,n )| | N k(Ck,

d,n )|

for all k 0. This can be done in exactly the same way as in the proof of Proposition 4.3from [6].

Now we consider the map h : Dk,

d,n Dk

d,n defined by forgetting the Partially ordered

set structure on Dk,

d,n (X). Applying the cocycle construction induces a map given by the


42/64


composition

(43) Dk,

d,n

h // Dk

d,n

// Dk

d,n

where the second map is the forgetful projection which is a weak equivalence, see remark 5.2.

Proposition 5.3. The above map (43) Dk,

d,n Dk

d,n , is a weak equivalence of sheaves.

Thus there is a weak homotopy equivalence B|Dk,

d,n | |Dk

d,n|.

Proof. The proof is essentially the same as the proof of [6, 4.2]. A proof is also writtenexplicitly in [7] for the case = k , for manifolds with corners.

The results from this section yield a zig-zag of weak homotopy equivalences

(44) B|Ck,

d,n | B|Dk,

d,n |oo // |D

k

d,n|.

Theorems 3.4 and 3.3 yield a zig zag,

(45) BCobk

d,n B|Ck

d,n|oo // B|C

k,

d,n | .

Combining the above yields a weak homotopy equivalence,

(46) BCobk

d,n |Dk

d,n|.

5.3. Proof of Theorems 1.2 and 1.1. Recall the functor

k : Cobkd,n Cob

k1dpkm,nmk

defined by sending a k -cobordism W to kW and the map of sheaves by the same name

k : Dkd,n D

k1dpk1,nmk

from Section 4.2. Since the weak homotopy equivalences from the above zig-zag diagrams areall induced by natural transformations of sheaves, the diagram

BCobkd,n

Bk

oo // |Dkd,n|

|k|

BCobk1dpk1,nmk

oo // |Dk1dpk1,nmk

|

commutes, where the horizontal arrows come from the zig-zags of weak equivalences from(44) and (45). Commutativity of this diagram implies that the vertical maps have weaklyequivalent homotopy-fibres. Applying Theorem 4.1, we see that the homotopy-fibre of

B(k) : BCobkd,n BCob

k1dpkm,nmk


43/64


is weakly equivalent to |Dk1d,n | and thus by (46), weakly equivalent to BCob

k1d,n . This holds

for all n. Taking the direct limit as n yields the homotopy fibre sequence,

BCobk1d BCob

kd BCob

k1dpkmk

.

This proves Theorem 1.2 stated in the introduction. We in fact just proved the stronger resultthat the above is a fibre sequence prior to letting n run to .

The above zig-zags from (44) and (45), since they are induced by natural transformations,yield a commutative diagram,

|D1k

d,n |44

tt

// |Dk

d,n|99

yy

k+1

BCob1k

d,n//

BCobk

d,n

k+1

|D1k1

dpk+11,nmk+1|

55

uu

Pk+1 // |D1k1

d1,n|99

yy

BCob1k1

dpk+11,nmk+1

Pk+1 // BCob1k1

d1,n

where the front and back squares are the cartesian squares (27) and (40) respectively. ByTheorem 4.5, the back square is homotopy cartesian. Since the diagonal zig-zags are weakequivalences, the front square is homotopy cartesian as well. This proves Theorem 1.1 from

the introduction.

6. Cubical Diagrams

6.1. k -Cubic Spaces. As in previous sections, let k denote the set {1, , k} of k -elements. We denote by 2k the Power Set of k made into a category with objects thesubsets of k and morphisms given by the inclusion maps. We call a contravarient functorfrom 2k to Top (the category of topological spaces), a k -cubic space. In order to define ak -cubic space one needs to associate to each subset J k a space XJ and to any pair ofsubsets I J, a map

fJ,I : XJ XI

such that for any triple K I J, the equation fJ,K = fI,K fJ,I holds. In this case wewill generally denote the resulting k -cubic space by X . The spaces XJ and maps fJ,I fittogether to form a k -dimensional cubical commutative diagram. We will often times refer tothe spaces XJ as vertices of the cube and the maps fJ,I as edges.


44/64


We define a map of k -cubic spaces, F : X Y , to be a natural transformation of thefunctors X and Y . Or in other words a collection of maps

FJ : XJ YJ

such that for any pair of subsets I J, the diagram

XIFI // YI

XJ

OO

FJ // YJ

OO

commutes where the vertical maps are the edges in the cubes X and Y .

We denote the set of all such k -cubic space maps by

C0k(X, Y).

This set is topologized naturally as a subset of the product Jk C0(XJ, YJ) where C

0(XJ, YJ)

is the space of continuous maps from XJ to YJ with the compact-open topology.It is easy to see that a k -manifold W determines a k -cubic space by the the correspondenceI IW for I k . Also notice that the spaces Rk+ and R

p+m from the previous sectionsdetermine k -cubic spaces via the correspondences,

I Rk+I = {(t1, . . . , tk) | ti = 0 if i / I}

I Rp+mI = {(x1, . . . , xk) Rp1+m1 Rpk+mk | xi = 0 i / I}.

6.2. k -Cubic Spectra. In addition to k -cubic spaces we will also have to consider k -cubicspectra. A k -cubic spectrum is a functor

X : 2k Specwhere Spec is the category of spectra. It is required that for each I J k the edge (inthe cubical diagram), XJ XIis defined to be a map of spectra of degree 0. Let X be a k -cubic spectrum. Then foreach integer n, there is a k -cubic space (X)n defined by sending each J k to the nthspace of the spectrum XJ. The operations of suspending, X X , and de-suspendingX 1X still make sense in the context of k -cubic spectra.6.3. The Total Homotopy Cofibre of a k -Cubic Space. We define an important type of

homotopy colimit associated to a k -cubic space called the Total Homotopy Cofibre. This willbe defined inductively on k . We first introduce some new notation to enable us to carry outthe induction. Let X be a k -cubic space. For j k we define X,j to be the k 1-cubicspace defined by restricting the functor

X : 2k Top


45/64


to the full subcategory of 2k on all subsets of k that contain j . Similarly, we define X,jto be the restriction of X to the full subcategory of 2

k on all subsets disjoint from j . Thereis a natural map of k 1-cubic spaces

(47) ikj : X,j X,j

induced by these inclusions.

Definition 6.1. Let X be a k -cubic

cobordism category of manifolds with baas-sullivan singularities, part 2

Documents