THE MODULI SPACE OF COMPACT BUNDLE STRUCTURES ON A

THE MODULI SPACE OF COMPACT BUNDLE STRUCTURES

ON A FIBRATION

STACY HOEHN

Contents

1. Introduction 12. Preliminary Definitions 33. Framed Bundle Structures 74. Framed Bundle Structures as a Space of Lifts 165. Relationship with h-cobordisms 216. Constructing Sections with the Grothendieck Construction 297. Retractive Space Models 328. Parametrized Euler Characteristics 369. Key Theorem 4110. Proof of Lemma 9.6 4511. Conclusion 54Appendix A. Results about Sections of Fibrations 55References 65

1. Introduction

A classical question asks when a finitely dominated topological space is homotopyequivalent to a finite CW complex. One way to answer this question was provided byWall in [Wal65, Theorem F]. Namely, to any finitely dominated CW complex X, heassociated an element σ(X), called the finiteness obstruction for X, in the reduced

projective class group K0(Zπ1X) of the integral group ring of the fundamentalgroup of X. Wall showed that σ(X) vanishes if and only if X is homotopy equivalentto a finite CW complex.

The vanishing of Wall’s finiteness obstruction can be turned into a statementabout compact topological manifolds. Any finite CW complex is homotopy equiva-lent to a finite simplicial complex ([LW69, Proposition 7.1]), and any finite simplicialcomplex is homotopy equivalent to the compact manifold (with boundary) givenby a regular neighborhood of the simplicial complex in Euclidean space ([RS82,Proposition 3.10]). Thus, every finite CW complex K is homotopy equivalent toa compact manifold. Conversely, work of Kirby and Siebenmann [KS69, TheoremIII] shows that every compact topological manifold is homotopy equivalent to a fi-nite CW complex. Therefore, Wall’s finiteness obstruction for a finitely dominated

The author was partially supported by the National Science Foundation.

1

2 STACY HOEHN

space X vanishes if and only if X is homotopy equivalent to a compact topologicalmanifold.

Instead of asking when a single finitely dominated space is homotopy equivalentto a compact topological manifold, we can ask a similar question for families offinitely dominated spaces. Namely, suppose p : E → B is a fibration over a con-nected finite CW complex B whose fibers are all homotopy equivalent to a finitelydominated CW complex X. We can ask when p admits a fiber homotopy equiva-lence to a topological fiber bundle whose fibers are all compact manifolds, and wecan also try to classify all such compact topological fiber bundle structures on p,provided that at least one exists. The former question was addressed in [DWW03]and [KW09], and the latter question will be the subject of this paper.

An obvious necessary condition for p to be fiber homotopy equivalent to a topo-logical fiber bundle with compact manifold fibers is that each individual fiber of p ishomotopy equivalent to a finite CW complex (or, equivalently, to a compact man-ifold), which can be checked using Wall’s classical finiteness obstruction. However,this is not a sufficient condition; even if the fibers of p are all homotopy equivalentto a finite CW complex, p still might not admit a compact topological fiber bundlestructure. An example of such a fibration p is provided in [KW09, Example 1.3].

In [DWW03], Dwyer, Weiss, and Williams construct a space-level version ofWall’s classical finiteness obstruction, which we will now describe, to provide a con-dition that must be satisfied in order for p to admit a compact topological fiber bun-dle structure. Given a space X, there is an associated based space Wh(X), called

the Whitehead space of X, which has the property that π0(Wh(X)) ∼= K0(Zπ1X),the home of Wall’s classical finiteness obstruction. Given a fibration p : E → B,there is an associated fibration

WhB(p) : WhB(E)→ B;

the fiber of this fibration over any point b in B is simply the Whitehead spaceWh(p−1(b)) of the fiber of p over b. Since each of the fibers of WhB(p) are based,the fibration WhB(p) has a zero section zero : B →WhB(E). In [DWW03], Dwyer,Weiss, and Williams construct another section of WhB(p), which we will denote by

Wall(p) : B →WhB(E).

By Corollary 10.18 in [DWW03], the fibration p : E → B admits a compact topo-logical fiber bundle structure if and only if Wall(p) is homotopic (through sections)to the zero section of WhB(p). In [KW09, Theorem B], Klein and Williams givealternate sufficient homotopy-theoretic criteria for a fibration p : E → B to admita compact topological fiber bundle structure.

Remark 1.1. Dwyer, Weiss, and Williams actually construct a section

χ(p) : B → AB(E)

of another fibration associated to p; Wall(p) is simply the composition of χ(p) witha natural map AB(E) → WhB(E) over B which fits into a homotopy pullbackdiagram of the form

A%B(E) //

B

zero

AB(E) //WhB(E)

THE MODULI SPACE OF COMPACT BUNDLE STRUCTURES ON A FIBRATION 3

where A%B(E) → B is yet another fibration associated to p. Corollary 10.18 of

[DWW03] states that p : E → B admits a compact topological fiber bundle struc-

ture if and only if χ(p) : B → AB(E) lifts to give a section χ%(p) : B → A%B(E).

Because of the above homotopy pullback diagram, χ(p) lifts to a section χ%(p) ifand only if Wall(p) is homotopic to the zero section.

In this paper, we will assume that the fibration p : E → B does admit at leastone compact topological fiber bundle structure; we aim to classify all such compacttopological fiber bundle structures on p. In other words, we will compute the modulispace S(p) of all such structures on p, which will be formally defined in Section 2.We will show that this structure space has a nice decomposition as the product oftwo other spaces, one related to a space of microbundles and the other related toalgebraic K-theory. In particular, we will prove the following theorem.

Theorem 1.2 (Main Theorem). Suppose that p : E → B is a fibration, whereB is a finite CW complex and the fibers of p are homotopy equivalent to a finiteCW complex, and suppose further that p admits a compact topological fiber bundlestructure. Then the moduli space of all such bundle structures on p is given by

S(p) ' Micro(E)×MapB(B,ΩBWhB(E)),

where Micro(E) denotes the space of all stable microbundles over E and

MapB(B,ΩBWhB(E))

denotes the space of all sections of a fibration ΩBWhB(E) → B associated to pwhose fiber over any point b in B is the loop space ΩWh(p−1(b)) of the Whiteheadspace of the fiber of p over b.

The proof of this theorem occupies the rest of this paper. We first use fiberwisetangent microbundles to define a map S(p) → Micro(E) and show that S(p) ishomotopy equivalent to Micro(E)×Sfr(p), where Sfr(p) is a suitably defined modulispace of framed bundle structures on p. We then extend ideas from [DWW03] anduse various ideas relating to Waldhausen’s algebraicK-theory of spaces to transitionfrom Sfr(p) to the space of sections of the fibration ΩBWhB(E)→ B.

2. Preliminary Definitions

Throughout this paper, we will let R∞ denote the set of sequences (x1, x2, x3, ...)in the real numbers such that xi = 0 for all but finitely many i. Note that forany s > 0, Rs includes in R∞ by sending a point (x1, x2, ..., xs) to the point(x1, x2, ..., xs, 0, 0, ...). We will let R∞ have the direct limit topology.

The space Bun(B) of bundle structures on a space B. Let B be a finite,connected CW complex, and fix an integer n ≥ 0. We will begin by defining asimplicial set of n-dimensional compact fiber bundle structures on B. A k-simplexof the simplicial set Bunn(B) consists of a space E′ embedded in Rs × B ×∆k ⊂R∞ ×B ×∆k for some s > 0 such that the projection

q : E′ → B ×∆k

is a topological fiber bundle projection whose fibers are compact n-dimensionalmanifolds.

4 STACY HOEHN

There is a stabilization map Bunn(B) → Bunn+1(B) which sends a k-simplexE′ ⊂ Rs ×B ×∆k to

I × E′ ⊂ I × Rs ×B ×∆k ⊂ Rs+1 ×B ×∆k ⊂ R∞ ×B ×∆k.

Let Bun(B) denote the homotopy colimit

Bun(B) := hocolimn

Bunn(B).

Then Bun(B) is a stabilized space of bundle structures over B.

The space S(p) of bundle structures on a fibration p. Suppose p : E → B isa fibration whose fibers are homotopy equivalent to a finite CW complex, and onceagain fix an integer n ≥ 0. We will now define a simplicial set of n-dimensionalcompact fiber bundle structures on p. A k-simplex in the simplicial set Sn(p)consists of a pair (E′, φ), where E′ is a k-simplex in Bunn(B) and φ : E′ → E×∆k

is a fiber homotopy equivalence.

E′

q ##

φ // E

p×id∆k

B ×∆k

Remark 2.1. Two vertices (E′0, φ0) and (E′1, φ1) in Sn(p) lie in the same path com-ponent if and only if there exists a bundle isomorphism h : E′0 → E′1 such thatq1 h = q0 and such that φ1 h is homotopic to φ0.

Once again, there is a stabilization map Sn(p) → Sn+1(p), which sends (E′, φ)to (I ×E′, φ π), where π : I ×E′ → E′ denotes projection onto the second factor.Let

S(p) := hocolimn

Sn(p)

denote the simplicial set of stabilized structures on p.

As a first step in proving Theorem 1.2, we will show that the space S(p) definedabove is related to a space Micro(E) of stable microbundle structures on the totalspace E of p. In particular, we will show that there is a map S(p) → Micro(E)which is obtained by essentially sending a compact bundle structure on p to itsstable fiberwise tangent microbundle, which we will now describe.

Recall from [Mil64, Section 2] that an n-dimensional microbundle X is a diagram

Bi // E(X)

j // B

of topological spaces such that ji = idB and such that the following local trivialitycondition is satisfied: For each b ∈ B, there exists an open neighborhood U of b inB and an open neighborhood V of i(b) in E = E(X) such that i(U) ⊂ V , j(V ) ⊂ U ,and V is homeomorphic to U ×Rn under a homeomorphism h : V → U ×Rn which


makes the following diagram commute:

V

h

j|V

##U

i|U

;;

×0 ##

U

U × Rnπ1

;;

Suppose X1 is an n-dimensional microbundle over a space B1, X2 is an n-dimensional microbundle over a space B2, and f : B1 → B2 is a map. By a

microbundle map f : X1 → X2 covering f , we will mean a map f : V → E(X2)defined on a neighborhood V of the zero section i1(B1) in E(X1) which makes thefollowing diagram commute:

B1i1 //

f

V

f

j1 // B1

f

B2

i2// E(X2)

j2// B2

We will call a microbundle map f : X1 → X2 a microbundle isomorphism if the

map f : V → E(X2) restricts to an open embedding on each fiber.Given an n-dimensional manifold M without boundary, the tangent microbundle

τ(M) of M is defined to be

M∆ // M ×M π1 // M,

where ∆ : M →M ×M is the diagonal map m 7→ (m,m), and π1 : M ×M →M isthe projection map (m1,m2) 7→ m1 [Mil64, Lemma 2.1]. Given a continuous mapf : M → M ′ between n-dimensional manifolds, there is an induced microbundlemap df : τ(M) → τ(M ′), called the derivative map df of f , which is given byf × f : M ×M →M ′ ×M ′.

If M is an n-dimensional manifold with boundary, then we can define the tangentmicrobundle of M to be the microbundle τ(M+)|M over M , where

M+ = M ∪∂M (∂M × [0, 1))

is M with an external collar added [RS70, p. 389]. Explicitly, if M is a manifoldwith boundary, τ(M) is the microbundle over M with total space M ×M+.

The fiberwise tangent microbundle τfib(p). There is a natural generalizationof tangent microbundles to families of manifolds parametrized over a base space B[BS98, p. 599]. First, suppose that p : E → B is a topological fiber bundle whosefibers are n-dimensional manifolds without boundary. Then the fiberwise tangentmicrobundle of p, τfib(p), is the n-dimensional microbundle over E given by

E∆ // E ×B E

π1 // E,

6 STACY HOEHN

where E ×B E is the pullback of the following diagram

E

p

E

p// B.

Proposition 2.0.2. If p : E → B is a topological fiber bundle whose fibers aren-dimensional manifolds without boundary, then τfib(p) is an n-dimensional mi-crobundle with base E.

Proof. Let e ∈ E be given, and let b := p(e). Since p : E → B is a topologicalfiber bundle, there exists an open neighborhood Vb of b in B and a homeomorphismh : Vb×p−1(b)→ p−1(Vb) such that ph = π1, where π1 : Vb×p−1(b)→ Vb denotesprojection onto the first factor. Since p−1(b) is an n-dimensional manifold withoutboundary and e ∈ p−1(b), there exists an open neighborhood Ue of e in p−1(b) anda homeomorphism φe : Ue → Rn. Then h restricts to give an open embeddingh| : Vb × Ue → p−1(Vb). Let We ⊂ E denote the image h(Vb × Ue) of this openembedding.

Since We is an open neighborhood of e in E, We ×B We is an open subset ofE ×B E. Note that

We ×B We = (h(b1, u1), h(b2, u2)) ∈We ×We| p(h(b1, u1)) = p(h(b2, u2))= (h(b1, u1), h(b2, u2)) ∈We ×We| b1 = b2= (h(b1, u1), h(b1, u2))| b1 ∈ Vb, u1, u2 ∈ Ue

The map H : We ×B We →We × Rn given by

(h(b1, u1), h(b1, u2)) 7→ (h(b1, u1), φe(u2)− φe(u1))

is a homeomorphism which makes the following diagram commute.

We ×B We

H

π1

%%We

∆

99

×0 %%

We

We × Rnπ1

99

Since such a diagram exists for all e ∈ E, τfib(p) is an n-dimensional microbundlewith base E.

Remark 2.3. The restriction of τfib(p) to any fiber p−1(b) ⊂ E of p is just the usualtangent microbundle of p−1(b).

Now consider the case when p : E → B is a topological fiber bundle whose fibersare n-dimensional manifolds with boundary. Then p contains a subbundle

∂B(p) : ∂BE → B

whose fiber over a point b ∈ B is just ∂(p−1(b)). Let π1 : ∂BE × [0, 1) → ∂BEdenote projection onto the first factor. Then

∂B(p) π1 : ∂BE × [0, 1)→ B


is another fiber bundle, whose fiber over a point b ∈ B is ∂(p−1(b)) × [0, 1), anexternal collar for ∂(p−1(b)). If we glue the fiber bundles p and ∂B(p) π1 along∂B(p), we obtain a new fiber bundle,

p+ : E ∪∂BE (∂BE × [0, 1))→ B

whose fibers are manifolds without boundary obtained by adding a collar to eachof the fibers of p. We will denote the total space of p+ by (E+)B . Since thefibers of p+ are manifolds without boundary, the fiberwise tangent microbundleτfib(p+) of p+ has already been defined as a microbundle over (E+)B . The fiberwisetangent microbundle τfib(p) of p is then defined to be the restriction of τfib(p+) toE ⊂ (E+)B .

Remark 2.4. The restriction of τfib(p) to a fiber p−1(b) of p is just the usual tangentmicrobundle of the manifold with boundary p−1(b), which was defined above usingcollars. Also, if p is a fiber bundle whose fibers are manifolds without boundary,then (E+)B = E, and the two definitions of τfib(p) agree.

3. Framed Bundle Structures

In the previous section, we saw how a fiber bundle q : E′ → B with n-dimensionalmanifold fibers determines an n-dimensional microbundle τfib(q) over E′. We willuse this construction to define a simplicial map from Sn(p) (actually a thickened-upversion of it) to a suitable space Micron(E) of n-dimensional microbundles over E,which we will now define.

The space Micro(E) of microbundles over a space E. Fix an integer n ≥ 0.For a topological space E, a k-simplex of the simplicial set Micron(E) of n-dimensionalmicrobundle structures on E consists of a space Z embedded in

Rs × E ×∆k ⊂ R∞ × E ×∆k

for some s > 0 such that

i) 0 × E ×∆k ⊂ Zii)

E ×∆k i // Zj // E ×∆k

is an n-dimensional microbundle, where i : E ×∆k → Z is the map givenby the inclusion of 0 × E ×∆k into Z, and j : Z → E ×∆k is given byprojection.

Remark 3.1. Two vertices of Micron(E) are in the same path component if andonly if they are microbundle isomorphic.

There is a stabilization map Micron(E)→ Micron+1(E) which sends a k-simplexZ ⊂ Rs × E ×∆k to ε1 ⊕ Z ∼= R × Z ⊂ Rs+1 × E ×∆k ⊂ R∞ × E ×∆k, whereε1 = R×E×∆k denotes the trivial one-dimensional microbundle over E×∆k. LetMicro(E) denote the homotopy colimit

Micro(E) := hocolimn

Micron(E).

Given a k-simplex (E′, φ : E′ → E ×∆k) in Sn(p), as defined in Section 2, thefiberwise tangent microbundle of q : E′ → B×∆k is an n-dimensional microbundleover E′. Given any choice of homotopy inverse ψ for φ, we obtain an n-dimensional

8 STACY HOEHN

microbundle over E ×∆k by pulling back along ψ. We will now define a simplicialset Sn(p) which incorporates these choices of homotopy inverses. A k-simplex in

the simplicial set Sn(p) simply consists of a k-simplex

(E′, φ : E′ → E ×∆k)

in Sn(p), along with a map

ψ : E ×∆k → E′

which is a fiber homotopy inverse for φ. Since the space of all fiber homotopyinverses of φ is contractible, Sn(p) is homotopy equivalent to Sn(p). There is astabilization map

Sn(p)→ Sn+1(p),

which sends a k-simplex

(E′, φ : E′ → E ×∆k, ψ : E ×∆k → E′)

to the k-simplex

(I × E′, φ π : I × E′ → E ×∆k, i0 ψ : E ×∆k → I × E′),

where π : I × E′ → E′ is the projection map and i0 : E′ → I × E′ is the mape′ 7→ (0, e′). Let

S(p) := hocolimn

Sn(p)

denote the stabilized space.As described above, a k-simplex (E′, φ : E′ → E×∆k, ψ : E×∆k → E) in Sn(p)

determines an n-dimensional microbundle ψ∗(τfib(q)) over E ×∆k. Using the factthat E′ ⊂ Rs ×B ×∆k for some s > 0, we see that points in the total space of

ψ∗(τfib(q)) = (E ×∆k)×E′ (E′ ×B×∆k (E′+)B)

are either of the form ((e, t), (ψ(e, t), (x, p(e), t))) where e ∈ E, t ∈ ∆k, and x ∈ Rs,or of the form ((e, t), (ψ(e, t), ((x, p(e), t), t′))) where e ∈ E, t ∈ ∆k, x ∈ Rs, andt′ ∈ [0, 1), so there is a canonical embedding of the total space of ψ∗(τfib(q)) inRs+1×E×∆k so that the projection map of ψ∗(τfib(q)) is just projection onto thesecond two factors. Thus, ψ∗(τfib(q)) determines a k-simplex in Micron(E). Thisgives a map

τfib : Sn(p)→ Micron(E),

and after stabilizing, we obtain a map

τfib : S(p)→ Micro(E).

The space Sfr(p, γ) of framed bundle structures on a fibration. Now fix aninteger n ≥ 0 and a vertex γ in Micron(E). For any k, there is a projection map

E ×∆k → E

that can be used to pull γ back to an n-dimensional microbundle γk over E ×∆k

(The total space of γk is isomorphic to E(γ) × ∆k, where E(γ) denotes the totalspace of γ.) A k-simplex in the simplicial set Sfrn (p, γ) of γ-framed n-dimensional


compact bundle structures on p then consists of a k-simplex (E′, φ) in Sn(p), along

with a microbundle isomorphism φ : τfib(q)→ γk which covers φ.

τfib(q)

φ // γk

E′

φ //

q%%

E ×∆k

p×id∆kyy

B ×∆k

Note that there is a forgetful map

Sfrn (p, γ)→ Sn(p).

As in the unframed case, crossing with the interval I gives a stabilization map

Sfrn (p, γ)→ Sfrn+1(p, γ ⊕ ε1),

where ε1 is a trivial 1-dimensional microbundle over E. Then

Sfr(p, γ) := hocolimj

Sfrn+j(p, γ ⊕ ε

j),

where on the left-hand side, γ is viewed as a stable microbundle over E. There isa forgetful map

Sfr(p, γ)→ S(p).

We can also define a closely related simplicial set Sfr

n (p, γ) which incorporateschoices of homotopy inverses. A k-simplex in this simplicial set consists of a k-simplex

(E′, φ : E′ → E ×∆k, φ : τfib(q)→ γk)

in Sfrn (p, γ), along with a fiber homotopy inverse

ψ : E ×∆k → E′

for φ. The stabilized version of this simplicial set will be denoted by Sfr

(p, γ). Note

that Sfr

(p, γ) ' Sfr(p, γ).

Proposition 3.0.2. Fix an integer n ≥ 0, and let γ be an n-dimensional microbun-dle over E. Then the homotopy fiber of the map

τfib : Sn(p)→ Micron(E)

over γ is homotopy equivalent to Sfr

n (p, γ).

Proof. We will apply Proposition 8.2 in [HTW90] to the functors

Structn(p) : R∞ − Topop → CAT

and

Micron(E) : R∞ − Topop → CAT,

where R∞ − Top denotes the category of subspaces of Rs ⊂ R∞ for some s > 0,where the morphisms are the continuous functions. The functor Structn(p) takes asubspace X of Rs ⊂ R∞ to the category Structn(p×idX) of all n-dimensional fiberbundle structures on p× idX : E×X → B×X. An object in this category consists

10 STACY HOEHN

of a triple (E′, φ, ψ), where E′ is a space embedded in Rs ×B ×X ⊂ R∞ ×B ×Xfor some s > 0 such that the projection

q : E′ → B ×Xis a topological fiber bundle projection whose fibers are compact n-dimensionalmanifolds, where φ : E′ → E × X is a fiber homotopy equivalence, and whereψ : E × X → E′ is a fiber homotopy inverse for φ. A morphism h between twoobjects (E′1, φ1) and (E′2, φ2) is a microbundle isomorphism h : E′1 → E′2 such thatφ1 = φ2 h and ψ2 = h ψ1. Meanwhile, the functor Micron(E) takes a subspaceX of Rs ⊂ R∞ to the category Micron(E ×X). An object in Micron(E ×X) isa subset Z of Rs × E ×X ⊂ R∞ × E ×X containing 0× E ×X such that

E ×∆k i // Zj // E ×∆k

is an n-dimensional microbundle, where i : E ×∆k → Z is the map given by theinclusion of 0 × E ×∆k into Z, and j : Z → E ×∆k is given by projection. Amorphism between two objects Z1 and Z2 is a microbundle isomorphism h : Z1 →Z2.

There is a canonical functor ∆op → R∞ − Top which takes [k] to the standardk-simplex ∆k ⊂ Rk+1 ⊂ R∞. When the functors Structn(p) and Micron(E) arecomposed with this functor, we obtain functors ∆op → CAT (i.e. simplicial ob-jects in the category of small categories, or simplicial categories), which by abuseof notation will also be denoted by Structn(p) and Micron(E). From each ofthese simplicial categories, we can obtain a simplicial set whose k-simplices arejust the objects of the simplicial category in degree k. For the simplicial categoryStructn(p), the corresponding simplicial set is Sn(p), and for the simplicial cate-gory Micron(E), the corresponding simplicial set is Micron(E). Since the functorMicron(E) satisfies the Amalgamation, Straightening, and Fill-in properties (seeSection 7 in [HTW90]), Proposition 8.2 in [HTW90] implies that

Sfr

n (p, γ) // Sn(p) // Micron(E)

is a homotopy fibration sequence.

By stabilizing, we obtain the following corollary.

Corollary 3.3. Let γ denote a stable microbundle over E. Then the homotopy

fiber of S(p)→ Micro(E) over γ is homotopy equivalent to Sfr

(p, γ).

In particular, if we let γ be the stable trivial microbundle ε over E, then theprevious corollary implies that

Sfr

(p, ε)→ S(p)→ Micro(E)

is a homotopy fibration sequence. The next series of propositions and corollaries

will be used to show that S(p) actually is the product of Sfr

(p, ε) and Micro(E).

Proposition 3.0.4. Suppose that p : E → B is a fibration over a connected finiteCW complex B whose fibers are homotopy equivalent to a finite CW complex. Thenthe forgetful map

i : Sfr(p, ε)→ S(p)

has a left homotopy inverse

j : S(p)→ Sfr(p, ε),


i.e. the composition

j i : Sfr(p, ε)→ S(p)→ Sfr(p, ε)

is homotopic to the identity.

Before proving Proposition 3.0.4, we introduce yet another simplicial set in or-der to avoid the technical issues associated with the fiberwise tangent bundles of

bundles of manifolds with boundary. Let Sfrn (p, εn) denote the simplicial set wherea k-simplex consists of a k-simplex (E′, φ) in Sn(p), along with a microbundle iso-morphism

φ| : τfib(q|intBE′)→ εn

that coversφ|int

B×∆kE′ : intB×∆kE′ → E ×∆k,

where q : E′ → B × ∆k denotes the projection map. For brevity, we will let q|denote q|int

B×∆kE′ , and we will let φ| denote φ|int

B×∆kE′ . Thus, a k-simplex in

Sfrn (p, εn) is of the form

(E′, φ : E′ → E ×∆k, φ| : τfib(q|)→ εn).

There is also a stabilized version Sfr(p, ε). There are natural restriction maps

Sfrn (p, εn)→ Sfrn (p, εn), Sfr(p, ε)→ Sfr(p, ε).

The following lemma follows from the fact that each inclusion map intB×∆kE′ → E′

is a fiber homotopy equivalence.

Lemma 3.5. The natural restriction maps

Sfrn (p, εn)→ Sfrn (p, εn), Sfr(p, ε)→ Sfr(p, ε)

are homotopy equivalences.

Because of this result, to show that the forgetful map Sfr(p, ε)→ S(p) has a left

homotopy inverse, it suffices to show that the forgetful map Sfr(p, ε)→ S(p) has aleft homotopy inverse. We will prove a special case of this now.

Lemma 3.6. Suppose that p : E → B is a fibration over a compact connectedtopological manifold B whose fibers are homotopy equivalent to a finite CW complex.

Then the map Sfr(p, ε)→ S(p) has a left homotopy inverse S(p)→ Sfr(p, ε).

Proof. We will begin by defining the proposed left homotopy inverse on the levelof 0-simplices. Suppose that

(q : E′ → B,φ : E′ → E)

is a 0-simplex in Sn(p) for some n ≥ 0. Since E′ is the total space of a fiber bundlewhose base space and fibers are compact manifolds, E′ is also a compact manifold.Therefore, the microbundle τfib(q) over E′ has a stable inverse, i.e. there is amicrobundle η over E′ and a microbundle isomorphism t : η ⊕ τfib(q)→ ε, where εis a trivial microbundle over E′, by [Mil64, Thm. 4.1]. Such inverses η are uniqueup to stable microbundle isomorphism, and such trivializations t are unique up tobundle homotopy [RS70, p. 389].

Let D = D(η) denote the total space of a disk bundle of η; if η does not contain adisk bundle, then replace η by η⊕ε1, which will always contain a disk bundle [Bro66,Corollary (Mazur), p. 220]. There is a projection map π : D → E′, which is a fiber

12 STACY HOEHN

bundle projection and a homotopy equivalence. The composite q π : D → B is atopological fiber bundle with compact manifold fibers, and the composition φ πgives a fiber homotopy equivalence from D to E.

We will now show that (φ π)| : intBD → E is covered by a stable microbun-

dle isomorphism (φ π)| : τfib((q π)|) → ε, where ε denotes the stable trivial

microbundle over E. This would imply that (D,φ π, (φ π)|) is a 0-simplex in

Sfr(p, ε).Since π| : intBD → intBE

′ and q| : intBE′ → B are both topological fiber

bundles whose fibers are manifolds without boundary, Lemma 3.7 below impliesthat there exists a microbundle isomorphism

σ : τfib(q| π|)→ τfib(π|)⊕ (π|)∗(τfib(q|)).

Note that

τfib(π|) = intBD ×intBE′ intBD,

while

(π|)∗(η|intBE′) = intBD ×intBE′ η|intBE′ .

Since these two bundles agree on the open neighborhood intBD ×intBE′ intBDof the zero section in each bundle, τfib(π|) and (π|)

∗(η|intBE′) are equivalent asmicrobundles. Thus, the map σ from above really gives us a stable microbundleisomorphism

σ : τfib(q| π|)→ (π|)∗(η|)⊕ (π|)

∗(τfib(q|)).

Recall that η was chosen in such a way so that there was a stable microbundleisomorphism η ⊕ τfib(q) → ε over E′. By pulling back along the inclusion mapintBE

′ → E′, we see that there is a stable microbundle isomorphism

η| ⊕ τfib(q|)→ ε

over intBE′. Pulling back along π| : intBD → intBE

′ yields a stable microbundleisomorphism

(π|)∗(η|)⊕ (π|)

∗(τfib(q|)→ ε

over intBD. The composition of σ : τfib(q| π|) → (π|)∗(η|) ⊕ (π|)

∗(τfib(q|)) withthis map yields a stable microbundle isomorphism from τfib(q| π|) to the stabletrivial microbundle ε over intBD. The composition of this map with the naturalmap from the stable trivial microbundle over intBD to the stable trivial microbun-dle over E, which is induced by (φ π)|, gives us our desired stable microbundle

isomorphism (φ π)| : τfib((q π)|) → ε which covers (φ π)|. This allows us to

pick out a 0-simplex (D,φ π, (φ π)|) in Sfr(p, ε). Thus, we have defined a map

Sn(p) → Sfr(p, ε) on the level of 0-simplices for any fixed integer n ≥ 0. The

map Sn(p)→ Sfr(p, ε) is inductively defined for k-simplices in such a way that allchoices made for k-simplices agrees with choices previously made for lower degreesimplices. (Compare with the proof of [Nic82, Theorem 2.3.4].)

Next, we note that the maps Sn(p) → Sfr(p, ε), where n varies, induce a map

S(p) → Sfr(p, ε). This follows from the fact that the following diagram commutes


up to homotopy for any n > 0:

Sn(p) //

Sfr(p, ε)

Sn+1(p)

99

Finally, we note that the composition of the forgetful map Sfr(p, ε)→ S(p) with

the map S(p) → Sfr(p, ε) is homotopic to the identity. This follows from the factthat for any fixed n ≥ 0, the following diagram commutes up to homotopy:

Sfrn (p, εn) //

))

Sn(p) // S(p)

Sfr(p, ε)

This is because the inverse bundle η in the construction of the map S(p)→ Sfr(p, ε)is stably isomorphic to the trivial bundle when the fiber bundle structure is already

framed. Thus, the map S(p)→ Sfr(p, ε) that we just constructed is a left homotopy

inverse to the forgetful map Sfr(p, ε)→ S(p).

Lemma 3.7. Suppose that π : D → E′ and q : E′ → B are topological fiber bundleswhose fibers are finite-dimensional manifolds without boundary, where B is a com-pact manifold with boundary. Then there exists a stable microbundle isomorphism

σ : τfib(q π)→ τfib(π)⊕ π∗(τfib(q)).

Proof. First, note that

• τfib(π) = D ×E′ D,• τfib(q π) = D ×B D, and• π∗(τfib(q)) = D ×E′ (E′ ×B E′).

Using similar arguments as in the proof of Proposition 2.0.2, we see that τfib(π) isa sub-microbundle of τfib(q π), so the inclusion map gives us a microbundle mapτfib(π)→ τfib(q π). There is also a microbundle map

dπ : τfib(q π)→ π∗(τfib(q))

which is analogous to the derivative of a map between manifolds without boundary.It sends a point (d1, d2) in τfib(q π) to the point (d1, (π(d1), π(d2))) in π∗(τfib(q)).

Define the kernel of dπ to be the subset of τfib(q π) consisting of those pointswhich map into the image of the zero section map

D → π∗(τfib(q)) = D ×E′ (E′ ×B E′)under dπ. Then the kernel of dπ is τfib(π) = D ×E′ D. Thus, we have a sequence

τfib(π) // τfib(q π)dπ // π∗(τfib(q))

where the microbundle map on the right is surjective and has kernel equal to themicrobundle on the left.

Since τfib(π) is a sub-microbundle of τfib(q π), we claim that there existsanother microbundle ξ over D which is complementary to τfib(π) in the sense thatτfib(π)⊕ ξ is stably isomorphic to τfib(q π). To see this, note that D is homotopy

14 STACY HOEHN

equivalent to a finite-dimensional simplicial complex since B is a compact manifoldand since the fibers of q and π are manifolds. Thus, since τfib(π) is a microbundleover D, there exists an inverse microbundle ξ′ for τfib(π) and a stable microbundleisomorphism τfib(π)⊕ ξ′ → ε, where ε is the stable trivial microbundle over D. Letξ equal ξ′ ⊕ τfib(q π). Then

τfib(π)⊕ ξ = τfib(π)⊕ (ξ′ ⊕ τfib(q π))

∼= (τfib(π)⊕ ξ′)⊕ τfib(q π)

∼= ε⊕ τfib(q π),

so there is a stable microbundle isomorphism τfib(π)⊕ ξ → τfib(q π).

Let ξ denote the image of ξ ⊂ τfib(π)⊕ ξ under the stable microbundle isomor-phism τfib(π)⊕ ξ → τfib(q π). Then we claim that dπ : τfib(q π)→ π∗(τfib(q))restricts to give a microbundle isomorphism

dπ|ξ : ξ → π∗(τfib(q)).

To see this, it is enough to check it locally around each point d in D. Since fiberbundle projections are topological submersions and since the fibers of q and πare manifolds without boundary, it is enough to consider the special case whereq : Rk × U → U and π : Rn × Rk × U → Rk × U are the natural projection mapsand U is an open neighborhood of q(π(d)) in B. This situation is straightforwardto check.

Combining the above results, we have that there is a stable microbundle isomor-phism σ : τfib(q π)→ τfib(π)⊕ π∗(τfib(q)).

Now that we completed the proof of Lemma 3.6 and the lemma necessary for itsproof, we are in a position to prove Proposition 3.0.4 that the forgetful map

Sfr(p, ε)→ S(p)

has a left homotopy inverse

S(p)→ Sfr(p, ε)

whenever p : E → B is a fibration over a connected finite CW complex B whosefibers are homotopy equivalent to a finite CW complex F .

Proof. (Proof of Proposition 3.0.4:) Since B is finite CW complex, there exists acodimension 0, compact submanifold UN of RN for some N >> 0 such that thereis a homotopy equivalence r : U → B; such a manifold U is a trivial thickening of Bas in [Wal66]. The homotopy equivalence r can be used to pull back the fibrationp : E → B to a fibration r∗(p) : r∗(E) → U whose fibers are also homotopyequivalent to the finite CW complex F . Moreover, r can also be used to pullback compact topological fiber bundle structures on p to analogous structures onr∗(p) and similarly for framed fiber bundle structures. By Lemma 3.6, the forgetful

map Sfr(r∗(p), ε) → S(r∗(p)) has a left homotopy inverse. It then follows from

the following commutative diagram that the map Sfr(p, ε) → S(p) also has a lefthomotopy inverse.


Sfr(p, ε)

'

// S(p)

'

Sfr(r∗(p), ε) // S(r∗(p))

Finally, the commutative diagram

Sfr(p, ε)

'

// S(p)

scrSfr

(p, ε)

::

implies that the map Sfr(p, ε)→ S(p) also has a left homotopy inverse.

Corollary 3.8. Suppose that p : E → B is a fibration over a connected finite CWcomplex B whose fibers are homotopy equivalent to a finite CW complex. ThenS(p) ' Micro(E)× Sfr(p, ε).

Proof. Note that we have the following commutative diagram, where the verticalmaps are homotopy equivalences and the top row is a homotopy fibration sequence.

Sfr

(p, ε) //

'

S(p) //

'

Micro(E)

Sfr(p, ε) // S(p)

By Proposition 3.0.4, the map Sfr(p, ε)→ S(p) has a left homotopy inverse. Since

the vertical maps are homotopy equivalences, the map Sfr

(p, ε) → S(p) also has aleft homotopy inverse. It follows via a long exact sequence argument that

S(p) ' Micro(E)× Sfr

(p, ε).

This, in turn, implies that

S(p) ' Micro(E)× Sfr(p, ε).

Remark 3.9. For any stable microbundle γ, another disk bundle argument showsthat Sfr(p, γ) ' Sfr(p, ε), so Corollary 3.8 can be reformulated to say that S(p) 'Micro(E)×Sfr(p, γ) for any stable microbundle γ over E. This reformulation couldbe particularly favorable when p : E → B is actually a compact topological fiberbundle instead of just a fibration; in this situation, there is a preferred, possiblynon-trivial, stable microbundle γ over E given by the fiberwise tangent microbundleof p.

16 STACY HOEHN

4. Framed Bundle Structures as a Space of Lifts

We just saw that to understand the moduli space S(p) of compact topologicalfiber bundle structures on p, we must understand the moduli space Sfr(p, ε) offramed compact topological fiber bundle structures on p. In this section, we willshow that Sfr(p, ε) is homotopy equivalent to a space of lifts. In order to do this,we will begin by defining several simplicial categories.

Remark 4.1. By a simplicial category, we will typically mean a category enrichedover simplicial sets, in other words, a category with a set of objects and a simplicialset of morphisms between any two objects. There is a more general notion of asimplicial category defined as a simplicial object in the category of small categories,i.e. a functor ∆op → CAT , in which we would also have a simplicial set of objects.We will specify when we mean this more general notion of a simplicial category.

Throughout this section, F will be a finite CW complex.

The simplicial categories T(F ) and Gfr(F ). Fix an integer n ≥ 0.

• The simplicial category Tn(F ) has as objects compact n-dimensional man-ifolds M , where M is homotopy equivalent to F and M ⊂ Rs ⊂ R∞ forsome s > 0. A 0-simplex in the simplicial set of morphisms from M to M ′

in Tn(F ) is a homeomorphism h : M →M ′, while a k-simplex is a map

h : M ×∆k →M ′ ×∆k

over ∆k which restricts to give a homeomorphism on each fiber.• The simplicial category Gfrn (F ) has as objects pairs (X, γ), where X is a

compact ENR homotopy equivalent to F , X ⊂ Rs ⊂ R∞ for some s > 0,and γ is an n-dimensional microbundle over X whose total space E(γ) isembedded in Rk × X ⊂ R∞ × X for some k > 0 so that the projectionmap for γ is just given by the inclusion of E(γ) into Rk ×X, followed byprojection to X. A k-simplex in the simplicial set of morphisms from (X, γ)to (X ′, γ′) is a homotopy equivalence

f : X ×∆k → X ′ ×∆k

over ∆k which is covered by a microbundle isomorphism

f : γ ×∆k → γ′ ×∆k,

where for a microbundle γ over X, γ × ∆k denotes the microbundle overX ×∆k whose total space is E(γ)×∆k and whose projection map is givenby the identity on the ∆k factor.

There is a stabilization map

Tn(F )→ Tn+1(F )

which sends a compact n-dimensional manifoldM to the compact (n+1)-dimensionalmanifold M × I. There is also a stabilization map

Gfrn (F )→ Gfrn+1(F )

which sends an object (X, γ) to the object (X × I, (γ × I)⊕ ε1), where ε1 denotesthe trivial 1-dimensional microbundle over X × I. Call the resulting stabilizedcategories T(F ) and Gfr(F ), respectively. Gfr(F ), for example, is the simplicialcategory whose objects are triples (n,X, γ), where X is a compact ENR homotopy


equivalent to F and γ is an n-dimensional microbundle over X. The simplicial setof maps from (n,X, γ) to (n′, X ′, γ′) is empty if n > n′ and is the simplicial set

of homotopy equivalences X × In′−n → X ′ covered by microbundle isomorphisms(γ × In

′−n) ⊕ εn′−n → γ′ if n ≤ n′, where εn

′−n denotes the trivial (n′ − n)-

dimensional microbundle over X × In′−n.We will now define a simplicial functor

Tn(F )→ Gfrn (F ).

This functor takes an object M in Tn(F ) to the pair (M, τ(M)), where τ(M) =M ×M+ is the tangent microbundle of M . (Recall that M+ = M ∪ ∂M × [0, 1)denotes M with a half-open external collar added.) The functor takes a 0-simplexh : M ×∆k →M ′×∆k in the simplicial set of maps from M to M ′ in Tn(F ) to thek-simplex (h, h′) in Gfrn (F ), where h′ : τ(M) = M ×M+ → M ′ ×M ′+ = τ(M ′) isgiven by h×h+, where h+ is the map on M+ which is given by h on M and is givenby h(m, t) = (h(m), t) for (m, t) ∈ ∂M × [0, 1). The functor is defined similarly fork-simplices.

We will now relate the simplicial categories introduced above to certain simplicialsets of bundles or fibrations. Once again, fix n ≥ 0 and a finite CW complex F . Forany topological space B, let Bunn(B;F ) denote the sub-simplicial set of Bunn(B)consisting of those simplices E′ ⊂ Rs×B×∆k ⊂ R∞×B×∆k where the projectionmap q : E′ → B × ∆k is a fiber bundle whose fibers are compact n-dimensionalmanifolds homotopy equivalent to F .

Lemma 4.2. There is a zig-zag of homotopy equivalences

|Bunn(∗;F )| ' BTn(F ).

Proof. We will introduce a new simplicial category, BUNn(∗;F ) (here, we mean themore general notion of simplicial category which has simplicial sets of both objectsand morphisms). Its objects in degree k are topological fiber bundles E → ∆k

whose fibers are compact n-dimensional manifolds M homotopy equivalent to F ,where once again we assume E ⊂ Rs × ∆k for some s and the projection mapE → ∆k is just given by the inclusion of E into Rs×∆k and then projecting to ∆k.A morphism in degree k is a bundle isomorphism h : E → E′. There is a simplicialfunctor I : Tn(F )→ BUNn(∗;F ) which takes an object M in degree k to the trivialbundle M×∆k over ∆k and which takes a homeomorphism h : M×∆k →M ′×∆k

over ∆k to itself, viewed as a bundle isomorphism between the two correspondingtrivial bundles.

Note that every object E → ∆k in BUNn(∗;F ) is isomorphic to a trivial bundleM × ∆k → ∆k since ∆k is contractible. We will use the simplicial version ofQuillen’s Theorem A (with addendum) from [Wal82, Theorem A’, Addendum] toshow that the simplicial functor I : Tn(F ) → BUNn(∗;F ) induces a homotopyequivalence on classifying spaces. This version of Quillen’s Theorem A states that Iinduces a homotopy equivalence on classifying spaces if the simplicial over-categoriesI/M ′ are contractible for any object M ′ in degree 0 in BUNn(∗;F ). Recall that thesimplicial over-category I/M ′ is the simplicial category (the more general notion)whose objects in degree k are pairs (M,h) where M is an object in degree k in Tn(F )and h : I(M) → M ′ ×∆k is a morphism in BUNn(∗;F ). A morphism in I/M ′ ofdegree k from (M,h) to (N, i) is a morphism j : M ×∆k → N ×∆k in degree k ofTn(F ) such that iI(j) = h. Note that there is exactly one morphism j of degree k

18 STACY HOEHN

between any two objects (M,h) and (N, i) in I/M ′, namely j = i−1 h. It followsthat I/M ′ is contractible, and hence by the simplicial version of Quillen’s TheoremA, I : Tn(F )→ BUNn(∗;F ) induces a homotopy equivalence on classifying spaces.

Now it remains to show that the classifying space B(BUNn(∗;F )) of the sim-plicial category BUNn(∗;F ) is homotopy equivalent to the geometric realizationof the simplicial set Bunn(∗;F ). Note that the classifying space B(BUNn(∗;F ))of the simplicial category BUNn(∗;F ) is given by the geometric realization of thediagonal nerve ∆N(BUNn(∗;F )) of BUNn(∗;F ), i.e. the geometric realization ofthe simplicial set whose k-simplices are of the form E0 ← E1 ← ... ← Ek, whereeach of the Ei’s are objects in degree k in BUNn(∗;F ) and the maps are all maps indegree k in BUNn(∗;F ). Meanwhile, the simplicial set Bunn(∗;F ) is the simplicialset which in degree k just consists of the objects in degree k in BUNn(∗;F ). Thereis a natural inclusion map Bunn(∗;F )→ ∆N(BUNn(∗;F )) of simplicial sets, whichjust takes a k-simplex E → ∆k in Bunn(∗;F ) to the k-simplex E ← E ← ... ← Ein ∆N(BUNn(∗;F )), where all of the maps are identity maps. It follows from[HTW90, Theorem 7.11] that the map Bunn(∗;F ) → ∆N(BUNn(∗;F )) induces ahomotopy equivalence |Bunn(∗;F )| → B(BUNn(∗;F )) on classifying spaces sinceamalgamation, straightening, and fill-in properties are satisfied.

Hence, there is a zig-zag of homotopy equivalences

BTn(F )' // B(BUNn(∗;F )) |Bunn(∗;F )|'oo .

We can use this lemma to show that maps from a finite CW complex B toBTn(F ) classify certain types of fiber bundles over B. (Compare with, for example,[HTW90].) In particular, we will show that if B is the geometric realization of afinite simplicial set B, then there is a simplicial isomorphism

D : Bunn(B;F )→ SSet(B,Bunn(∗;F )),

where SSet(B,Bunn(∗;F )) denotes the simplicial set whose k-simplices are simpli-cial maps B×∆k → Bunn(∗;F ).

We will begin by defining the simplicial map

D : Bunn(B;F )→ SSet(B,Bunn(∗;F )).

Suppose q : E → B is a 0-simplex in Bunn(B;F ). We need to associate to q asimplicial map D(q) : B→ Bunn(∗;F ). Suppose α is an l-simplex in B representedby a simplicial map α : ∆l → B. Then the pullback |α|∗(E) is an l-simplex inBunn(∗;F ). Hence, we can define the simplicial map D(q) : B→ Bunn(∗;F ) to bethe map which sends an l-simplex α : ∆l → B to the l-simplex |α|∗(E). The mapD is defined similarly for k-simplices q : E → B ×∆k. This map D will be calledthe disassembly map.

There is also an assembly map

A : SSet(B,Bunn(∗;F ))→ Bunn(B;F ).

Under this map, a k-simplex B × ∆k → Bunn(∗;F ), which sends an l-simplexα : ∆l → B×∆k to the bundle Eα → ∆l, is mapped to the bundle

∪αEα → B ×∆k,

where the union is taken over all non-degenerate simplices α : ∆l → B×∆k of B×∆k. Since B is assumed to be finite, this is, in fact, a bundle, by an amalgamation


property, and its total space can be taken to be a subset of R∞ × B × ∆k, so itdetermines a k-simplex in Bunn(B;F ).

It is clear after working through the definitions of the disassembly and assemblymaps that the following lemma is true. Compare with [HTW90, Theorem 4.6].

Lemma 4.3. Fix an integer n ≥ 0 and a finite CW complex F . If B is thegeometric realization of a finite simplicial set B, then the disassembly map

D : Bunn(B;F )→ SSet(B,Bunn(∗;F ))

is a simplicial isomorphism, with inverse the assembly map

A : SSet(B,Bunn(∗;F ))→ Bunn(B;F ).

Corollary 4.4. If B is the geometric realization of a simplicial set B, there is ahomotopy equivalence

D′ : Bunn(B;F )→ S(Map(B, |Bunn(B;F )|)),

where S(Map(B, |Bunn(B;F )|)) denotes the singular simplicial set correspondingto the space of maps from B to |Bunn(B;F )|.

Proof. There is a homotopy equivalence

SSet(B,Bunn(∗;F ))→ S(Map(|B|, |Bunn(∗;F )|))

which sends a k-simplex φ : B×∆k → Bunn(∗;F ) to |φ| : |B|×∆k → |Bunn(∗;F )|.(Compare with the beginning of the proof of Proposition 1.4.2 in [Nic82].) Thecomposition of the simplicial isomorphism D from the previous lemma with thishomotopy equivalence gives the desired homotopy equivalence.

D′ : Bunn(B;F )D

'// SSet(B,Bunn(∗;F )) '

// S(Map(B, |Bunn(∗;F )|)).

Now for any topological space B, we will let

Fibfrn (B;F )

denote the simplicial set whose k-simplices are framed fibrations over B × ∆k.Explicitly, a k-simplex in Fibfrn (B;F ) consists of a pair (E′, η), where E′ is asubset of Rs ×B ×∆k ⊂ R∞ ×B ×∆k such that the projection q : E′ → B ×∆k

is a fibration whose fibers are homotopy equivalent to F and η is an n-dimensionalmicrobundle over E′. Just as in the bundle case, we have the following lemma.

Lemma 4.5. There is a zig-zag of homotopy equivalences

|Fibfrn (∗;F )| ' BGfrn (F ).

Similarly, we can also define disassembly and assembly maps for framed fibra-tions. Suppose once again that B is the geometric realization of a finite simplicialset B. Then the disassembly map

D : Fibfrn (B;F )→ SSet(B,Fibfrn (∗;F ))

takes a k-simplex which consists of a fibration q : E′ → B×∆k and an n-dimensionalmicrobundle η over E′ to the k-simplex

φq : B×∆k → Fibfrn (∗;F ),

20 STACY HOEHN

where φq maps an l-simplex α : ∆l → B × ∆k of B × ∆k to the l-simplex whichis given by the fibration |α|∗(E′)→ ∆l along with the n-dimensional microbundleη′ over |α|∗(E′) which is obtained by pulling η back along the projection map|α|∗(E′)→ E′. The assembly map

A : SSet(B,Fibfrn (∗;F ))→ Fibfrn (B;F )

takes a k-simplexB×∆k → Fibfrn (∗;F ),

which sends an l-simplex α : ∆l → B×∆k to the l-simplex given by the fibrationEα → ∆l along with the microbundle ηα over Eα, to the k-simplex which is givenby the fibration

∪αEα → B ×∆k,

along with the microbundle ∪αηα over ∪αEα, where both unions are taken over allnondegenerate simplices of B×∆k. (Note that the microbundles ηα agree on anyoverlap.) Just as in the bundle case, these assembly and disassembly maps can beused to prove the following lemma.

Lemma 4.6. Fix an integer n ≥ 0 and a finite CW complex F . If B is the geo-metric realization of a finite simplicial set B, then there is a homotopy equivalence

D′ : Fibfrn (B;F )→ S(Map(B, |Fibfrn (∗;F )|)).Proposition 4.0.7. Suppose that B is the geometric realization of a finite sim-plicial set. Further, suppose that p : E → B is a fibration whose fibers areall homotopy equivalent to a finite CW complex F and whose total space E isequipped with the n-dimensional trivial microbundle εn. Then p is classified by amap p : B → |Fibfrn (∗;F )| and

Sfrn (p, εn) ' Lift

|Bunn(∗;F )|

B

p//

::

|Fibfrn (∗;F )|

.

Remark 4.8. Here and in the following, whenever we refer to a lift or to a spaceof lifts, we will always implicitly assume that the vertical map has been convertedinto a fibration.

Proof. If we equip E with the trivial n-dimensional microbundle εn over E, thefibration p : E → B is a vertex in Fibfrn (B;F ). We will denote its image D′(p) in

S(Map(B, |Fibfr(∗;F )|)) by p.From Corollary 4.4 and Lemma 4.6, we have the following commutative diagram:

Bunn(B;F )D′

'//

I

S(Map(B, |Bunn(∗;F )|))

J

Fibfrn (B;F )D′

'// S(Map(B, |Fibfrn (∗;F )|)).

Since the horizontal maps are both homotopy equivalences, the homotopy fiber ofthe left-hand vertical map I over p is homotopy equivalent to the homotopy fiberof the right-hand vertical map J over p. We can convert the map

|Bunn(∗;F )| → |Fibfrn (∗;F )|


into a fibration. Then the homotopy fiber of the right-hand map J over p is homo-topy equivalent to

Lift

|Bunn(∗;F )|

B

p//

::

|Fibfrn (∗;F )|

.

Meanwhile, by an argument similar to Proposition 8.2 in [HTW90], the homotopyfiber of the left-hand vertical map over p (which is equipped with the trivial mi-crobundle εn) is homotopy equivalent to Sfrn (p, εn). Therefore,


|Bunn(∗;F )|

B

p//

::

|Fibfrn (∗;F )|

.

Corollary 4.9. Suppose that B is the geometric realization of a finite simplicialset. Further, suppose that p : E → B is a fibration whose fibers are all homotopyequivalent to a finite CW complex F and whose total space E is equipped with the n-dimensional trivial microbundle εn. Then p is classified by a map p : B → BGfrn (F )and


BTn(F )

B

p//

;;

BGfrn (F )

.

Similarly, we could equip E with the stable trivial microbundle ε. Then p isclassified by a map p : B → BGfr(F ), and by stabilizing our earlier observations,we see that

Sfr(p, ε) ' Lift

BT(F )

B

p//

;;

BGfr(F )

.

Remark 4.10. The above corollaries also hold true when B is only homotopy equiv-alent to a finite simplicial complex (for example, when B is homotopy equivalentto a finite CW complex [LW69, Proposition 6.1]).

5. Relationship with h-cobordisms

In the previous section, we saw that

Sfr(p, ε) ' Lift

BT(F )

B

p//

;;

BGfr(F )

,

22 STACY HOEHN

where p : E → B is a fibration over a finite CW complex with fibers homotopyequivalent to the finite CW complex F . In this section, we will take a step towardsunderstanding this space of lifts by showing that the homotopy fiber of the verticalmap

BT(F )→ BGfr(F )

over the point determined by an object (n, (Nn, τ(N))), where N is a compactn-dimensional manifold with boundary, is related to a stabilized space HCob(N)of h-cobordisms on N .

In order to understand the homotopy fiber of the map BT(F ) → BGfr(F ),we will factor the map as the composition of two other maps, one of which willbe a homotopy equivalence. It will then follow that the homotopy fiber of ouroriginal map is homotopy equivalent to the homotopy fiber of the other map in ourcomposition. To define the first map in the composition, we will introduce a newsimplicial category (see Remark 4.1).

The simplicial category E(F ) of invertible embeddings. Once again, fix aninteger n ≥ 0. The simplicial category En(F ) has the same objects as the simpli-cial category Tn(F ) from Section 4, namely compact n-dimensional manifolds Mhomotopy equivalent to F which are embedded as subsets of Rs ⊂ R∞ for somes > 0. A 0-simplex in the simplicial set of morphisms from M to N is an invertibleembedding e : M → N over ∆k. By an invertible embedding, we will mean a con-tinuous injective map e : M → N between compact manifolds which satisfies twoadditional properties:

i) There exists an extension e+ : M+ → N+ of e to an embedding from theopen manifold M+ (obtained by adding a half-open external collar to M)to the open manifold N+.

ii) There exists another embedding f : N → M such that f e and e f areisotopic to idM , idN , respectively. Further, we assume that there exists anextension f+ : N+ → M+ of f and that the isotopies f e ' idM ande f ' idN extend to give isotopies f+ e+ ' idM+ and e+ f+ ' idN+ .

A k-simplex in the simplicial set of maps between M and N is a map

e : M ×∆k → N ×∆k

over ∆k which restricts to give an invertible embedding over each fiber.There is once again a stabilization map En(F )→ En+1(F ) obtained by sending

M to M × I; we will let E(F ) denote the resulting stabilized category. Thereare obvious functors Tn(F ) → En(F ) and T(F ) → E(F ) obtained by viewing ahomeomorphism as an invertible embedding.

The space hcob(∂N) of h-cobordisms on ∂N . The homotopy fiber of the mapson classifying spaces induced by the functors Tn(F ) → En(F ) and T(F ) → E(F )are related to spaces of h-cobordisms, as we will now explain. Let N be a compactn-dimensional manifold homotopy equivalent to F . Fix an external collar ∂N×[0, 1]of ∂N and an embedding of N ∪∂N ∂N× [0, 1] in Rs for some s. Then H(∂N) is thesimplicial category of h-cobordisms on ∂N . The objects of H(∂N) are h-cobordismsZ on ∂N , where

Z ⊂ ∂N × [0, 1] ⊂ N ∪∂N ∂N × [0, 1] ⊂ Rs ⊂ R∞.


A k-simplex in the simplicial set of morphisms from Z to Z ′ is a homeomorphismh : Z ×∆k → Z ′ ×∆k over ∆k which restricts to the identity on ∂N ×∆k.

Note that Z ∪∂N N has a canonical embedding in Rs since it is just a subsetof N with its fixed external collar ∂N × [0, 1] added. Furthermore, Z ∪∂N N ishomotopy equivalent to N , and hence to F , since Z is an h-cobordism. Thus, thereis a functor

H(∂N)→ Tn(F )

which takes an object Z to Z ∪∂N N , and which takes a morphism

h : Z ×∆k → Z ′ ×∆k

over ∆k to the morphism

(Z ∪∂N N)×∆k → (Z ′ ∪∂N N)×∆k

which takes a point (z, t) ⊂ Z × ∆k to the point h(z, t), and which takes a point(x, t) ⊂ N ×∆k to itself via the identity. Note that this map is well-defined since hrestricts to the identity on ∂N ×∆k. The functor H(∂N)→ Tn(F ) induces a maphcob(∂N)→ BTn(F ), where the h-cobordism space hcob(∂N) is defined to be theclassifying space of the simplicial category H(∂N).

Proposition 5.0.1. For n ≥ 6, the homotopy fiber of BTn(F )→ BEn(F ) over thepoint determined by an object Nn is homotopy equivalent to the h-cobordism spacehcob(∂N).

Before giving the proof of Proposition 5.0.1, we will state and prove the fol-lowing lemma which indicates how h-cobordisms arise in the context of invertibleembeddings.

Lemma 5.2. Suppose that M and N are compact n-dimensional manifolds and thate : M → N is an invertible embedding such that e(M) ⊂ int N . Then N− int e(M)is an h-cobordism.

Proof. Suppose e : M → N is an invertible embedding such that e(M) ⊂ int N .Then condition i) of being an invertible embedding guarantees that

W := N − int e(M)

is a cobordism from e(∂M) to ∂N . We claim that W is actually an h-cobordism.Recall that the cobordism (W ; e(∂M), ∂N) is invertible if there exist cobordisms(Z1; ∂N,L1) and (Z2;L2, e(∂M)) such that W ∪∂N Z1 and Z2∪e(∂M)W are productcobordisms. It follows from [Sta65, Theorem 2] that invertible cobordisms are h-cobordisms (and the s-cobordism theorem of Barden, Mazur, and Stallings impliesthat h-cobordisms are invertible cobordisms if n ≥ 6). Thus, to show that W is anh-cobordism, it suffices to show that (W ; e(∂M), ∂N) is an invertible cobordism.

Condition ii) of e : M → N being an invertible embedding guarantees that thereexists another embedding f : N → M such that f e and e f are isotopic toidM and idN , respectively, and it also guarantees that there exists an extensionf+ : N+ → M+ of f . We will assume that f(N) ⊂ int M . (If not, f is isotopic toan embedding f ′ for which f ′(N) ⊂ int M , so replace f by f ′.)

Let W ′ = M − int f(N); then W ′ is a cobordism from f(∂N) to ∂M . LetZ2 = e(W ′). Then Z2 is a cobordism from e(f(∂N)) to e(∂M). We claim that

Z2 ∪e(∂M) W

24 STACY HOEHN

is a product cobordism. Let N+ = N ∪∂N (∂N × [0, 1]). The isotopy from idN toe f restricts to give an isotopy

ht : ∂N → N

which sends ∂N to e(f(∂N)). Extend this isotopy via the identity on ∂N+ to anisotopy

h′t : ∂N t ∂N+ → N+

which transforms ∂N t∂N+ to e(f(∂N))t∂N+. Note that this isotopy extends toan isotopy on a neighborhood of ∂Nt∂N+ in N+ by condition ii) in the definition ofinvertible embeddings. Thus, the Isotopy Extension Theorem in [EK71, Corollary1.2] implies that this isotopy extends to give an isotopy

Ht : N+ → N+.

This isotopy sends the closed collar between ∂N+ and ∂N to the cobordism be-tween ∂N+ and e(f(∂N)). Thus, the cobordism between ∂N+ and e(f(∂N)) ishomeomorphic to the closed collar, which is a product cobordism. Moreover, thecobordism between ∂N+ and e(f(∂N)) is homeomorphic to W∪∂NZ2, so W∪∂NZ2

is a product cobordism, as desired.Similar arguments show thatW has a left inverse Z1. Thus, W is an h-cobordism.

Proof. (Proof of Proposition 5.0.1) Let F : Tn(F ) → En(F ) denote the inclusionfunctor, and let F/N denote the simplicial over-category of F over the object N inEn(F ). (See the proof of Lemma 4.2 for the definition of a simplicial over-category.)We will begin by showing that B(F/N) ' hcob(∂N) by showing that these spacesrepresent equivalent functors F1,F2. Let CWfin denote the category whose objectsare finite CW complexes and whose morphisms are homotopy classes of maps.Define a contravariant functor F1 : CWfin → Set which sends a finite CW complexB to the set F1(B) of equivalence classes of pairs (E, e), where E is embedded inRs×B for some s > 0 such that the projection map q : E → B is a topological fiberbundle whose fibers are compact n-dimensional manifolds homotopy equivalent toF and e : E → N × B is a bundle map from q to the trivial bundle over B withfiber N which restricts to an invertible embedding on each fiber. Two such pairs(E, e) and (E′, e′) represent the same equivalence class in F1(B) if and only if thereis a bundle isomorphism h : E → E′ so that e′ h = e. Note that every equivalenceclass contains a pair (E, e) where e(E) ⊂ (int N)×B.

We will now define a functor F2 : CWfin → Set which takes a finite CW complexB to the set F2(B) of equivalence classes of spaces Z, which are embedding in Rs×Bfor some s > 0, such that the projection map π : Z → B is topological fiber bundlewhose fibers are h-cobordisms on ∂N and such that π : Z → B contains the trivialbundle ∂N × B → B as a subbundle. Two such bundles Z and Z ′ represent thesame equivalence class if there is a bundle isomorphism j : Z → Z ′ which restrictsto the identity on ∂N ×B.

Recall that if e : N ′ → N is an invertible embedding such that e(N ′) ⊂ int N ,then N − int e(N ′) is an h-cobordism between ∂N and e(∂N ′) by Lemma 5.2.Similarly, if (E, e) is a representative of an element in F1(B) such that e(E) ⊂(int N)×B, then

Ze = N ×B − intBe(E)


is the total space of an h-cobordism bundle over B. (The fact that Ze is actuallya fiber bundle follows from [BS98, Section 3, A2’].) We will use this informationto define a natural transformation η : F1 → F2. For a finite CW complex B, letηB : F1(B) → F2(B) denote the functor which takes an equivalence class [(E, e)],where (E, e) is a pair such that e(E) ⊂ (int N) × B, to the the equivalence class[Ze], where Ze is the h-cobordism bundle given by N ×B − intB(e(E)).

We can claim that ηB is a bijection for all B, making η is a natural equivalencebetween F1 and F2. To see this, we will define a map ζB : F2(B) → F1(B)for any finite CW complex B. (Compare with the proof of Proposition 5.1 in[Wal82].) Let [Z] be an element in F2(B). Using the invertibility of h-cobordisms,we see that there is an fiberwise embedding Z → ∂N × [0, 1] × B of Z into abundle of closed collar neighborhoods of ∂N which restricts to the identity on∂N ×B = ∂N ×0×B; it follows from the fact that the space of collars of ∂N iscontractible that the space of all such choices of embeddings Z → ∂N× [0, 1]×B iscontractible. Meanwhile, fix an internal collar bundle C of ∂N×B inN×B. In otherwords, fix a subbundle C of N ×B which contains ∂N ×B and which is a bundle ofclosed collar neighborhoods of ∂N in N ; such a bundle exists by [BS98, Section 3,A1’]. Once again, since the space of collars is contractible, it is a contractible choiceto send the previous collar bundle ∂N × [0, 1] × B to our fixed collar bundle C,fixing ∂N ×0×B = ∂N ×B. Thus, it is a contractible choice to find a fiberwiseembedding j : Z → C of Z into our fixed internal collar bundle which restricts to theidentity on ∂N×B. Define ζB by ζB([Z]) = [(j(Z), i : N×B−intB(j(Z)) → N×B],where i is the inclusion map. The fact that j is determined up to a contractiblechoice tells us that ζB is well-defined. It is then easy to see that ζB is an inverseto ηB , for any finite CW complex B. Thus, η is a natural equivalence between F1

and F2.We will now relate the functors F1 and F2 to the categories I/N and H(∂N),

respectively. As in Section 4, we see that for any finite CW complex B, there isa natural bijection between F1(B) and [B,B(F/N)], where [B,B(F/N)] denotesthe set of homotopy classes of maps f : B → B(F/N). Similarly, for any finiteCW complex B, there is a natural bijection between F2(B) and [B, hcob(∂N)].Combining these bijections, we see that there is a natural equivalence between thefunctors F1 and the functor [ , B(F/N)], which in turn is naturally equivalent tothe functor [ , hcob(∂N)]. Thus, B(F/N) and hcob(∂N) both represent the functorF1. By [Bro62, Theorem I], B(F/N) ' hcob(∂N).

So far, we have shown thatB(F/N) is homotopy equivalent to hcob(∂N). Now wewish to show that B(F/N) is homotopy equivalent to the homotopy fiber of the mapinduced by F. To do this, we will use the simplicial version of Quillen’s Theorem Bfrom [Wal82, Section 4, Addendum], which we will now recall. (Compare with theproof of Lemma 4.2.) Suppose C and D are simplicial categories (in the restrictedsense, see Remark 4.1), and suppose that F : C→ D is a simplicial functor. Supposethat for any 0-morphism e : D → D′ in D, the transition functor

e∗ : F/D → F/D′

between simplicial over-categories induces a homotopy equivalence on classifyingspaces. Then the simplicial version of Quillen’s Theorem B (with addendum) from[Wal82] implies that B(F/D)→ BC→ BD is a homotopy fibration sequence.

We will apply this version of Quillen’s Theorem B for the case when C = Tn(F ),D = En(F ), and F : Tn(F ) → En(F ) is the inclusion functor. Suppose that

26 STACY HOEHN

e : N → N ′ is a morphism in En(F ); without loss of generality, we can assume thate(N) ⊂ int N ′ since the homotopy class of the transition map

e∗ : B(F/N)→ B(F/N ′)

only depends on the connected component of e. Then as above, Ze = N ′− int e(N)is an h-cobordism between ∂N ′ and e(∂N). We will use the homotopy equivalencesB(F/N) ' hcob(∂N) and B(F/N ′) ' hcob(∂N ′) to identify e∗ with a map

e∗ : hcob(∂N)→ hcob(∂N ′).

e∗ is induced by the functor H(∂N) → H(∂N ′) which sends an h-cobordism Zbetween some manifold P and ∂N to the h-cobordism Z ∪∂N Ze between P and∂N ′. Since n ≥ 6, the h-cobordism Ze is invertible by the s-cobordism theorem, soe∗ is a homotopy equivalence. Thus, the simplicial version of Quillen’s Theorem Bimplies that

B(F/N) ' hcob(∂N)→ BTn(F )→ BEn(F )

is a homotopy fibration sequence, as desired.

The stabilized space HCob(N) of h-cobordisms on N . There are upper sta-bilization maps hcob(N × Ik) → hcob(N × Ik+1) described in [Wal82, Section 1].Define the stabilized space of h-cobordisms to be the homotopy colimit

HCob(N) := hocolimk

hcob(N × Ik).

Corollary 5.3. Let Nn (n ≥ 6) be a compact topological manifold homotopy equiv-alent to F . Then the homotopy fiber of BT(F )→ BE(F ) over the point determinedby N is homotopy equivalent to the stabilized h-cobordism space HCob(N).

Proof. By Proposition 5.0.1, the homotopy fiber of BT(F ) → BE(F ) is given bythe homotopy colimit of

hcob(∂N)→ hcob(∂(N × I))→ hcob(∂(N × I2))→ ...

where N is a compact manifold homotopy equivalent to F and the map

hcob(∂(N × Ik))→ hcob(∂(N × Ik+1))

is given as the composition of upper stabilization

hcob(∂(N × Ik))→ hcob(∂(N × Ik)× I)

with the inclusion-induced map

hcob(∂(N × Ik)× I)→ hcob(∂(N × Ik+1)).

Since upper stabilization commutes with the inclusion-induced maps, this homotopycolimit is homotopy equivalent to

hocolimk

hocolimj

hcob(∂(N × Ik)× Ij) = hocolimk

HCob(∂(N × Ik)).

Since the inclusion of ∂(N × Ik) → N × Ik becomes highly connected as k goes toinfinity and since HCob(−) is a homotopy-invariant functor,

hocolimk

HCob(∂(N × Ik)) ' hocolimk

HCob(N × Ik) ' HCob(N).


So far, we have shown that the homotopy fiber of the map BT(F ) → BE(F )is the stabilized space of h-cobordisms HCob(∂N). What we would like to showis that the homotopy fiber of the map BT(F ) → BGfr(F ) is also HCob(∂N). To

do this, we will introduce another new simplicial category Gfrn (F ) for any fixedn ≥ 0, which is a variation of the category Gfrn (F ), and a new stabilized category,

Gfr(F ), which is a variation of Gfr(F ). The category Gfrn (F ) has objects of theform (M,γ), where M is a compact manifold homotopy equivalent to F which isembedded in Rs ⊂ R∞ for some s > 0 and γ is an n-dimensional microbundle overM , where the total space of γ is embedded in Rk ×M ⊂ R∞ ×M for some k > 0and the projection map for γ is just given by inclusion into Rk ×M , followed byprojection to M . A k-simplex in the simplicial set of maps from (M,γ) to (M ′, γ′)

consists of a pair of maps (f, f), where f : M × ∆k → M ′ × ∆k is a homotopy

equivalence over ∆k and f : i∗(γ ×∆k)→ γ′ ×∆k is a microbundle isomorphism,where i : int M × ∆k → M × ∆k is the inclusion map. Note that a morphism

(f, f) in Gfrn (F ) is similar to a morphism in Gfrn (F ), except f is only defined on therestriction of γ to int M instead of on all of γ.

There is a stabilization functor Gfrn (F )→ Gfrn+1(F ) which takes (M,γ) to

(M × I, (γ × I) ⊕ ε1), where ε1 denotes a trivial 1-dimensional microbundle over

M × I. Let Gfr(F ) denote the resulting stablilized simplicial category. We claim

that BGfr(F ) ' BGfr(F ). This follows from the fact that there is a third simplicial

category C and simplicial functors C → Gfr(F ) and C → Gfr(F ) which inducehomotopy equivalences on classifying spaces. The objects of C are of the form(M,γ), where M is a compact manifold and γ is an n-dimensional microbundle overM for some n (with M and E(γ) subspaces of R∞ and R∞×M , respectively, as in

Gfrn (F ). A morphism in degree k in C is of the form (f, f ′), where f : M ×∆k →M ′ × ∆k is a homotopy equivalence over ∆k and f ′ : γ × ∆k → γ′ × ∆k is amicrobundle isomorphism covering f . The simplicial functor C→ Gfr(F ) is given bythe inclusion of a subcategory. Since every compact ENR X homotopy equivalent tothe finite CW complex F is homotopy equivalent to a compact manifold M of somedimension, it follows that this functor induces a homotopy equivalence on classifyingspaces. Meanwhile, the simplicial functor C → Gfr(F ) is obtained by restricting

the microbundle maps f ′ : γ ×∆k → γ′ ×∆k to maps f : i∗(γ ×∆k) → γ′ ×∆k,where i : int M × ∆k → M × ∆k is the inclusion map; this functor induces ahomotopy equivalence since the inclusion map i : int M × ∆k → M × ∆k is ahomotopy equivalence for any compact manifold M and any k. A homotopy inverse

BGfr(F )→ BC for this map is induced by pulling back along a homotopy inversej : M → int M for i.

We have the following sequence of spaces and continuous maps which commutesup to homotopy:

BT(F ) //

%%

BE(F ) // BGfr(F )

'

BC

'

II

BGfr(F ).

28 STACY HOEHN

Therefore, to show that the homotopy fiber of BT(F )→ BGfr(F ) is the stabilizedspace of h-cobordisms HCob(∂N), by Corollary 5.3 it suffices to show that the map

BE(F )→ BGfr(F ) is a homotopy equivalence.

Lemma 5.4. The map BE(F )→ BGfr(F ) is a homotopy equivalence.

Proof. The proof uses general position and topological immersion theory. First,we wish to show that the map induces a bijection on components. Let N , N ′ betwo objects of E(F ); without loss of generality, we can assume dim N = dim N ′

by replacing N by N × Ij , for example. Suppose N and N ′ map into the same

component of Gfr(F ). Then for sufficiently large k, there is a homotopy equivalencef : N × Ik → N ′ × Ik which is partially covered by a microbundle isomorphismf : i∗(τ(N×Ik))→ τ(N ′×Ik), where i : int (N×Ik)→ N×Ik is the inclusion map.For later purposes, we want to ensure that k is at least as large as the dimensionof N and N ′; if not, stabilize more so that it is.

Since τ(int N × (0, 1)k) ∼= i∗(τ(N × Ik)) and since there is a homeomorphismN+ → int N , it follows that there is a microbundle isomorphism

τ(N+ × (0, 1)k)→ i∗(τ(N × Ik)).

There is also a microbundle isomorphism

τ(N ′+ × Rk)→ τ(int (N ′ × Ik)).

Combining these microbundle isomorphisms with the microbundle isomorphismf : i∗(τ(N×Ik))→ τ(N ′×Ik) and the microbundle map τ(N ′×Ik)→ τ(N ′+×Rk)which is induced by inclusion, we obtain a microbundle map g′ : τ(V ) → τ(V ′)which covers a homotopy equivalence g : V → V ′, where V := N+ × (0, 1)k andV ′ := N ′+×(0, 1)k. This pair of maps (g, g′) determines a point in the space R(V, V ′)of representations from τ(V ) to τ(V ′) (see [Gau70, Section 1]). By [Gau70, Theorem2] (see also [Lee69, Immersion Theorem]), there exists an immersion j : V → V ′

which is homotopic to g; moreover, the pair (j, dj), where dj : τ(V )→ τ(V ′) is thederivative map, lies in the same homotopy class in R(V, V ′) as (g, g′).

Since k > dim N = dim N ′, the General Position Theorem from [KS77, p. 147]implies that the immersion j : V → V ′ is regularly homotopic to an immersionj′ : V → V ′ such that j′ restricts to an embedding on N ×0 ⊂ N+× (0, 1)k = V .In fact, j restricts to an embedding on a neighborhood of N × 0 in V as well.Since N × 0 has arbitrarily small neighborhoods homeomorphic to N × Ik, j

restricts to give an embedding e : N × Ik → V ′ = N ′+ × (0, 1)k → N′+ × Ik. By

combining e with a homeomorphism N′+ = N ′ ∪ ∂N ′ × [0, 1] → N ′, we obtain an

embedding e′ : N×Ik → N ′×Ik. By our construction, e′ extends to an embeddinge′+ : (N × Ik)+ → (N ′ × Ik)+.

To show that e′ is actually an invertible embedding (and thus determines a mapin E(F ) between N × Ik and N ′ × Ik), we must show that there exists anotherembedding

h′ : N ′ × Ik → N × Ik

which has an extension to an embedding

h′+ : (N ′ × Ik)+ → (N × Ik)+

such that h′ e′ and e′ h′ are isotopic to idN×Ik and idN ′×Ik , respectively (and sothat the isotopies also have extensions). The embedding h′ will be the embedding


obtained by performing the above process on a homotopy inverse r to our homotopyequivalence g; the isotopies will come from performing the above process on thehomotopies r g ∼ id and g r ∼ id. (We actually need a space-level generalposition argument.)

Since e′ : N × Ik → N ′ × Ik is an invertible embedding, it is a map in E(F )between N ×Ik and N ′×Ik. This shows that N and N ′ lie in the same componentof E(F ), so the map E(F ) → Gfr(F ) induces a bijection on components. Similararguments show that the map from any component of BE(F ) to its image compo-

nent in BGfr(F ) is a homotopy equivalence. (Compare with the proof of TheoremA in [CG77].)

Corollary 5.5. The homotopy fiber of BT(F ) → BGfr(F ) over the point deter-mined by an object (n, (Nn, τN )) (n ≥ 6) is homotopy equivalent to the stabilizedh-cobordism space HCob(N).

6. Constructing Sections with the Grothendieck Construction

In the last section, we saw that the homotopy fiber of the map between classifyingspaces induced by the simplicial functor T(F )→ Gfr(F ) from Section 4 is a spaceof h-cobordisms. There is another map, which for now we will denote by A% → A,between certain spaces related to algebraic K-theory which has the same homotopyfiber. To prove Theorem 1.2, we need to reformulate the space of lifts

Lift

BT(F )

B

p//

;;

BGfr(F )

from Corollary 4.9 in terms of algebraic K-theory. To do this, we will ultimatelyconstruct a homotopy pullback diagram of the form

BT(F )χ%

//

A%

BGfr(F )

χfr// A,

where A% → A is the map mentioned above. In this section, we will describe anabstract procedure for how to construct parametrized characteristics like χfr andχ% out of a space which is the classifying space of a category. The constructionof these parametrized characteristics is similar to a construction from Section 1 of[DWW03], although we take a slightly different perspective here to avoid having tomake contractible choices for as long as possible.

Suppose that we have a functor F : D → CAT, where D is a small categoryand CAT is the category of all small categories, such that the composition of F

with the classifying space functor B : CAT → Spaces sends all morphisms in D toweak homotopy equivalences. Let ∗ : D→ Spaces denote the functor which sends

30 STACY HOEHN

every object of D to the one-point space ∗. Then hocolimD

∗ ∼= BD, so the obvious

natural transformation BF → ∗ induces a map

hocolimD

BF → BD.

This map is not necessarily a fibration, but it is at least a quasi-fibration by [Dwy96,Prop. 3.12]. Let q : E(q) → BD denote the associated fibration. We want asystematic way of constructing sections of fibrations which are obtained in thisway. We will use the Grothendieck construction to do this.

The Grothendieck construction D∫F . The Grothendieck construction for a

functor F : D→ CAT is the category

D

∫F

whose objects are pairs (d, x), where d is an object in D and x is an object in F(d).A morphism from (d, x) to (d′, x′) is a pair (f, f ′), where f : d→ d′ is a morphismin D, and f ′ : F(f)(x)→ x′ is a morphism in F(d′).

Let ϕ : D∫F → D denote the forgetful functor which takes (d, x) to d. By a

section of ϕ, we mean a functor ψ : D → D∫F such that ϕ ψ = idD. We will

now give a way of picking out a preferred section of ϕ.

Definition 6.1. Suppose that we have a rule (!) for F : D→ CAT which picks out

(i) a preferred object d! in F(d) for each object d in D and(ii) a preferred morphism f ! in F(d′) from F(f)(d!) to (d′)! for every morphism

f : d→ d′ in D.

We will call such a rule well-behaved if (id : d→ d)! = id : d! → d! for each identitymap in D and (f ′ f)! = (f ′)! F(f ′)(f !) for all composable morphisms f : d→ d′

and f ′ : d′ → d′′ in D (i.e. if the rule (!) satisfies a one-cocycle condition).

Lemma 6.2. A rule (!) for a functor F : D → CAT which is well-behaved inthe sense of Definition 6.1 determines a preferred section ψ! : D → D

∫F of the

forgetful functor ϕ.

Proof. Define ψ! on objects by ψ!(d) = (d, d!) and on maps by ψ!(f : d → d′) =(f, f !). The compatibility conditions in Definition 6.1 guarantee that this assign-ment actually defines a functor.

Theorem 6.3 (Homotopy Colimit Theorem,[Tho79, Thm. 1.2]). There is a naturalhomotopy equivalence η : hocolim

DBF → B(D

∫F).

Thus, given a well-behaved rule (!) as in Definition 6.1, we obtain the followingzigzag of maps, where the map ψ!

∗ : BD→ B(D∫F) is induced by the section ψ!.

BDψ!∗ // B(D

∫F) hocolim

DBF

η

'oo

The map ϕ∗ : B(D∫F) → BD induced by the forgetful functor can be converted

into a fibration p : E(p) → BD. After this conversion, the map ψ!∗ corresponds

to a section of p, which we will also call ψ!∗. The fibrations p and q fit into the

following commutative diagram, where the map η∗ : E(q) → E(p) induced by


Thomason’s map η from Theorem 6.3 is a homotopy equivalence; it follows from[May99, Proposition 7.5] that η∗ is actually a fiber homotopy equivalence.

E(p)

p

E(q)'η∗oo

q

BD

Let MapBD(BD, E(p)) (resp. MapBD(BD, E(q))) denote the space of sections ofp (resp. q). Given a section of q, composing with the map η∗ gives a section of p.Thus, there is a natural map

MapBD(BD, E(q))→ MapBD(BD, E(p)).

Since η∗ is a fiber homotopy equivalence, Lemma A.1 implies that this map is ahomotopy equivalence. Thus, the section ψ!

∗ of p determines, up to a contractiblechoice, a section (ψ!

∗)′ of q such that the composition η∗ (ψ!

∗)′ is homotopic to ψ!

∗.In summary, a well-behaved rule (!) for a functor F : D → CAT strictly deter-

mines a section ψ!∗ of the fibration p associated to the map

B(D

∫F)→ BD,

which in turn determines, up to a contractible choice, a section (ψ!∗)′ of the fibration

q associated to the quasi-fibration hocolimD

BF → BD. To get around the fact that

(ψ!∗)′ is only defined up to a contractible choice, we will use the intermediate section

ψ!∗ when trying to prove that (ψ!

∗)′ has certain properties.

Finally, we include here a lemma which will be used later to understand thehomotopy fibers of certain maps induced by functors between the Grothendieckconstructions for two different functors.

Lemma 6.4. Suppose D is a small category and F%, F : D → CAT are twofunctors from D to the category of small categories. Suppose further that we havea natural transformation α : F% → F , so that we get an induced functor

A : D

∫F% → D

∫F

which takes an object (d, x) to the object (d, αd(x)). Then for any object (d0, x0) inD∫F , there is a functor

ψ : αd0/x0 → A/(d0, x0)

between over-categories which induces a homotopy equivalence on classifying spaces.

Proof. Explicitly, an object in αd0/x0 is a pair (x, f ′), where x is an object in

F%(d0) and f ′ : αd0(x)→ x0 is a map in F (d0). Meanwhile, an object in A/(d0, x0)

is of the form ((d, x), (f, f ′)), where d is an object in D, x is an object in F%(d),f : d → d0 is a map in D, and f ′ : F (f)(αd(x)) → x0 is a map in F (d0). A mapfrom ((d, x), (f, f ′)) to ((d′, x′), (g, g′)) is given by a map (h, h′) : (d, x) → (d′, x′)

32 STACY HOEHN

in D∫F% which makes the following diagram in D

∫F commute.

(d, αd(x))(h,αd′ (h

′)) //

(f,f ′) %%

(d′, αd′(x′))

(g,g′)xx(d0, x0)

The functor ψ : αd0/x0 → A/(d0, x0) takes an object (x, f ′) in αd0

/x0 to the object((d0, x), (id, f ′)) in A/(d0, x0). There is also a functor

φ : A/(d0, x0)→ αd0/x0

which takes an object ((d, x), (f, f ′)) to the object (F%(f)(x), f ′). The compositionφψ is the identity functor on αd0/x0. Meanwhile, there is a natural transformationfrom the identity functor on A/(d0, x0) to the composition ψ φ. Namely, thisnatural transformation is given on an object ((d, x), (f, f ′)) by the map (f, id) inA/(d0, x0) from ((d, x), (f, f ′)) to (ψ(φ((d, x), (f, f ′)))). Thus, when we pass toclassifying spaces, ψ∗ is a homotopy equivalence with homotopy inverse φ∗.

7. Retractive Space Models

We will apply the material from the previous section to obtain sections of certainfibrations whose fibers are homotopy equivalent to spaces related to algebraic K-theory. In this section, we will recall some retractive space models for these fibers.Recall from [Wal85] that a Waldhausen category is a category C with a zero ob-ject, together with a choice of two subcategories, coC (cofibrations) and wC (weakequivalences), subject to certain axioms. To any Waldhausen category C, we canassociate an infinite loop space K(C). An exact functor F : C→ D between Wald-hausen categories (that is, a functor that preserves all of the relevant structures)induces a map K(C)→ K(D).

Of particular note is the fact that given any Waldhausen category, there is map

|wC| → K(C).

We will informally refer to a map of this type as a group completion-like map. Wewill repeatedly make use of these group completion-like maps to give us a nice wayto pick out points and paths between points in K(C). In particular, using this map,any object c in wC determines a preferred point 〈c〉 in K(C), and any map f : c→c′in wC determines a preferred path from 〈c〉 to 〈c′〉 in K(C).

Our primary examples of Waldhausen categories will come from retractive spacesover a topological space X. A retractive space over X is a space Y equipped withmaps

Y

r

X

s

OO

such that r s = idX and s is a cofibration in the usual sense. If Y and Y ′ aretwo retractive spaces over X, a retractive map Y → Y ′ is a map f : Y → Y ′ of


spaces over and relative to X. In other words, f must make both the inner andouter triangles in the following diagram commute.

Y

r

f // Y ′

r′~~X

s′>>

s

``

The retractive spaces over X form a category, R(X), with zero object the re-tractive space X over X whose retraction and section maps are both given by theidentity. A morphism in R(X) is a cofibration if the underlying map of spaces isa cofibration in the usual sense, and a morphism in R(X) is a weak equivalence ifthe underlying map of spaces is a homotopy equivalence. (Note that the homotopyinverse of a map f : Y → Y ′ between retractive spaces is not required to be aretractive map.) With these notions of cofibrations and weak equivalences, R(X)is a Waldhausen category.

There is a subcategory Rfd(X) of homotopy finitely dominated retractive spacesover X. Using the same notions of cofibrations and weak equivalences as above, itis a Waldhausen category. The algebraic K-theory of a space X, A(X), is definedto be the infinite loop space obtained by taking the K-theory of this Waldhausencategory:

A(X) := K(Rfd(X)).

We can use the group completion-like map |wRfd(X)| → A(X) to pick out pointsand paths between points in A(X). For example, if X itself is finitely dominated,then XtX is a finitely dominated retractive space over X which determines a pointin |wRfd(X)| and hence a point in A(X). This point will be denoted by χ(X) andwill be called the A-theory Euler characteristic for X.

Remark 7.1. Waldhausen defines A(X) in [Wal82] using the category Rhf (X) ofhomotopy finite retractive spaces over X instead of Rfd(X). The two resultingmodels for A(X) agree except possibly on the level of path components. The pathcomponents of A(X) using the model above are in one-to-one correspondence withthe elements of the group K0(Zπ1(X)), whereas the path components of A(X)using Waldhausen’s model are in one-to-one correspondence with the elements of

Z ⊂ Z⊕ K0(Zπ1X) ∼= K0(Zπ1X). It turns out that this difference will not matterfor us.

Given a map f : X → X ′ and a retractive space Y over X, we can produce aretractive space f∗(Y ) over X ′ by taking the pushout of the following diagram.

Y // f∗(Y )

X

s

OO

f// X ′

OO

Such a map f induces an exact functor f∗ : Rfd(X) → Rfd(X ′) and hence aninduced map f∗ : A(X) → A(X ′). (Refer to Section 6 in [DWW03] to see whatexactly we mean by taking the pushout to ensure that this assignment actuallygives a functor.)

34 STACY HOEHN

The functor X 7→ A(X) is homotopy invariant by [Wal85, Prop. 2.17], but it isnot excisive. Theorem 1.1 in [WW95], however, implies that there does exists anexcisive functor X 7→ A%(X) and a natural transformation

α : A% → A

such that the map α : A%(∗) → A(∗) is a homotopy equivalence. For any spaceX, the map α : A%(X) → A(X) will be called the assembly map; it is an infiniteloop space map. We want to create a way to pick out points in A%(X) that usesretractive spaces, at least in the case when X is a compact ENR.

In order to do this, we will have to introduce control on our retractive spaces.We will follow [DWW03, Sect. 7], but we will restrict our attention to the controlspace

J(X) :=(X × [0,∞], X × [0,∞)

),

where X is a compact ENR. (See also [ACFP94, CP95, CPV98].) Given two re-tractive spaces Y1, Y2 over X × [0,∞), a retractive map f : Y1 → Y2 is, as before,a map both over X × [0,∞) and relative to X × [0,∞). Meanwhile, a controlledmap f : Y1 → Y2 is still a map relative to X × [0,∞), but it is not necessarily amap over X × [0,∞). However, f ’s behavior with respect to the retractions muststill satisfy a certain control condition.

Definition 7.2. A map f : Y1 → Y2 between retractive spaces satisfies the controlcondition if for every (x,∞) ∈ X × ∞ and every neighborhood L of (x,∞) inX× [0,∞], there exists a smaller neighborhood L′ ⊂ L of (x,∞) in X× [0,∞] suchthat r2(f(y)) ∈ L whenever y ∈ Y1 and r1(y) ∈ L′.

Let R(J(X)),C(J(X)) denote the categories of retractive spaces over X × [0,∞)whose maps are retractive maps and controlled maps, respectively. Since everyretractive map f : Y1 → Y2 is a controlled map, R(J(X)) ⊂ C(J(X)). We aremainly interested in R(J(X)), but we will use C(J(X)) as an auxiliary categoryof retractive spaces in order to define what we mean by cofibrations and weakequivalences in R(J(X)).

We need to define a notion of homotopy in C(J(X)). If Y is a retractive spaceover X × [0,∞), then Y × [0, 1] is naturally a retractive space over

X × [0,∞)× [0, 1].

We will let π : X× [0,∞)× [0, 1]→ X× [0,∞) denote the projection map; pushingout along π takes the retractive space Y ×[0, 1] over X×[0,∞)×[0, 1] to a retractivespace π∗(Y × [0, 1]) over X × [0,∞). Two morphisms f, g : Y1 → Y2 in C(J(X))(i.e. controlled maps) are controlled homotopic if there exists a controlled map

λ : π∗(Y1 × [0, 1])→ Y2

such that λ i0 = f and λ i1 = g, where i0, i1 : Y1 → π∗(Y1 × [0, 1]) are themaps induced by the maps j0, j1 : Y1 → Y1× [0, 1] defined by y 7→ (y, 0), y 7→ (y, 1),respectively. A morphism f : Y1 → Y2 in C(J(X)) is a weak equivalence if thereexists a controlled map g : Y2 → Y1 such that gf and f g are controlled homotopic(in the above sense) to idY1

and idY2, respectively. A morphism in C(J(X)) is a

cofibration if it is a closed embedding and has the homotopy extension property forhomotopies in C(J(X)).

Using the inclusion R(J(X)) ⊂ C(J(X)), we say that a morphism f : Y1 → Y2

in R(J(X)) is a weak equivalence (resp. cofibration) if it is a weak equivalence


(resp. cofibration) in C(J(X)). Note that a retractive map f : Y1 → Y2 is a weakequivalence if there is a controlled map g : Y2 → Y1 (not necessarily a retractivemap) such that gf and fg are controlled homotopic to idY1 and idY2 , respectively.

Just as there is the notion of homotopy finitely dominated retractive spaces inthe uncontrolled setting, there is similarly a notion of homotopy locally finitelydominated retractive spaces in the controlled setting. (See [DWW03, Sect. 7] formore details.) Let Rld(J(X)) denote the full subcategory of R(J(X)) consistingof those retractive spaces which are homotopy locally finitely dominated. It isa Waldhausen category whose notions of weak equivalences and cofibrations areinherited from those of R(J(X)).

We will also consider one more Waldhausen category. This Waldhausen category,which we will denote by V (X), is going to be a full subcategory of Rld(J(X)). Itsobjects are those retractive spaces Y over X × [0,∞) for which Y is an ENR andthe retraction map r is proper. The morphisms in V (X) are retractive maps.

For any real number i ≥ 0, there is a shift map

φi : X × [0,∞)→ X × [0,∞)

which sends (x, t) to (x, t + i) for t ≥ 0. Pushing out along φi induces a functorφi∗ : V (X) → V (X). A morphism f : Y1 → Y2 in V (X) is a micro-equivalence if,for every sequence of positive integers k0, k1, k2, ..., the induced morphism⊔

kiφi∗(Y1)→

⊔kiφ

i∗(Y2)

is a weak equivalence in Rld(J(X)), as defined above. Here, kiφi∗(Yj) is short for the

disjoint union of ki copies of φi∗(Yj), for j = 1, 2. The primary example of micro-equivalences that we will care about will be homeomorphisms between retractivespaces in V (Y ). Cell-like maps between ENRS are also micro-equivalences. (Amap f : Y1 → Y2 between ENRS is cell-like if g−1(y) is homeomorphic to a cellularsubset of some Euclidean space for all y ∈ Y2.) A morphism in V (X) is a weakequivalence if it is a micro-equivalence, and it is a cofibration if it is a cofibrationin Rld(J(X)). With these notions of cofibrations and weak equivalences, V (X) is aWaldhausen category.

There is a functor i : Rfd(X)→ Rld(JX) which takes a retractive space Y over Xto the retractive space over X× [0,∞) given by taking the pushout of the followingdiagram.

Y

X = X × 0

s

OO

// X × [0,∞)

This functor is exact, so it induces a map i : A(X)→ A(J(X)) , where

A(J(X)) := K(Rld(J(X))).

The natural inclusion functor α′ : V (X)→ Rld(J(X)) is also exact, so it induces amap α′ : K(V (X))→ A(J(X)).

36 STACY HOEHN

Given a compact ENR X, a model for A%(X) is given around Lemma 8.7 in[DWW03] by taking the homotopy pullback of the following diagram.

A%(X)α //

A(X)

i

K(V (X))

α′// A(J(X))

The natural map α : A%(X)→ A(X) is a model for the assembly map.Let R%(X) be defined to be the pullback of the following diagram.

Rfd(X)

i

V (X)

α′// Rld(J(X)).

An object in this category consists of an object Y in Rfd(X) and an object Y ′

in V (X) which map to the same object in Rld(J(X)). This category inheritsa natural Waldhausen category structure from Rfd(X) and V (X), so it makessense to talk about its algebraic K-theory, K(R%(X)). There is a natural mapK(R%(X))→ A%(X). (Remark: We are not claiming that this map is necessarilya homotopy equivalence.) If we combine this map with the group completion-likemap |wR%(X)| → K(R%(X)), we obtain a natural map |wR%(X)| → A%(X).Thus any object in R%(X) determines a point in A%(X), and any weak equiva-lence in R%(X) determines a path between points in A%(X). In particular, for acompact ENR X, the pair (XtX,XtX× [0,∞)) is an object in wR%(X); for easeof notation, we will often just denote this object by XtX because the second objectin the pair is determined by the first. This object determines a point in A%(X),which we will denote by χ%(X); this is the excisive A-theory Euler characteristicof X.

8. Parametrized Euler Characteristics

We will now use the material from the previous two sections to construct sec-tions of certain universal fibrations whose fibers are homotopy equivalent to A(X) orA%(X). These are the parametrizedA-theory Euler characteristic and parametrizedexcisive A-theory Euler characteristic from [DWW03] (or at least closely relatedvariations of them). They will be used in the next section to relate the space offramed fiber bundle structures on a fibration p to a space of lifts involving algebraicK-theory.

Let G(F ) denote the simplicial category obtained from Gfrn (F ) by forgetting themicrobundle information. Namely, the objects in G(F ) are compact ENRs X ' F ,and the k-morphisms are homotopy equivalences f : X × ∆k → X × ∆k over∆k. (Note that BG(F ) ' BG(F ), where G(F ) denotes the simplicial monoid ofself-homotopy equivalences of F .)


Discrete Euler Characteristics. By the discrete category Cδ underlying a sim-plicial category C, we will mean the (small) category obtained by ignoring the sim-plicial structure on the morphisms. Let Gδ(F ) be the discrete category underlyingG(F ), and let

wRfd : Gδ(F )→ CAT

be the functor which sends an objectX to the Waldhausen category wRfd(X) whoseobjects are finitely dominated retractive spaces over X and whose morphisms arehomotopy equivalences. We will define a rule (!) for this functor as in Definition 6.1.For any object X in Gδ(F ), we will let X ! be the retractive space X tX over X,and for any map f : X → X ′ in Gδ(F ), we will let

f ! : wRfd(f)(X tX) = X tX ′ → X ′ tX ′

be the map f t id in wRfd(X ′). It is easily verified that this rule is well-behavedin the sense of Definition 6.1, so it determines a section

χδ : Gδ(F )→ Gδ(F )

∫wRfd

of the forgetful functor for the Grothendieck construction. The induced map onclassifying spaces fits into the following zigzag, where the last map is induced bythe group completion-like maps |wRfd(X)| → A(X).

BGδ(F )χδ−→ B(Gδ(F )

∫wRfd)

'←− hocolimX∈Gδ(F )

|wRfd(X)| −→ hocolimX∈Gδ(F )

A(X).

Similarly, let Tδn(F ) denote the discrete category underlying the simplicial cate-gory Tn(F ). Let

wR% : Tδn(F )→ CAT

be the functor which takes an object M to the Waldhausen category wR%(M).(Compare with Section 7 in [DWW03] to see why a homeomorphism h : M →M ′ induces a functor R%(M) → R%(M ′) which takes weak equivalences to weakequivalences.) We will define a rule (!) as in Definition 6.1 for this functor bytaking excisive Euler characteristics. In particular, for an object M in Tδn(F ), wewill let M ! be the object in wR%(M) determined by the retractive space M tMover M , and for a map h : M → M ′ in Tδn(F ), we will let h! be the map inwR%(M) determined by the map h t id : M tM ′ →M ′ tM ′. (The fact that h isa homeomorphism and not just a homotopy equivalence ensures that h! is actuallya weak equivalence in wR%(M ′).) This rule (!) determines a section

Tδn(F )→ Tδn(F )

∫wR%

of the forgetful functor for the Grothendieck construction.There is also a functor wR% : Gδ(F ) → CAT which takes an object X to the

Waldhausen category wR%(X). (Once again, compare with Section 7 in [DWW03]to see why a map f : X → X ′ between compact ENRs induces a functor

R%(X)→ R%(X ′)

which actually takes weak equivalences to weak equivalences.) There is a functor(Tδn(F )

∫wR%

)→(Gδ(F )

∫wR%

)

38 STACY HOEHN

induced by the forgetful functor Tδn(F ) → Gδ(F ). Let χ%,δn denote the map on

classifying spaces which is induced by the following composition:

Tδn(F )→(Tδn(F )

∫wR%

)→(Gδ(F )

∫wR%

).

This map fits into the following zigzag.

BTδn(F )χ%,δn−→ B(Gδ(F )

∫wR%)

'←− hocolimX∈Gδ(F )

|wR%(X)| −→ hocolimX∈Gδ(F )

A%(X).

The two zigzags of maps that we have constructed thus far fit into the followingcommutative diagram, where the vertical maps are induced by the functor Tδn(F )→Gδ(F ) and the exact functor α : wR%(X)→ wRfd(X).

BTδn(F )χ%,δn //

B(Gδ(F )∫wR%)

hocolimX∈Gδ(F )

|wR%(X)| //

'oo hocolimX∈Gδ(F )

A%(X)

BGδ(F )

χδ// B(Gδ(F )

∫wRfd) hocolim

X∈Gδ(F )|wRfd(X)|'

oo // hocolimX∈Gδ(F )

A(X)

Note that the left-hand vertical map factors through BGfr,δn (F ); we will let

χfr,δn : BGfr,δn (F )→ B(Gδ(F )

∫wRfd)

denote the composition of the forgetful functor-induced map from BGfr,δn (F ) toBGδ(F ) with χδ.

Let qδ : E(qδ)→ BGδ(F ) denote the fibration associated to the quasi-fibration

hocolimX∈Gδ(F )

A(X)→ BGδ(F ),

and similarly, let q%,δ : E(q%,δ) → BGδ(F ) denote the fibration associated to thequasi-fibration

hocolimX∈Gδ(F )

A%(X)→ BGδ(F ).

After converting the relevant maps in the above zigzag diagram into fibrations,Proposition A.0.4 implies that the section χδ and the map χ%,δ

n over BGδ(F ) de-termine (up to contractible choice) a section

χδ : BGδ(F )→ E(qδ)

of qδ and a map

χ%,δn : BTδn(F )→ E(q%,δ)

over BGδ(F ) which make the following diagram commute up to a preferred homo-topy.

BTδn(F )

χ%,δn // E(q%,δ)

BGδ(F )

χδ// E(qδ)


We will let χfr,δn : BGfr,δn (F ) → E(qδ) denote the composition of the forgetfulfunctor-induced map from BGfr,δn (F ) to BGδ(F ) with χδ. Then the diagram

BTδn(F )

χ%,δn // E(q%,δ)

BGfr,δn (F )

χfr,δn

// E(qδ)

also commutes up to a preferred homotopy.

Non-Discrete Euler Characteristics. So far we have only considered the dis-crete categories Gδ(F ) and Tδn(F ), but we can also consider the corresponding sim-plicial categories. Recall that a simplicial category C can be viewed as a functorC : ∆op → CAT . Thus, to any simplicial category C, we can associate an ordinarycategory

Simp(C) := ∆op

∫C

using the Grothendieck construction. There is a natural chain of homotopy equiv-alences between the classifying space of Simp(C) and the classifying space of thesimplicial category C. (See, for example, [Wal82, Sect. 4].)

There is a functor

A : Simp(G(F ))→ Spaces

(resp. A% : Simp(G(F )) → Spaces) which takes an object ([k], X) in Simp(G(F ))to the infinite loop space A(X) (resp. A%(X)). Let q : E(q)→ BG(F ) denote thefibration associated to the quasi-fibration

hocolim([k],X)∈Simp(G(F ))

A(X)→ B(Simp(G(F ))),

where we identify BG(F ) with B(Simp(G(F ))) via the above chain of homotopyequivalences. Similarly, let q% : E(q%)→ BG(F ) denote the fibration associated tothe quasi-fibration

hocolim([k],X)∈Simp(G(F ))

A%(X)→ BG(F ),

where we once again identify BG(F ) with B(Simp(G(F ))).The inclusion-induced map BGδ(F ) → BG(F ) is a homotopy equivalence by

[DK80] (see also [Fie84]). Meanwhile, the inclusion-induced mapBTδn(F )→ BTn(F )is not necessarily a homotopy equivalence, but it is an acyclic map (i.e. its homotopyfiber over any point in BTn(F ) is acyclic) by [McD80]. Since both inclusion-inducedmaps are at least acyclic maps and since the fibers of E(q) and E(q%) are infiniteloop spaces and hence componentwise nilpotent, Proposition A.0.8 implies that themaps χδ and χ%,δ

n determine (up to contractible choice) a section

X : BG(F )→ E(q)

of q and a map

X%n : BTn(F )→ E(q%)

40 STACY HOEHN

over BG(F ) which make the following diagram commute up to a preferred homo-topy.

BTn(F )X%n //

E(q%)

BG(F )

X// E(q)

Once again, the left-hand vertical map factors through BGfrn (F ). We will let

Xfrn : BGfrn (F )→ E(q)

denote the composition of the forgetful functor-induced map BGfrn (F ) → BG(F )with X. Then we have a diagram

BTn(F )X%n //

E(q%)

BGfrn (F )

Xfrn

// E(q)

which also commutes up to a preferred homotopy.

Stabilized Euler Characteristics. The above constructions can also be donein the stabilized setting. There is a simplicial category G∞(F ) whose objects arepairs (n,X), where n is a non-negative integer and X is a compact ENR homotopyequivalent to F × In ' F . The simplicial set of maps from (n,X) to (n′, X ′) isempty if n > n′. If n ≤ n′, a 0-simplex in the simplicial set of maps from (n,X) to

(n′, X ′) is a homotopy equivalence from X × In′−n to X ′, with k-simplices definedsimilarly. We will let Gδ∞(F ) denote the discrete category underlying this simplicialcategory. There is a functor

wRfd : Gδ∞(F )→ CAT,

which takes an object (n,X) to the Waldhausen category wRfd(X) of finitely dom-inated retractive spaces over X. It takes a morphism f : (n,X) → (n′, X ′) inGδ∞(F ) to the exact functor which is given by the composition

wRfd(X)×In

′−n

−→ wRfd(X × In′−n)

f∗−→ wRfd(X ′).

Similarly, there is also a functor

wR% : Gδ∞(F )→ CAT.

Just as above, we can use rules (!) as in Definition 6.1, as well as Proposi-tion A.0.4 and Proposition A.0.8, to construct (up to contractible choice) a stabi-lized parametrized Euler characteristic and a stabilized excisive Euler characteristicwhich make the following diagram commute up to a preferred homotopy

BT(F )X%

//

E(q%∞)

BGfr(F )

Xfr// E(q∞),


where q∞ : E(q∞) → BG∞(F ) denotes the fibration associated to the quasi-fibration

hocolimG∞(F )

A(X)→ BG∞(F )

and q%∞ : E(q%

∞)→ BG∞(F ) denotes the fibration associated to the quasi-fibration

hocolimG∞(F )

A%(X)→ BG∞(F ).

The maps Xfr and X% are essentially obtained by stabilizing the maps Xfrn andX%n constructed earlier.

9. Key Theorem

In the last section, we constructed a diagram

BT(F )X%

//

E(q%∞)

BGfr(F )

Xfr// E(q∞)

which commutes up to a preferred homotopy; this diagram allows us to relate themap BT(F ) → BGfr(F ), which we have already shown is related to the modulispace Sfr(p, ε) of framed fiber bundle structures on p, to the map E(q%

∞)→ E(q∞),which is related to the A-theory assembly map. This section will be devoted toproving the following theorem, which will be the key step in proving Theorem 1.2.

Theorem 9.1 (Key Lemma). The diagram

BT(F )X%

//

E(q%∞)

BGfr(F )

Xfr// E(q∞)

is a homotopy pullback square.

Corollary 9.2. (to Theorem 9.1) Let p : E → B be a fibration over a finiteconnected CW complex B whose fibers are homotopy equivalent to a finite CWcomplex F . Then

Lift

BT(F )

B

p//

;;

BGfr(F )

' Lift

E(q%

∞)

B

<<

Xfrp// E(q∞)

where p : B → BGfr(F ) is the map which classifies p when it is viewed as a framedfibration by equipping E with the stable trivial microbundle.

42 STACY HOEHN

The proof of Theorem 9.1 will occupy the rest of this section. Note that everycomponent of Gfr(F ) contains an object of the form (n, (Nn, τ(N))), where N issome n-dimensional compact manifold homotopy equivalent to F and τ(N) denotesthe tangent bundle of N . Without loss of generality, we can assume that n ≥6. From Corollary 5.5, we know that the homotopy fiber of the left-hand mapover the point determined by the object (n, (Nn, τ(N))) is homotopy equivalentto the stabilized h-cobordism space HCob(N). Meanwhile, the homotopy fiberof the right-hand map over the corresponding point is homotopy equivalent tohofibχ(N)(α : A%(N)→ A(N)), the homotopy fiber of the A-theory assembly mapfor N over the point determined by the retractive space N tN . Since the diagramfrom Theorem 9.1 is commutative up to a preferred homotopy, we get an inducedmap between the vertical homotopy fibers,

X : HCob(N)→ hofibχ(N)(α : A%(N)→ A(N)).

We need to show that this map is a homotopy equivalence.In [Wal82, Theorem 1], Waldhausen showed that there is a homotopy equivalence

Wald : HCob(N)→ hofib0(α : A%(N)→ A(N)),

where this homotopy fiber is taken over the point determined by the zero object Nin Rfd(N). (Note that the proof in [Wal82] that this map is a homotopy equivalenceis not self-contained; it relies on material in [Wal85] as well as forward references tomaterial which is now contained in [JRW08].) Since the spaces A%(N) and A(N)are infinite loop spaces, the homotopy fiber of α over any two points in A(N) arehomotopy equivalent. Thus, using Waldhausen’s map, we know that the domainand codomain of the map

X : HCob(N)→ hofibχ(N)(α : A%(N)→ A(N))

are abstractly homotopy equivalent. We just need to show that X actually gives ahomotopy equivalence between the two spaces. To do this, we will relate the mapX to Waldhausen’s map.

We will begin by giving a description of an unstabilized version

hcob(∂N)→ hofib0(α : A%(∂N)→ A(∂N))

of Waldhausen’s map, or at least a map homotopic to it. (Note that this unsta-bilized map is not a homotopy equivalence in general, but it becomes a homo-topy equivalence after stabilizing.) We will first introduce a new categorical modelfor the space of h-cobordisms. Let Hret(∂N) denote the simplicial category of h-cobordisms equipped with retractions. Its objects in degree 0 are pairs (Z, r), whereZ ⊂ ∂N × [0, 1] is an h-cobordism on ∂N = ∂N × 0 and r : Z → ∂N is a choiceof retraction. An object in degree k is a ∆k-parametrized family of such pairs. A0-morphism h : (Z, r)→ (Z ′, r′) is a homeomorphism h : Z → Z ′ which restricts tothe identity on ∂N such that r = r′h; a k-morphism is defined similarly. The clas-sifying space hcobret(∂N) of this simplicial category is homotopy equivalent to theclassifying space hcob(∂N) of H(∂N) since the space of retractions is contractible.

Remark 9.3. Waldhausen actually uses a slightly different model for hcob(∂N) in[Wal82]. He defines a simplicial set P (∂N) of partitions of ∂N × [0, 1]. A 0-simplexin P (∂N) is a triple (Z,F, Z ′), where Z is a submanifold of ∂N × [0, 1] containing∂N×0, Z ′ is the closure of the complement of Z in ∂N× [0, 1], and F = Z∩Z ′. His


model for hcob(∂N) is the simplicial subset of P (∂N) consisting of those partitions(Z,F, Z ′) where Z is an h-cobordism between ∂N × 0 and F .

A Discrete Model for Waldhausen’s Map. We will now give a descriptionof Waldhausen’s map restricted to hcobret,δ(∂N), the classifying space of the dis-crete category Hret,δ(∂N) underlying Hret(∂N). Recall that we are using theWaldhausen category wR%(∂N) to pick out points and paths between points inA%(∂N), where wR%(∂N) was defined to be the strict pullback of the followingdiagram.

wR%(∂N)

α // wRfd(∂N)

i

wV (∂N) // wRld(J(∂N))

We will essentially make use of the fact that an h-cobordism Z with a choice ofretraction r : Z → ∂N can also be viewed as a retractive space over ∂N . Inparticular, an object (Z, r) in Hret,δ(∂N) determines an object

((Z,Z ∪∂N ∂N × [0,∞)), r : Z → ∂N)

in the over-category α/(∂N). A morphism h : (Z, r) → (Z ′, r′) in Hret,δ(∂N)determines a morphism (h, h∪ id∂N×[0,∞)) in α/(∂N) since h is a homeomorphism.Thus, there is a functor

Wald′δ : Hret,δ(∂N)→ α/(∂N).

Upon passing to classifying spaces and using the group completion-like maps, thisgives us a map

Waldδ : hcobret,δ(∂N)→ hofib0(α : A%(∂N)→ A(∂N)).

This map agrees with the restriction of Waldhausen’s map to hcobret,δ(∂N). (See[JRW08, Wal82, Wal85] for more details. Compare also with Sections 9 and 10 in[DWW03] which use relative excisive Euler characteristics to give a model for amap homotopic to Waldhausen’s map.)

A Non-Discrete (Simplicial) Model for Waldhausen’s Map. To define ananalogous map from hcobret(∂N) instead of hcobret,δ(∂N), Waldhausen turns theappropriate categories of retractive spaces into simplicial categories. In order toavoid having to introduce these simplicial categories, we will make use of the fol-lowing lemma to obtain a map

Wald : hcobret(∂N)→ hofib0(α : A%(∂N)→ A(∂N))

which is homotopic to an unstable version of Waldhausen’s map.

Lemma 9.4. Suppose that f : X → Y is an acyclic map and that Z is componen-twise nilpotent. Then the natural map [Y,Z]→ [X,Z] between homotopy classes ofmaps which is induced by pre-composing with f is a bijection.

Proof. This follows from [May83, Thm. 5#] and the Universal Coefficient Theorem.

Remark 9.5. This implies that, under the hypotheses of the lemma, two mapsg1, g2 : Y → Z are homotopic if and only if g1 f, g2 f : X → Z are homotopic.

44 STACY HOEHN

We can once again use [McD80] to show that the inclusion-induced map

i : hcobret,δ(∂N)→ hcobret(∂N)

is an acyclic map. Since

hofib0(α : A%(∂N)→ A(∂N))

is an infinite loop space (and hence componentwise nilpotent), Lemma 9.4 thenimplies that the map Waldδ determines, up to contractible choice, a map

Wald : hcobret(∂N)→ hofib0(α : A%(∂N)→ A(∂N))

such that the composition i Wald is homotopic to Waldδ. Since Waldδ agreeswith the restriction of Waldhausen’s map to hcobret,δ(∂N), the remark followingLemma 9.4 implies that the map Wald must be homotopic to the unstable versionof Waldhausen’s map from [Wal82]. Upon stabilization (replacing N by N × Ik forlarge k), the unstable maps induce a homotopy equivalence

Wald : HCob(N)→ hofib0(α : A%(N)→ A(N)).

(As in the proof of Corollary 5.3, we are using the facts that the inclusion of∂(N × Ik) in N × Ik becomes highly connected as k goes to infinity and that thefunctors HCob(−), A%(−), and A(−) are homotopy-invariant functors.)

Reduction of Theorem 9.1 to Lemma 9.6. As mentioned earlier, to proveTheorem 9.1 that

BT(F )X%

//

E(q%∞)

BGfr(F )

Xfr// E(q∞)

is a homotopy pullback square, we will show that the induced map

X : HCob(N)→ hofibχ(N)(α : A%(N)→ A(N))

between vertical homotopy fibers is related to Waldhausen’s homotopy equivalence

Wald : HCob(N)→ hofib0(α : A%(N)→ A(N))

that we just described.

Lemma 9.6. The maps X and Wald fit into a homotopy commutative diagram

HCob(N)X //

Wald ))

hofibχ(N)(α : A%(N)→ A(N))

hofib0(α : A%(N)→ A(N)),

+χ(N)

33

where the map +χ(N) is also a homotopy equivalence.

As a corollary to this lemma, which is proven in the next section, X is a homotopyequivalence. Thus, the diagram from Theorem 9.1 is a homotopy pullback square.


10. Proof of Lemma 9.6

Constructing the diagram in Lemma 9.6 is not difficult; the map +χ(N) is justgiven by infinite loop space addition. What is more difficult is to show that theresulting diagram is actually homotopy commutative since the maps X and Waldwere both only defined up to contractible choice. To get around this problem,we will show that analogous diagrams in the discrete setting, where fewer (or no)choices were involved, are homotopy commutative, and then we will show that thisimplies that the corresponding diagram in the simplicial setting is also homotopycommutative.

We will first consider the unstable case. In particular, we will return to thefollowing zigzag diagram which was related to the construction of the maps Xfr

and X% in the diagram from Theorem 9.1.

BTδn(F )χ%,δn //

F δ1

B(Gδ(F )∫wR%)

F δ2

hocolimX∈Gδ(F )

|wR%(X)| //

F δ3

'oo E(q%,δ)

F δ4

BGfr,δn (F )

χfr,δn

// B(Gδ(F )∫wRfd) hocolim

X∈Gδ(F )|wRfd(X)|'

oo // E(qδ),

Recall that E(qδ) ' hocolimX∈Gδ(F )

A(X) and E(q%,δ) ' hocolimX∈Gδ(F )

A%(X) are the total

spaces of fibrations over BGδ(F ). The left-most square in this diagram was inducedby the following commutative diagram of categories and functors.

Tδn(F )χ%,δn //

F δ1

Gδ(F )∫wR%

F δ2

Gfr,δn (F )χfr,δn

// Gδ(F )∫wRfd

For an object (Nn, τ(N)) in Gfr,δn (F ), we get an induced functor

χδn : F δ1 /(N, τ(N))→ F δ2 /(N,N tN)

between over-categories which takes an object

(N ′, (f, f∗) : (N ′, τ(N ′))→ (N, τ(N)))

in F δ1 /(N, τ(N)) to the object

((N ′, N ′ tN ′), (f : N ′ → N, f t id : N ′ tN → N tN))

in F δ2 /(N,N tN).Recall from the construction of our model for Waldhausen’s map that there is a

functorWald′δ : Hret,δ(∂N)→ α∂N/(∂N)

from the discrete category of h-cobordisms over ∂N to the over-category for thefunctor

α∂N : wR%(∂N)→ wRfd(∂N)

over the zero object ∂N in wRfd(∂N) which essentially views an h-cobordism Zover ∂N (with choice of retraction r) as a retractive space over ∂N . We wantto relate this functor to the functor χδn between over-categories. Recall that if

46 STACY HOEHN

Z is an h-cobordism with retraction r : Z → ∂N , then r ∪ id : Z ∪∂N N →N is a homotopy equivalence between compact topological manifolds homotopyequivalent to F . Moreover, the derivative of this map induces a microbundle mapτ(Z ∪∂N N)→ τ(N) covering r ∪ id. Thus, we obtain a functor

β : Hret,δ(∂N)→ F δ1 /(N, τ(N))

which takes an h-cobordism Z over ∂N with retraction map r to the object

(Z ∪∂N N, (Z ∪∂N N, τ(Z ∪∂N N))→ (N, τ(N)))

in F δ1 /(N, τ(N)).So far, we have constructed the following diagram.

Hret,δ(∂N)

β

Wald′δ // α∂N/(∂N)

F δ1 /(N, τ(N))χδn

// F δ2 /(N,N tN)

We want to complete the right side of this diagram. Pushing out along the inclusion∂N → N induces an exact functor

α∂N/(∂N)→ αN/(N),

where αN : wR%(N) → wRfd(N) and N denotes the zero object in wRfd(N). Inorder to transition from over-categories over the zero object N to over-categoriesover the retractive space N t N , we need to introduce shift functors. The shiftfunctor

shift : wRfd(N)→ wRfd(N)

takes a retractive space Y over N with section map s to the pushout of the diagram

N //

s

N tN

Y,

which is just the retractive space Y tN over N . The shift functor

shift : wR%(N)→ wR%(N)

is defined similarly and fits into the following commutative diagram.

wR%(N)shift //

αN

wR%(N)

αN

wRfd(N)

shift// wRfd(N)

These shift functors induce another shift functor

shift : αN/(N)→ αN/(N tN)

between over-categories. Finally, there is a functor

φ : F δ2 /(N,N tN)→ αN/(N tN)


which takes an object

((X,Y ), (f : X → N, f ′ : wRfd(f)(Y )→ N tN))

to the object

(wRfd(f)(Y ), f ′ : wRfd(f)(Y )→ N tN).

This functor induces a homotopy equivalence on classifying spaces by Lemma 6.4.By examining where an object (Z, r) in Hret,δ(∂N) maps to under both the

clockwise and counterclockwise compositions of functors in the following diagram,we obtain the following lemma.

Lemma 10.1. The following diagram of categories and functors is commutative.

Hret,δ(∂N)Wald′δ //

β

α∂N/(∂N)

i∗

αN/(N)

shift

αN/(N tN)

F δ1 /(N, τ(N))χδn

// F δ2 /(N,N tN)

φ

OO

Corollary 10.2. The following diagram of spaces and continuous maps is alsocommutative.

hcobret,δ(∂N)Wald′δ //

β

B(α∂N/(∂N))

i∗

B(αN/(N))

shift

B(αN/(N tN))

B(F δ1 /(N, τ(N)))χδn

// B(F δ2 /(N,N tN))

φ

OO

Instead of working with classifying spaces of over-categories, we now want totransition to talking about homotopy fibers. Recall from [Qui73, p. 88] that when-ever a map between spaces is induced by applying the classifying space functor to afunctor G : C→ D between categories, there is a canonical map from the classifyingspace B(G/D) of the over-category over any object D in D to the homotopy fiber ofthe induced map BG on classifying spaces over the point determined by the objectD. (Moreover, this map is a homotopy equivalence if the hypotheses of Quillen’sTheorem B are satisfied.) More generally, we have the following lemma.

48 STACY HOEHN

Lemma 10.3. Suppose that we have a commutative diagram

C1φC //

G1

C2

G2

D1

φD

// D2

of categories and functors, and let D1 be any object in D1. Then the canonicalmaps

B(G1/D1)→ hofibD1(BG1), B(G2/φD(D1))→ hofibφD(D1)(BG2)

fit into the following homotopy commutative diagram.

B(G1/D1) //

B(G2/φD(D1))

hofibD1(BG1) // hofibφD(D1)(BG2)

Let Fδ1 denote the homotopy fiber of F δ1 over the point determined by the object(N, τ(N)) in Gfr,δn (F ), and let Fδ2 denote the homotopy fiber of F δ2 over the corre-sponding point in B(Gδ(F )

∫wRfd). Then by Lemma 10.3, we have the following

homotopy commutative diagram.

B(F δ1 /(N, τ(N))) //

B(F δ2 /(N,N tN))

Fδ1

// Fδ2

Similarly, we can use Lemma 10.3 to obtain the following diagram in which bothsquares are homotopy commutative,

B(α∂N/(∂N)) //

i∗

hofib0(B(α∂N ))

i∗

B(αN/(N)) //

shift

hofib0(B(αN ))

shift

B(αN/(N tN)) // hofibχ(N)(B(αN )),

where hofib0(B(αN )) (resp. hofibχ(N)(B(αN ))) denotes the homotopy fiber of the

map αN : |wR%(N)| → |wRfd(N)| over the point determined by the zero objectN over N (resp. over the point determined by the retractive space N t N) andsimilarly for hofib0(B(α∂N )).

Note that we also have the following commutative diagram.

|wR%(N)| //

B(αN )

B(Gδ(F )∫wR%)

F2

|wRfd(N)| // B(Gδ(F )

∫wRfd)


Thus, we get an induced map hofibχ(N)(B(αN ))→ Fδ2 between vertical homotopyfibers. This map fits into the following diagram

B(αN/(N tN))ψ //

B(F δ2 /(N,N tN))

hofibχ(N)(B(αN )) // Fδ2,

where ψ is the homotopy inverse for φ : B(F δ2 /(N,N t N)) → B(αN/(N t N))given in Lemma 6.4. Similarly to above, Lemma 10.3 implies that this diagram ishomotopy commutative. If we replace the upper horizontal map ψ by its homotopyinverse φ, the resulting diagram is also homotopy commutative.

Now let Fδ3 denote the homotopy fiber of the map

F δ3 : hocolimX∈Gδ(F )

|wR%(X)| → hocolimX∈Gδ(F )

|wRfd(X)|

over the point corresponding to the object (N,N tN). Similarly to above, we getan induced map

hofibχ(N)(B(αN ))→ Fδ3

between homotopy fibers, and one can check that the following triangle is alsohomotopy commutative.

hofibχ(N)(B(αN ))

xx Fδ2 Fδ3

'oo

So far, we have shown that each of the regions in the following diagram is ho-motopy commutative.

hcobret,δ(∂N) //

B(α∂N/(∂N))

i∗

// hofib0(B(α∂N ))

i∗

B(αN/(N)) //

shift

hofib0(B(αN ))

shift

B(αN/(N tN)) // hofibχ(N)(B(αN ))

B(F δ1 /(N, τ(N))) //

B(F δ2 /(N,N tN))

'

OO

Fδ1

// Fδ2 Fδ3'oo

If we choose a homotopy inverse for the wrong-way homotopy equivalence Fδ2'← Fδ3

in the bottom row, then we obtain the following corollary, by essentially goingaround the outside of the previous diagram.

50 STACY HOEHN

Corollary 10.4. The following diagram is homotopy commutative.

hcobret,δ(∂N) //

B(α∂N/(∂N)) // hofib0(B(α∂N ))

i∗

hofib0(B(αN ))

shift

B(F δ1 /(N, τ(N)))

hofibχ(N)(B(αN ))

Fδ1

// Fδ2 // Fδ3

We now want to use the group completion-like maps to pass from the right-hand vertical column in the above diagram to an analogous column that involvesalgebraic K-theory. Let Fδ4 denote the homotopy fiber of the map

F δ4 : E(q%,δ)→ E(qδ)

over the image of the point in BGfr,δn (F ) determined by the object (N, τ(N)) underthe map χfr,δn . Similarly to above, there is a natural map

hofibχ(N)(K(αN ))→ Fδ4,

where hofibχ(N)(K(αN )) denotes the homotopy fiber of the map

K(αN ) : A%(N)→ A(N)

over the point determined by the object N t N . This map fits into the followingdiagram where all of the regions are homotopy commutative.

hofib0(B(α∂N ))

i∗

// hofib0(K(α∂N ))

i∗

hofib0(B(αN ))

shift

// hofib0(K(αN ))

+χ(N)

hofibχ(N)(B(αN ))

// hofibχ(N)(K(αN ))

Fδ3

// Fδ4

The map +χ(N) : hofib0(K(αN )) → hofibχ(N)(K(αN )) in this diagram is givenby the infinite loop space sum.


Corollary 10.5. The following diagram is homotopy commutative.

hcobret,δ(∂N) //

B(α∂N/(∂N)) // hofib0(B(α∂N )) // hofib0(K(α∂N ))

i∗

hofib0(K(αN ))

+χ(N)

B(F δ1 /(N, τ(N)))

hofibχ(N)(K(αN ))

Fδ1

// Fδ2 // Fδ3 // Fδ4

The composition of the maps in the upper horizontal row of this diagram is justthe discrete version Waldδ of Waldhausen’s map that we constructed earlier, andthe composition of the maps in the lower horizontal row of the diagram is homotopicto the map χδn : Fδ1 → Fδ4 between vertical homotopy fibers which is induced by thediagram

BTδn(F )

χ%,δn // E(q%,δ)

BGfr,δn (F )

χfr,δn

// E(qδ).

In the non-discrete setting, we have a similar diagram

BTn(F )X%n //

E(q%)

BGfrn (F )

Xfrn

// E(q)

which commutes up to a preferred homotopy. Let F1 denote the homotopy fiber ofBTn(F )→ BGfrn (F ) over the point determined by the object (N, τ(N)), and let F4

denote the homotopy fiber of E(q%)→ E(q) over the corresponding point in E(q).Then we get an induced map Xn : F1 → F4 between vertical homotopy fibers. This

52 STACY HOEHN

map fits into the following diagram.

hcobret(∂N)Wald //

hofib0(K(α∂N ))

i∗

hofib0(K(αN ))

+χ(N)

hofibχ(N)(K(αN ))

F1

Xn

// F4

We will use the following lemma to show that this diagram is homotopy commuta-tive as well.

Lemma 10.6. Suppose that we have a cube

A′

f ′

i′ // D′

h′

Ai //

f

φA

>>

DφD

>>

h

B′g′ // C ′

B

φB

>>

g// C

φC

>>

of topological spaces and continuous maps, where φA is an acyclic map between CWcomplexes and D′ is componentwise nilpotent. Suppose further that the top, bottom,front, left, and right faces of the cube are homotopy commutative. Then the backface is also homotopy commutative.

Proof. We want to show that [g′ f ′] = [h′ i′]. By Lemma 9.4, it suffices toshow that [g′ f ′ φA] = [h′ i′ φA]. By the homotopy commutativity of theother faces, we know that [φD i] = [i′ φA], [φC g] = [g′ φB ], [g f ] = [h i],[f ′ φA] = [φB f ], and [h′ φD] = [φC h]. Therefore,

[g′ f ′ φA] = [g′ φB f ]

= [φC g f ]

= [φC h i]= [h′ φD i]= [h′ i′ φA].


We will apply this lemma to the following cube.

hcobret(∂N)

Wald // hofib0(K(α∂N ))

hcobret,δ(∂N)

77

Waldδ //

hofib0(K(α∂N ))

id

55

F1Xn // F4

Fδ1

66

χδn

// Fδ4

55

The map hcobret,δ(∂N)→ hcobret(∂N) in this cube is an acyclic map by argumentssimilar to those in [McD80], and the space hofib0(K(α∂N )) is componentwise nilpo-tent since it is an infinite loop space. The front face of this cube is homotopy com-mutative by Corollary 10.5. By our construction of Waldδ and Wald, the top face ofthe cube is homotopy commutative. Also by the construction of the parametrizedEuler characteristics, the bottom face of the cube is homotopy commutative. Itis straightforward to verify that this cube also satisfies the other requirements ofLemma 10.6, so the back face of the cube, which is given by the diagram

hcobret(∂N)Wald //

hofib0(K(α∂N ))

i∗

hofib0(K(αN ))

+χ(N)

hofibχ(N)(K(αN ))

F1

Xn

// F4,

is also homotopy commutative.The maps

hofib0(K(αN ))+χ(N)−→ hofibχ(N)(K(αN )) −→ F4

in the previous diagram are both homotopy equivalences. After stabilizing (i.e.replacing N by N × Ik for large k), the map i∗ becomes a homotopy equivalencebecause the inclusion map ∂(N × Ik) → N × Ik becomes highly connected as kgoes to infinity. The left-hand vertical map also becomes a homotopy equivalenceafter stabilizing by Corollary 5.5 and its proof. Moreover, the upper horizontalmap becomes the stabilized version of Waldhausen’s map which is a homotopyequivalence, and the lower horizontal map becomes X.

Thus, we have now proven the following statement.

54 STACY HOEHN

Corollary 10.7. The map X is a homotopy equivalence, and hence the desireddiagram from Theorem 9.1 is a homotopy pullback square.

11. Conclusion

By combining Corollary 3.8, Corollary 4.9, and Theorem 9.1, we have completedour proof of the following theorem that the space S(p) of compact topological fiberbundle structures on p decomposes into two pieces, one related to microbundleinformation and one related to algebraic K-theory through a space of lifts.

Theorem 11.1. Let p : E → B be a fibration over a finite connected CW complexB, whose fibers are homotopy equivalent to a finite CW complex F . Then

S(p) ' Micro(E)× Lift

E(q%

∞)

B

Xfrp//

<<

E(q∞)

.

This result can also be reformulated in terms of the parametrized Euler charac-teristic for p : E → B instead of for the universal fibration q∞ : E(q∞)→ BG∞(F ).The fibration p is classified by a map B → BG∞(F ), which we will call p by abuseof notation. Let AB(E) denote the pullback of the following diagram.

AB(E) //

E(q∞)

B

p// BG∞(F )

We have a section of the fibration q∞ : E(q∞) → BG∞(F ) on the right, and thissection determines a section χ(p) of the fibration AB(p) : AB(E) → B on the leftby pulling back. This is the parametrized Euler characteristic for p from [DWW03].

Similarly, let A%B(E) denote the pullback of the analogous diagram where E(q∞)

is replaced by E(q%∞). Then the above theorem can be restated in terms of lifts of

χ(p) : B → AB(E) to a map from B to A%B(E).

Theorem 11.2. Let p : E → B be a fibration over a finite connected CW complexB, whose fibers are homotopy equivalent to a finite CW complex F . Then

S(p) ' Micro(E)× Lift

A%B(E)

B

χ(p)//

<<

AB(E)

.

Now assume that p : E → B actually admits a compact topological fiber bundlestructure; without loss of generality, we will assume that p itself is a compacttopological fiber bundle. This implies that there is a canonical lift χ%(p) of χ(p)

to A%B(E). Given any two sections χ, χ′ : B → AB(E), there is another section

χ+ χ′ : B → AB(E), which is given by infinite loop space sum, and given any liftχ% of χ and any lift χ′% of χ′, the infinite loop space sum χ% +χ′% is a lift of χ+χ′.


Similarly, given any lift χ% of χ(p), we can use the infinite loop space structure on

A%B(E) to form the difference χ%−χ%(p) between χ% and our preferred lift χ%(p);

this difference will be a lift of the zero section zero : B → AB(E). These ideasshow how to obtain a homotopy equivalence from the space of lifts

Lift

A%B(E)

B

χ(p)//

<<

AB(E)

to the space of lifts

Lift

A%B(E)

B

zero//

<<

AB(E)

when we have a preferred lift of χ(p) (for example, when p is a compact topologicalfiber bundle). This latter space of lifts is homotopy equivalent to the space ofsections

MapB(B,ΩBWhB(E)),

where the fibration ΩBWhB(p) : ΩBWhB(E) → B is obtained analogously to the

fibrations AB(p), A%B(p) using the functor which associates to a space X, the space

ΩWh(X) which is the homotopy fiber of the A-theory assembly map A%(X) →A(X). The fact that the relevant space of lifts is homotopy equivalent to this spaceof sections is a consequence of the fact that the following maps fit into a homotopypullback square

ΩBWhB(E) //

A%B(E)

B

zero// AB(E).

Thus, we have finally proven Theorem 1.2, which is restated below.

Theorem 11.3 (Main Theorem). Suppose that p : E → B is a fibration, whereB is a finite CW complex and the fibers of p are homotopy equivalent to a finiteCW complex, and suppose further that p admits a compact topological fiber bundlestructure. Then the moduli space of all such bundle structures on p is given by

S(p) ' Micro(E)×MapB(B,ΩBWhB(E)).

Appendix A. Results about Sections of Fibrations

Throughout this section, our goal will be to state several results about spaces ofsections of fibrations. All of these results are concerned with how we can go fromhaving a section of one fibration to obtaining a section of a related fibration.

First suppose the we have maps

B2φB // B1

pB1 // B0,

56 STACY HOEHN

where pB1 and pB2 := pB1 φB are both fibrations. For i = 1, 2, let MapB0(B0, Bi)

denote the space of sections of pBi . By composing a section of pB2 with φB , weobtain a section of pB1 . Thus, we have a map

φ∗B : MapB0(B0, B2)→ MapB0

(B0, B1).

The following standard lemma gives us a condition under which this map is ahomotopy equivalence.

Lemma A.1. If φB is a fiber homotopy equivalence, then the map


(B0, B1)

is also a homotopy equivalence.

This implies that if φB is a fiber homotopy equivalence, a section sB1 of pB1determines, up to contractible choice, a section sB2 of pB2 which makes the followingdiagram commute up to a preferred homotopy.

B0sB1

//

sB2

B1 B2

φB

'oo

Thus, we can think of sB2 as having been obtained from sB1 by choosing a homotopyinverse for φB .

Remark A.2. By [May99, Proposition 7.5], every homotopy equivalence B2 → B1

over B0 is also a fiber homotopy equivalence. Therefore, we could have just assumedthat the map φB is a homotopy equivalence.

More generally, suppose that we have a diagram

(A.3) E2//

φE

B2

φB

E1

//

pE1

B1

pB1

E0f// B0

where the maps pE1 , pB1 , pE2 , and pB2 are all fibrations, where

pE2 := pE1 φE ,

pB2 := pB1 φB .

For i = 1, 2, let MapB0(B0, Bi) (resp. MapE0

(E0, Ei)) denote the space of

sections of pBi (resp. pEi ). The maps φB and φE induce maps


(B0, B1),

φ∗E : MapE0(E0, E2)→ MapE0

(E0, E1).


For i = 1, 2, let MapB0(E0, Bi) denote the space of maps from E0 to Bi over B0.

Recall that when we have a commutative diagram

Ei //

Bi

E0

f//

==

B0,

giving a map g : E0 → Bi over B0 is equivalent to giving a section of the fibration

f∗(Bi)→ E0

obtained by pulling back Bi → B0 along f . In fact,

MapB0(E0, Bi) ∼= MapE0

(E0, f∗(Bi)).

We will use this identification often to go back and forth from maps over a particularspace to a section of an associated fibration.

The map φB induces a map f∗(B2)f∗(φB)−→ f∗(B1). Composing with f∗(φB)

induces a map

MapE0(E0, f

∗(B2))→ MapE0(E0, f

∗(B1)),

which after the above identifications of section spaces with spaces of maps over B0,gives us a map

MapB0(E0, B2)→ MapB0

(E0, B1).

For i = 1, 2, let

MapE0,B0

E0

// Ei

B0

// Bi

denote the homotopy pullback of

MapE0(E0, Ei)

MapB0

(B0, Bi) // MapB0(E0, Bi).

A point in this homotopy pullback consists of a section sEi of pEi , a section sBi ofpBi , and a path between their images in MapB0

(E0, Bi). In other words, sEi and

sBi are two sections which make the following diagram commute up to a preferredhomotopy.

E0

sEi //

Ei

B0

sBi

// Bi

Since we have maps from three of the corners of the homotopy pullback diagramcorresponding to i = 2 to three of the corners of the homotopy pullback diagram

58 STACY HOEHN

corresponding to i = 1, we get an induced map

MapE0,B0

E0

// E2

B0

// B2

−→ MapE0,B0

E0

// E1

B0

// B1

.

Under certain conditions, this map is a homotopy equivalence.

Proposition A.0.4. Suppose that we have a commutative diagram as in (A.3). Ifthe maps φB and φE are fiber homotopy equivalences, then the map

MapE0,B0

E0

// E2

B0

// B2

−→ MapE0,B0

E0

// E1

B0

// B1

is a homotopy equivalence.

Proof. Since the maps φB and φE are fiber homotopy equivalences, the maps

φ∗B : MapB0(B0, B2)→MapB0(B0, B1),

φ∗E : MapE0(E0, E2)→ MapE0

(E0, E1)

are also homotopy equivalences by Lemma A.1.To see that the map


(E0, B1)

is a homotopy equivalence, recall that it was obtained via an identification with themap

MapE0(E0, f

∗(B2))→ MapE0(E0, f

∗(B1)),

which was induced by the map f∗(φB). Since φB is a fiber homotopy equivalence,the map f∗(φB) is also a fiber homotopy equivalence, so the map

MapE0(E0, f

∗(B2))→ MapE0(E0, f

∗(B1))

is a homotopy equivalence by Lemma A.1. Hence, the map


(E0, B1)

is also a homotopy equivalence.Since the 3 maps which induce the map

MapE0,B0

E0

// E2

B0

// B2

−→ MapE0,B0

E0

// E1

B0

// B1

are all homotopy equivalences, this map is also a homotopy equivalence.


In particular, this implies that, under the hypotheses of the proposition, sectionssB1 and sE1 of pB1 and pE1 which make the diagram

E0

sE1 //

E1

B0

sB1

// B1

commute up to a preferred homotopy determine, up to a contractible choice, sec-tions sB2 and sE2 of pB2 and pE2 which make the diagram

E0

sE2 //

E2

B0

sB2

// B2

commute up to a preferred homotopy. Moreover, all of these sections are compatiblein the sense that all of the regions in the diagram

E0sE1

//

sE2

E1

E2

φE

'oo

B0

sB1 //

sB2

>>B1 B2φB

'oo

commute up to a preferred homotopy. Thus, we can think of sB2 as being obtainedfrom sB1 by a choosing a homotopy inverse for φB , and similarly sE2 is obtainedfrom sE1 by choosing a homotopy inverse for φE . The above lemma is saying thatthe choices for these two homotopy inverses can be made in a compatible way.

Remark A.5. We apply Proposition A.0.4 in the first subsection of Section 8 toobtain a section

χδ : BGδ(F )→ E(qδ)

of qδ and a map

χ%,δn : BTδn(F )→ E(q%,δ)

over BGδ(F ) which make the following diagram commute up to a preferred homo-topy.

BTδn(F )

χ%,δn // E(q%,δ)

BGδ(F )

χδ// E(qδ)

60 STACY HOEHN

Now suppose that we have a commutative pullback square

K0//

pK0

K1

pK1

B0 gB// B1,

where pK0 and pK1 are fibrations. For i = 0, 1, let MapBi(Bi,Ki) denote the space

of sections of pKi . By pulling back a section of pK1 along the map gB , we obtain asection of pK0 . Thus, there is a map

g∗B : MapB1(B1,K1)→ MapB0

(B0,K0).

Under certain conditions, this map is a homotopy equivalence.

Proposition A.0.6. Suppose we have a commutative pullback square as above,where B0 and B1 are homotopy equivalent to CW complexes, gB : B0 → B1 is amap whose homotopy fiber over any point in B1 is acyclic, and φK1 : K1 → B1 is afibration whose fibers are componentwise nilpotent. Then the pullback-induced map


(B0,K0)

between section spaces is a homotopy equivalence.

Proof. See the proof of Corollary 2.7 in [DWW03].

In particular, this implies that, under the conditions of the proposition, thehomotopy fiber of


(B0,K0)

is contractible. Thus, a section sK0 of pK0 determines, up to contractible choice, asection sK1 of pK1 and a path in MapB0

(B0,K0) between g∗B(sK1 ) and sK0 . In other

words, sK0 determines, up to contractible choice, a section sK1 which makes thefollowing diagram commute up to a preferred homotopy.

K0// K1

B0 gB//

sK0

OO

B1,

sK1

OO

The previous proposition can be extended to the case where we have a commu-tative cube

(A.7) K0

pK0

// K1

pK1

L0//

pL0

φL,K0

==

L1

pL1

φL,K1

==

B0gB // B1

E0

φE,B0

==

gE// E1

φE,B1

==


where the maps pK0 , pK1 , p

L0 , p

L1 are fibrations and both the front and back faces of

the cube are pullback squares.Similarly to above, given a section sK1 of pK1 (resp. sL1 of pL1 ), pulling back along

gB (resp. gE) gives a section of pK0 (resp. pL0 ). Thus, we have maps


(B0,K0),

g∗E : MapE1(E1, L1)→ MapE0

(E0, L0).

For i = 0, 1, let MapBi(Ei,Ki) denote the space of maps f : Ei → Ki over Bi.Recall that

MapBi(Ei,Ki) ∼= MapEi(Ei, (φE,Bi )∗(Ki)).

The diagram

(φE,B0 )∗(K0) //

(φE,B1 )∗(K1)

E0 gE

// E1

is a pullback square, so by pulling back a section of the right-hand fibration alonggE , we obtain a section of the left-hand fibration. Thus, we have a map

MapE1(E1, (φ

E,B1 )∗(K1))→MapE0

(E0, (φE,B0 )∗(K0)).

After making the above identification between these section spaces and associatedspaces of maps over Bi, we obtain a map

MapB1(E1,K1)→ MapB0

(E0,K0).

For i = 0, 1, let

MapEi,Bi

Ei

// Li

Bi // Ki

denote the homotopy pullback of the following diagram of mapping spaces.

MapEi(Ei, Li)

MapBi(Bi,Ki) // MapBi(Ei,Ki).

A point in this homotopy pullback consists of a section sLi : Ei → Li of pLi , a sectionsKi : Bi → Ki of pKi , and a path in MapBi(Ei,Ki) which starts at

EisLi→ Li

φL,Ki→ Ki

and ends at

EiφE,Bi→ Bi

sKi→ Ki.

62 STACY HOEHN

In other words, sLi , sKi are two sections that are compatible in the sense that the

following diagram commutes up to a preferred homotopy.

EisLi //

φE,Bi

Li

φL,Ki

Bi

sKi

// Ki

Since we have maps between three of the corners in the homotopy pullbackdiagram for i = 1 to three of the corners in the corresponding diagram for i = 0,we get an induced map

MapE1,B1

E1

// L1

B1

// K1

−→ MapE0,B0

E0

// L0

B0

// K0

.

Proposition A.0.8. Suppose we have a commutative cube as in (A.7). Supposefurther that the spaces B0, B1, E0, and E1 are all homotopy equivalent to CWcomplexes, the maps gB and gE are both maps whose homotopy fiber over any pointis acyclic, and the fibers of pK1 and pL1 are componentwise nilpotent. Then the map

MapE1,B1

E1

// L1

B1

// K1

−→ MapE0,B0

E0

// L0

B0

// K0

is a homotopy equivalence.

Proof. By Proposition A.0.6, we know that the pullback maps

MapB1(B1,K1)→ MapB0

(B0,K0),

MapE1(E1, L1)→ MapE0

(E0, L0)

of the associated section spaces are homotopy equivalences.To see that the map MapB1

(E1,K1) → MapB0(E0,K0) is a homotopy equiva-

lence, recall that it was defined via an identification with a map

MapE1(E1, (φ

E,B1 )∗(K1))→ MapE0

(E0, (φE,B0 )∗(K0))

between section spaces which was induced by pulling back along the map gE whosehomotopy fibers are all acyclic. To apply Proposition A.0.6, we have to make sure

that the fibers of (φE,B1 )∗(K1)→ E1 are componentwise nilpotent, but this is clearfrom the fact that the fibers of the fibration pK1 : K1 → E1 are componentwisenilpotent. Thus the map

MapE1(E1, (φ

E,B1 )∗(K1))→ MapE0

(E0, (φE,B0 )∗(K0))

is a homotopy equivalence by Proposition A.0.6, and hence the map

MapB1(E1,K1)→ MapB0

(E0,K0)

is also a homotopy equivalence.


Since the 3 maps that induce the map

MapE1,B1

E1

// L1

B1

// K1

−→ MapE0,B0

E0

// L0

B0

// K0

are all homotopy equivalences, this map is also a homotopy equivalence.

In particular, this implies that, under the hypotheses of the proposition, sectionssK0 : B0 → K0 and sL0 : E0 → L0 of pK0 and pL0 which make the diagram

E0

sL0 //

L0

B0

sK0

// K0

commute up to a preferred homotopy determine, up to a contractible choice, sec-tions sK1 : B1 → K1 and sL1 : E1 → L1 of pK1 and pL1 which make the diagram

E1

sL1 //

L1

B1

sK1

// K1

commute up to a preferred homotopy. Moreover, these choices of sections are allcompatible in the sense that all of the faces of the following cube commute up to apreferred homotopy.

K0// K1

L0//

==

L1

==

B0//

sK0

OO

B1

sK1

OO

E0

sL0

OO

==

// E1

==sL1

OO

Remark A.9. We apply Proposition A.0.8 in the second subsection of Section 8 toobtain a section

X : BG(F )→ E(q)

of q and a map

X%n : BTn(F )→ E(q%)

64 STACY HOEHN

over BG(F ) which make the following diagram commute up to a preferred homo-topy.

BTn(F )X%n //

E(q%)

BG(F )

X// E(q)


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