Über die Autorin bzw. den Autor

Friedhelm Waldhausen is professor emeritus of mathematics at Bielefeld University. Bjorn Jahren is professor of mathematics at the University of Oslo. John Rognes is professor of mathematics at the University of Oslo.

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Spaces of PL Manifolds and Categories of Simple Maps

PRINCETON UNIVERSITY PRESS

Contents

Introduction.......................................................11. The stable parametrized h-cobordism theorem.....................72. On simple maps..................................................293. The non-manifold part...........................................994. The manifold part...............................................139Bibliography.......................................................175Symbols............................................................179Index..............................................................181

Chapter One

1.1. The manifold part

We write DIFF for the category of C^∞ smooth manifolds, PL for the category of piecewise-linear manifolds, and TOP for the category of topological manifolds. We generically write CAT for any one of these geometric categories. Let I = [0, 1] and J be two fixed closed intervals in R. We will form collars using I and stabilize manifolds and polyhedra using J.

In this section, as well as in Chapter 4, we let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the standard affine q-simplex.

By a CAT bundle π : E -> Δ^q we mean a CAT locally trivial family, i.e., a map such that there exists an open cover {U_α} of Δ^q and a CAT isomorphism over U_α (= a local trivialization) from π^-1 (U_α) -> U_α to a product bundle, for each α. For π to be a CAT bundle relative to a given product subbundle, we also ask that each local trivialization restricts to the identity on the product subbundle. We can always shrink the open cover to a cover by compact subsets {K_α}, whose interiors still cover Δ^q, and this allows us to only work with compact polyhedra in the PL case.

Definition 1.1.1. (a) Let M be a compact CAT manifold, with empty or nonempty boundary. We define the CAT h-cobordism space H(M) = H^CAT (M) of M as a simplicial set. Its 0-simplices are the compact CAT manifolds W that are h-cobordisms on M, i.e., the boundary

[partial derivative] W = M [union] N

is a union of two codimension zero submanifolds along their common boundary [partial derivative]M = [partial derivative]N, and the inclusions

M [subset] W [contains] N

are homotopy equivalences. For each q ≥ 0, a q-simplex of H(M) is a CAT bundle π: E -> Δ^q relative to the trivial subbundle pr : M × Δ^q -> Δ^q, such that each fiber W_p = π^-1(p) is a CAT h-cobordism on M [congruent to] M × p, for p [member of] Δ^q.

(b) We also define a collared CAT h-cobordism space H(M)^c = H^CAT (M)^c, whose 0-simplices are h-cobordisms W on M equipped with a choice of collar, i.e., a CAT embedding

c : M × I -> W

that identifies M × 0 with M in the standard way, and takes M × [0, 1) to an open neighborhood of M in W. A q-simplex of H(M)^c is a CAT bundle π : E -> Δ^q relative to an embedded subbundle pr : M × I × Δ^q -> Δ^q, such that each fiber is a collared CAT h-cobordism on M. The map H(M)^c -> H(M) that forgets the choice of collar is a weak homotopy equivalence, because spaces of collars are contractible.

Remark 1.1.2. To ensure that these collections of simplices are really sets, we might assume that each bundle E -> Δ^q is embedded in R^∞ × Δ^q -> Δ^q. The simplicial operator associated to α: Δ^p -> Δ^q takes E -> Δ^q to the image of the pullback [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] under the canonical identification [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. See [HTW90, 2.1] for a more detailed solution. To smooth any corners that arise, we interpret DIFF manifolds as coming equipped with a smooth normal field, as in [Wa82, §6]. The emphasis in this book will be on the PL case.

To see that the space of CAT collars on M in W is contractible, we note that [Ar70, Thm. 2] proves that any two TOP collars are ambient isotopic (relative to the boundary), and the argument generalizes word-for-word to show that any two parametrized families of collars (over the same base) are connected by a family of ambient isotopies, which proves the claim for TOP. In the PL category, the same proof works, once PL isotopies are chosen to replace the TOP isotopies F_s and G_s given on page 124 of [Ar70]. The proof in the DIFF case is different, using the convexity of the space of inward pointing normal fields.

Definition 1.1.3. (a) The stabilization map

σ : H(M) -> H(M × J)

takes an h-cobordism W on M to the h-cobordism W × J on M × J. It is well-defined, because M × J [subset] W × J and (N × J) [union] (W × [partial derivative] J) [subset] W × J are homotopy equivalences. The stable h-cobordism space of M is the colimit

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

over k ≥ 0, formed with respect to the stabilization maps. Each stabilization map is a cofibration of simplicial sets, so the colimit has the same homotopy type as the corresponding homotopy colimit, or mapping telescope.

(b) In the collared case, the stabilization map σ : H(M)^c -> H(M×J)^c takes a collared h-cobordism (W, c) on M to the h-cobordism W × J on M × J with collar

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Each codimension zero CAT embedding M -> M' induces a map H(M)^c -> H(M')^c that takes (W, c) to the h-cobordism

W' = M' × I [union] _M×I W,

with the obvious collar c' : M' × I -> W'. This makes H(M)^c and H^CAT (M)^c = colim_k H(M × J^k)^c covariant functors in M, for codimension zero embeddings of CAT manifolds. The forgetful map H^CAT (M)^c -> H^CAT (M) is also a weak homotopy equivalence.

We must work with the collared h-cobordism space when functoriality is required, but will often (for simplicity) just refer to the plain h-cobordism space. To extend the functoriality from codimension zero embeddings to general continuous maps M -> M' of topological spaces, one can proceed as in [Ha78, Prop. 1.3] or [Wa82, p. 152], to which we refer for details.

Remark 1.1.4. For a cobordism to become an h-cobordism after suitable stabilization, weaker homotopical hypotheses suffice. For example, let X [subset] V be a codimension zero inclusion and homotopy equivalence of compact CAT manifolds. Let c₀ : [partial derivative] X × I -> X be an interior collar on the boundary of X, let M₀ = c₀([partial derivative] X × 1) and W₀ = c₀ ([partial derivative] X × I) [union] (V\X). Then W₀ is a cobordism from M₀ to N₀ = [partial derivative]V, and the inclusion M₀ -> W₀ is a homology equivalence by excision, but W₀ is in general not an h-cobordism on M₀. However, if we stabilize the inclusion X [subset] V three times, and perform the corresponding constructions, then the resulting cobordism is an h-cobordism.

In more detail, we have a codimension zero inclusion and homotopy equivalence X × J³ [subset] V × J³. Choosing an interior collar c : [partial derivative] (X × J³)× I -> X × J³ on the boundary of X × J³, we let M = c([partial derivative] (X × J³)×1), N = [partial derivative] (V × J³) and

W = c([partial derivative] (X × J³) × I) [[union] (V × J³ \ X × J³) .

Then W is a cobordism from M to N. The three inclusions M [subset] X × J³, N [subset] V × J³ and W [subset] V × J³ are all π₁-isomorphisms (because any nullhomotopy in V × J³ of a loop in N can be deformed away from the interior of V times some interior point of J³, and then into N, and similarly in the two other cases). Since X × J³ [subset] V × J³ is a homotopy equivalence, it follows that both M [subset] W and N [subset] W are π₁-isomorphisms. By excision, it follows that M [subset] W is a homology equivalence, now with arbitrary local coefficients. By the universal coefficient theorem, and Lefschetz duality for the compact manifold W, it follows that N [subset] W is a homology equivalence, again with arbitrary local coefficients. Hence both M [subset] W and N [subset] W are homotopy equivalences, and W is an h-cobordism on M.

In the following definitions, we specify one model [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (M) for the stable PL h-cobordism space H^PL(M), based on a category of compact polyhedra and simple maps. In the next two sections we will re-express this polyhedral model: first in terms of a category of finite simplicial sets and simple maps, and then in terms of the algebraic K-theory of spaces.

Definition 1.1.5. A PL map f : K -> L of compact polyhedra will be called a simple map if it has contractible point inverses, i.e., if f^-1(p) is contractible for each point p [member of] L. (A space is contractible if it is homotopy equivalent to a one-point space. It is, in particular, then non-empty.)

In this context, M.Cohen [Co67, Thm. 11.1] has proved that simple maps (which he called contractible mappings) are simple homotopy equivalences. Two compact polyhedra are thus of the same simple homotopy type if and only if they can be linked by a finite chain of simple maps. The composite of two simple maps is always a simple map. This follows from Proposition 2.1.3 in Chapter 2, in view of the possibility of triangulating polyhedra and PL maps. Thus we can interpret the simple homotopy types of compact polyhedra as the path components of (the nerve of) a category of polyhedra and simple maps.

Definition 1.1.6. Let K be a compact polyhedron. We define a simplicial category [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of compact polyhedra containing K as a deformation retract, and simple PL maps between these. In simplicial degree 0, the objects are compact polyhedra L equipped with a PL embedding and homotopy equivalence K -> L. The morphisms f : L -> L' are the simple PL maps that restrict to the identity on K, via the given embeddings. A deformation retraction L -> K exists for each object, but a choice of such a map is not part of the structure.

In simplicial degree q, the objects of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are PL Serre fibrations (= PL maps whose underlying continuous map of topological spaces is a Serre fibration) of compact polyhedra π : E -> Δ^q, with a PL embedding and homotopy equivalence K × Δ^q -> E over Δ^q from the product fibration pr : K × Δ^q -> Δ^q. The morphisms f : E -> E' of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are the simple PL fiber maps over Δ^q that restrict to the identity on K × Δ^q, via the given embeddings.

Each PL embedding K -> K' of compact polyhedra induces a (forward) functor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that takes K -> L to K' -> K' [union]_K L, and similarly in parametrized families. The pushout K'?K L exists as a polyhedron, because both K -> K' and K -> L are PL embeddings. This makes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a covariant functor in K, for PL embeddings. There is a natural stabilization map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that takes K -> L to K × J -> L × J, and similarly in parametrized families. It is a homotopy equivalence by Lemma 4.1.12 in Chapter 4.

As in the following definition, we often regard a simplicial set as a simplicial category with only identity morphisms, a simplicial category as the bisimplicial set given by its degreewise nerve (Definition 2.2.1), and a bisimplicial set as the simplicial set given by its diagonal. A map of categories, i.e., a functor, is a homotopy equivalence if the induced map of nerves is a weak homotopy equivalence. See [Se68, §2], [Qu73, §1] or [Wa78a, §5] for more on these conventions.

Definition 1.1.7. Let M be a compact PL manifold. There is a natural map of simplicial categories

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

that takes (W, c) to the underlying compact polyhedron of the h-cobordism W, with the PL embedding and homotopy equivalence provided by the collar c : M × I -> W, and views PL bundles over Δ^q as being particular cases of

PL Serre fibrations over Δ^q. It commutes with the stabilization maps, and therefore induces a natural map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Here is the PL manifold part of the stable parametrized h-cobordism theorem.

Theorem 1.1.8. Let M be a compact PL manifold. There is a natural homotopy equivalence

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

More precisely, there is a natural chain of homotopy equivalences

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

and H^PL(M)^c [??] H^PL(M).

By the argument of [Wa82, p. 175], which we explain below, it suffices to prove Theorem 1.1.8 when M is a codimension zero submanifold of Euclidean space, or a little more generally, when M is stably framed (see Definition 4.1.2). The proof of the stably framed case will be given in Chapter 4, and is outlined in Section 4.1. Cf. diagram (4.1.13).

Remark 1.1.9 (Reduction of Theorem 1.1.8 to the stably framed case). Here we use a second homotopy equivalent model H(M)^r for the h-cobordism space of M, where each h-cobordism W comes equipped with a choice of a CAT retraction r : W -> M, and similarly in parametrized families. The forgetful map H(M)^r -> H(M) is a weak homotopy equivalence, because each inclusion M [subset] W is a cofibration and a homotopy equivalence. For each CAT disc bundle υ: N -> M there is a pullback map υ^! : H(M)^r -> H(N)^r, which takes an h-cobordism W on M with retraction r : W -> M to the pulled-back h-cobordism N ×_M W on N, with the pulled-back retraction.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If τ: M × J^k -> N is a second CAT disc bundle, so that the composite υτ equals the projection pr : M × J^k -> M, then (υτ)^! equals the k-fold stabilization map τ^!υ^! = σ^k. Hence there is a commutative diagram

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

According to Haefliger–Wall [HW65, Cor. 4.2], each compact PL manifold M admits a stable normal disc bundle υ: N -> M, with N embedded with co-dimension zero in some Euclidean n-space. Furthermore, PL disc bundles admit stable inverses. Let τ: M × J^k -> N be the disc bundle in such a stable inverse to υ, such that υτ is isomorphic to the product k-disc bundle over M, and τ (υ × id) is isomorphic to the product k-disc bundle over N. By the diagram above, pullback along υ and τ define homotopy inverse maps

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

after stabilization.

Likewise, there is a homotopy equivalent variant [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with a (contractible) choice of PL retraction r : L -> M for each polyhedron L containing M, and similarly in parametrized families. There is a simplicial functor [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by the pullback property of simple maps (see Proposition 2.1.3). It is a homotopy equivalence, because each stabilization map s is a homotopy equivalence by Lemma 4.1.12. Thus it suffices to prove Theorem 1.1.8 for N, which is stably framed, in place of M.

(Continues...)

Excerpted from Spaces of PL Manifolds and Categories of Simple Mapsby Friedhelm Waldhausen Bjørn Jahren John Rognes Copyright © 2013 by Princeton University Press. Excerpted by permission of PRINCETON UNIVERSITY PRESS. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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