Über die Autorin bzw. den Autor

Friedhelm Waldhausen is professor emeritus of mathematics at Bielefeld University. Bjørn 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...