Annals of Mathematics Studies Number 186 Spaces of PL Manifolds and Categories of Simple Maps Friedhelm Waldhausen, Bjørn Jahren and John Rognes PRINCETON UNIVERSITY PRESS PRINCETON AND OXFORD 2013 Copyright (cid:13)c 2013 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Waldhausen, Friedhelm, 1938– Spaces of PL manifolds and categories of simple maps / Friedhelm Wald- hausen, Bjørn Jahren, and John Rognes. pages cm. – (Annals of mathematics studies ; no. 186) Includes bibliographical references and index. ISBN 978-0-691-15775-7 (hardcover : alk. paper) – ISBN 978-0-691-15776-4 (pbk. : alk. paper) 1. Piecewise linear topology. 2. Mappings (Mathematics) I. Jahren, Bjørn, 1945– II. Rognes, John. III. Title. QA613.4.W35 2013 514’.22–dc23 2012038155 British Library Cataloging-in-Publication Data is available Thepublisherwouldliketoacknowledgetheauthorsofthisvolumeforprovid- ing the camera-ready copy from which this book was printed. This book has been composed in AMS-TEX. Printed on acid-free paper ∞ Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 Contents Introduction 1 1. The stable parametrized h-cobordism theorem 7 1.1. The manifold part 7 1.2. The non-manifold part 13 1.3. Algebraic K-theory of spaces 15 1.4. Relation to other literature 20 2. On simple maps 29 2.1. Simple maps of simplicial sets 29 2.2. Normal subdivision of simplicial sets 34 2.3. Geometric realization and subdivision 42 2.4. The reduced mapping cylinder 56 2.5. Making simplicial sets non-singular 68 2.6. The approximate lifting property 74 2.7. Subdivision of simplicial sets over ∆q 83 3. The non-manifold part 99 3.1. Categories of simple maps 99 3.2. Filling horns 108 3.3. Some homotopy fiber sequences 119 3.4. Polyhedral realization 126 3.5. Turning Serre fibrations into bundles 131 3.6. Quillen’s Theorems A and B 134 4. The manifold part 139 4.1. Spaces of PL manifolds 139 4.2. Spaces of thickenings 150 4.3. Straightening the thickenings 155 Bibliography 175 Symbols 179 Index 181 Spaces of PL Manifolds and Categories of Simple Maps Introduction We present a proof of the stable parametrized h-cobordism theorem, which we choose to state as follows: Theorem 0.1. There is a natural homotopy equivalence HCAT(M)≃ΩWhCAT(M) for each compact CAT manifold M, with CAT = TOP, PL or DIFF. Here HCAT(M) denotes a stable CAT h-cobordism space defined in terms of manifolds, whereas WhCAT(M) denotes a CAT Whitehead space defined in terms of algebraic K-theory. We specify functorial models for these spaces in Definitions 1.1.1, 1.1.3, 1.3.2 and 1.3.4. See also Remark 1.3.1 for comments about our notation. This is a stable range extension to parametrized families of the classical h- ands-cobordismtheorems. SuchatheoremwasfirststatedbyA.E.Hatcherin [Ha75, Thm.9.1], buthisproofswereincomplete. Theaimofthepresentbook istoprovide afull proofofthiskeyresult, which providesthelink between the geometric topology of high-dimensional manifolds and their automorphisms, and the algebraic K-theory of spaces and structured ring spectra. The book is based on a manuscript by the first author, with the same title, which was referredtoas“toappear”in[Wa82]andas“toappear(since’79)”in[Wa87b]. Wefirstrecalltheclassicalh-ands-cobordismtheorems. LetM beacompact manifold of dimension d, either in the topological, piecewise-linear or smooth differentiable sense. An h-cobordism on M is a compact (d+1)-manifold W whose boundarydecomposes asaunion ∂W ∼=M∪N of two codimension zero submanifolds along their common boundary, such that each inclusion M ⊂W and N ⊂ W is a homotopy equivalence. Two h-cobordisms W and W′ on M areisomorphicifthereisahomeomorphism, PLhomeomorphismordiffeomor- phismW ∼=W′, asappropriateforthegeometriccategory, thatrestrictstothe identity on M. An h-cobordism is said to be trivial if it is isomorphic to the producth-cobordismW =M×[0,1],containingM asM×0. (Thereisalittle technical point here about corners in the DIFF case, which we gloss over.) Assume for simplicity that M is connected and has a chosen base point, let π = π (M) be its fundamental group, and let Wh (π) = K (Z[π])/(±π) 1 1 1 be the Whitehead group of π, which is often denoted Wh(π). To each h- cobordism W on M there is associated an element τ(W,M) ∈ Wh (π), called 1 its Whitehead torsion [Mi66, §9]. Trivial h-cobordisms have zero torsion. In general, an h-cobordism with zero torsion is called an s-cobordism. The h- cobordism theorem of S.Smale (concerning the simply-connected case, when 2 INTRODUCTION the Whitehead group is trivial) and the s-cobordism theorem of D.Barden, B.MazurandJ.R.Stallings(forarbitraryfundamentalgroups),assertford≥5 that the Whitehead torsion defines a one-to-one correspondence {h-cobordisms on M}/(iso)−→∼= Wh (π) 1 [W]7→τ(W,M) betweentheisomorphismclassesofh-cobordismsonM andtheelementsofthe Whiteheadgroup. Thus,inthesedimensionsthes-cobordismsarepreciselythe trivial h-cobordisms. This result should be viewed as the computation of the set of path com- ponents of the space of h-cobordisms, and the aim of a parametrized h-cobordism theorem is to determine the homotopy type of this space. More precisely,forM aCATmanifoldthereisaspaceHCAT(M)thatclassifiesCAT bundlesofh-cobordismsonM,andthes-cobordismtheoremasserts(ford≥5) that there is a natural bijection π HCAT(M)∼=Wh (π). 0 1 Weshallhavetosettleforastableparametrizedh-cobordismtheorem,which provides a homotopy equivalent model for the stabilized h-cobordism space HCAT(M)=colimHCAT(M ×[0,1]k). k The model in question is defined in algebraic K-theoretic terms, much like the definitionoftheWhiteheadgroupbymeansofthealgebraicK -groupofthein- 1 tegralgroupringZ[π]. WewillexpresseachCATWhiteheadspaceWhCAT(M) in terms of the algebraic K-theory space A(M), which was introduced (largely forthispurpose)bythefirstauthor. Asstatedattheoutset, themodelforthe homotopy type of HCAT(M) will be the based loop space ΩWhCAT(M). The PL and TOP Whitehead spaces WhPL(M) = WhTOP(M) will be the same, because it is known by triangulation theory that HPL(M) ≃ HTOP(M) for PL manifolds of dimension d ≥ 5, but WhDIFF(M) will be different. (This wordplay is due to Hatcher.) By definition, the algebraic K-theory A(M) of the space M is the loop space Ω|hS R (M)| of the geometric realization of the subcategory of homo- • f topy equivalences in the S -construction on the category with cofibrations and • weak equivalences R (M) of finite retractive spaces over M. See [Wa85, §2.1]. f IterationoftheS -constructionspecifiesapreferredsequenceofhigherdeloop- • ings of A(M), so we may view that space as the underlying infinite loop space of a spectrum A(M), in the sense of algebraic topology. Letting ∗ denote a one-point space, the PL Whitehead space is defined so that there is a natural homotopy fiber sequence h(M;A(∗))−→α A(M)−→WhPL(M) for each space M, where h(M;A(∗)) = Ω∞(A(∗) ∧ M ) is the unreduced + generalizedhomologyofM withcoefficientsinthespectrumA(∗),andαisthe naturalassemblymaptothehomotopyfunctorA(M). Thestableparametrized PL h-cobordism theorem can therefore be restated as follows.