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Ends of Complexes Contents Bruce Hughes Andrew Ranicki Vanderbilt University University of Edinburgh Ends of complexes Contents Introduction page ix Chapter summaries xxii Part one Topology at in¯nity 1 1 End spaces 1 2 Limits 13 3 Homology at in¯nity 29 4 Cellular homology 43 5 Homology of covers 56 6 Projective class and torsion 65 7 Forward tameness 75 8 Reverse tameness 92 9 Homotopy at in¯nity 97 10 Projective class at in¯nity 109 11 In¯nite torsion 122 12 Forward tameness is a homotopy pushout 137 Part two Topology over the real line 147 13 In¯nite cyclic covers 147 14 The mapping torus 168 15 Geometric ribbons and bands 173 16 Approximate ¯brations 187 17 Geometric wrapping up 197 18 Geometric relaxation 214 19 Homotopy theoretic twist glueing 222 20 Homotopy theoretic wrapping up and relaxation 244 Part three The algebraic theory 255 21 Polynomial extensions 255 22 Algebraic bands 263 23 Algebraic tameness 268 24 Relaxation techniques 288 25 Algebraic ribbons 300 26 Algebraic twist glueing 306 27 Wrapping up in algebraic K- and L-theory 318 vii viii Ends of complexes Part four Appendices 325 Appendix A. Locally ¯nite homology with local coe±cients 325 Appendix B. A brief history of end spaces 335 Appendix C. A brief history of wrapping up 338 References 341 Index 351 Introduction We take `complex' to mean both a CW (or simplicial) complex in topology and a chain complex in algebra. An `end' of a complex is a subcomplex with a particular type of in¯nite behaviour, involving non-compactness in topology and in¯nite generation in algebra. The ends of manifolds are of greatest interest; we regard the ends of CW and chain complexes as tools in the investigation of manifolds and related spaces, such as strati¯ed sets. The interplay of the topological properties of the ends of manifolds, the homotopy theoretic properties of the ends of CW complexes and the algebraic properties of the ends of chain complexes has been an important theme in the classi¯cation theory of high dimensional manifolds for over 35 years. However, the gaps in the literature mean that there are still some loose ends to wrap up! Our aim in this book is to present a systematic exposition of the various types of ends relevant to manifold classi¯cation, closing the gaps as well as obtaining new results. The book is intended to serve both as an account of the existing applications of ends to the topology of high dimensional manifolds and as a foundation for future developments. We assume familiarity with the basic language of high dimensional man- ifold theory, and the standard applications of algebraic K- and L-theory to manifolds, but otherwise we have tried to be as self contained as possible. The algebraic topology of ¯nite CW complexes su±ces for the combinato- rial topology of compact manifolds. However, in order to understand the dif- ference between the topological and combinatorial properties it is necessary to deal with in¯nite CW complexes and non-compact manifolds. The clas- sic cases include the Hauptvermutung counterexamples of Milnor [96], the topological invariance of the rational Pontrjagin classes proved by Novikov [103], the topological manifold structure theory of Kirby and Siebenmann [84], and the topological invariance of Whitehead torsion proved by Chap- man [22]. The algebraic and geometric topology of non-compact manifolds has been a prominent feature in much of the recent work on the Novikov ix x Ends of complexes conjectures { see Ferry, Ranicki and Rosenberg [59] for a survey. (In these applications the non-compact manifolds arise as the universal covers of as- pherical compact manifolds, e.g. the Euclidean space Ri covering the torus T i = S1 £ S1 £ ::: £ S1 = BZi.) In fact, many current developments in topology, operator theory, di®erential geometry, hyperbolic geometry, and group theory are concerned with the asymptotic properties of non-compact manifolds and in¯nite groups { see Gromov [65], Connes [33] and Roe [135] for example. What is an end of a topological space? Roughly speaking, an end of a non-compact space W is a component of W nK for arbitrarily large compact subspaces K ⊆ W . More precisely : De¯nition 1. (i) A neighbourhood of an end in a non-compact space W is a subspace U ½ W which contains a component of W nK for a non-empty compact subspace K ½ W . (ii) An end ² of W is an equivalence class of sequences of connected open neighbourhoods W ⊃ U1 ⊃ U2 ⊃ ::: such that \1 cl (Ui) = ; i=1 subject to the equivalence relation (W ⊃ U1 ⊃ U2 ⊃ :::) » (W ⊃ V1 ⊃ V2 ⊃ :::) if for each Ui there exists j with Ui ⊆ Vj, and for each Vj there exists i with Vj ⊆ Ui. (iii) The fundamental group of an end ² is the inverse limit ¼ (²) = lim ¼ (U ) : 1 á 1 i i The theory of ends was initiated by Freudenthal [61] in connection with topological groups. The early applications of the theory concerned the ends of open 3-dimensional manifolds, and the ends of discrete groups (which are the ends of the universal covers of their classifying spaces). We are especially interested in the ends of manifolds which are `tame', and in extending the notion of tameness to other types of ends. An end of a manifold is tame if it has a system of neighbourhoods satisfying certain strong restrictions on the fundamental group and chain homotopy type. Any non-compact space W can be compacti¯ed by adding a point at in¯nity, W 1 = W [ f1g. A manifold end is `collared' if it can be compacti¯ed by a manifold, i.e. if the point at in¯nity can be replaced by a closed manifold boundary, allowing the end to be identi¯ed with the interior of a compact Introduction xi manifold with boundary. A high dimensional tame manifold end can be collared if and only if an algebraic K-theory obstruction vanishes. The theory of tame ends has found wide application in the surgery classi¯cation theory of high dimensional compact manifolds and strati¯ed spaces, and in the related controlled topology and algebraic K- and L-theory. Example 2. Let K be a connected compact space. (i) K £ [0; 1) has one end ², with connected open neighbourhoods Ui = K £ (i; 1) ½ K £ [0; 1) ; such that ¼1(²) = ¼1(K). (ii) K £ R has two ends ²+; ²¡, with connected open neighbourhoods + ¡ Ui = K £ (i; 1) ;Ui = K £ (¡1; ¡i) ½ K £ R ; § such that ¼1(² ) = ¼1(K). (iii) K £ R2 has one end ², with connected open neighbourhoods 2 2 2 2 Ui = K £ f(x; y) 2 R j x + y > i g ; such that ¼1(²) = ¼1(K) £ Z. Example 3. (i) Let W be a space with a proper map d : W ¡![0; 1) which ¡1 is onto, and such that the inverse images Ut = d (t; 1) ⊆ W (t ¸ 1) are connected. Then W has one end ² with connected open neighbourhoods 1 ¡1 T W ⊃ U1 ⊃ U2 ⊃ ::: such that cl(Ut) = d [t; 1), cl(Ui) = ;. i=0 (ii) Let (W; @W ) be a connected open n-dimensional manifold with con- nected compact boundary. Then W has one end ² if and only if there exists a proper map d :(W; @W )¡!([0; 1); f0g) which is transverse regular at N = f0; 1; 2;:::g ½ [0; 1), with the inverse images ¡1 (Wi; Mi;Mi+1) = d ([i; i + 1]; fig; fi + 1g)(i 2 N) connected compact n-dimensional cobordisms such that [1 (W; @W ) = ( Wi;M0) : i=0 (iii) Given connected compact n-dimensional cobordisms (Wi; Mi;Mi+1) (i 2 N) there is de¯ned a connected open n-dimensional manifold with 1S compact boundary (W; @W ) = ( Wi;M0). The union of Morse func- i=0 tions di :(Wi; Mi;Mi+1)¡!([i; i + 1]; fig; fi + 1g) de¯nes a proper map d :(W; @W )¡!([0; 1); f0g), and as in (ii) W has one end ². If the inclu- sions Mi¡!Wi, Mi+1¡!Wi induce isomorphisms in ¼1 then ¼1(M0) = ¼1(W0) = ¼1(M1) = ::: = ¼1(W ) = ¼1(²) : xii Ends of complexes De¯nition 4. An end ² of an open n-dimensional manifold W can be collared if it has a neighbourhood of the type M£[0; 1) ½ W for a connected closed (n ¡ 1)-dimensional manifold M. Example 5. (i) An open n-dimensional manifold with one end ² is (home- omorphic to) the interior of a closed n-dimensional manifold if and only if ² can be collared. More generally, if W is an open n-dimensional manifold with compact boundary @W and one end ², then there exists a compact n-dimensional cobordism (L; @W; M) with LnM homeomorphic to W rel @W if and only if ² can be collared. (ii) If (V; @V ) is a compact n-dimensional manifold with boundary then for any x 2 V n@V the complement W = V nfxg is an open n-dimensional mani- fold with a collared end ² and @W = @V , with a neighbourhood M£[0; 1) ½ W for M = Sn¡1. The one-point compacti¯cation of W is W 1 = V . The compacti¯cation of W provided by (i) is L = cl(V nDn), for any neigh- bourhood Dn ½ V n@V of x, with (L; @W; M) = (W [ Sn¡1; @V; Sn¡1). Stallings [154] used engul¯ng to prove that if W is a contractible open n-dimensional PL manifold with one end ² such that ¼1(²) = f1g and n ¸ 5 then W is PL homeomorphic to Rn { in particular, the end ² can be collared. Let (W; @W ) be an open n-dimensional manifold with compact boundary and one end ².
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