The Hauptvermutung Book A collection of papers on the topology of manifolds by A.A.Ranicki (editor), A.J.Casson, D.P.Sullivan, M.A.Armstrong, C.P.Rourke, and G.E.Cooke M. A. Armstrong Mathematics Department, University of Durham, Durham DH1 3LE, England, UK A. J. Casson Mathematics Department, University of California, Berkeley, CA 94720, USA G. E. Cookey A. A. Ranicki Department of Mathematics and Statistics, The University of Edinburgh, Edinburgh EH9 3JZ, Scotland, UK C. P. Rourke Mathematics Institute, University of Warwick, Coventry CV4 7AL, England, UK D. P. Sullivan Mathematics Department, City University of New York, Graduate Center, 33 West 42 Street, New York, NY10036, USA Contents Preface . 1 On the Hauptvermutung (by A.A.Ranicki, 1996) 1. Introduction . 3 2. The Polyhedral Hauptvermutung . 6 3. The Rochlin Invariant . 11 4. The Manifold Hauptvermutung . 13 5. Homology Manifolds . 23 References . 29 Generalisations and Applications of Block Bundles (by A.J.Casson, 1967) Introduction . 33 I. Block Bundles . 35 II. Homotopy Properties of Block Bundles . 43 III. Tangential Properties of Block Bundles . 49 IV. Periodicity of G=P L ....................55 V. Topologically Trivial Block Bundles . 59 References . 67 Triangulating Homotopy Equivalences and Homeomorphisms. Geometric Topology Seminar Notes (by D.P.Sullivan, 1967) Introduction . 69 I. Triangulating and Smoothing Homotopy Equivalences . 69 II. The Characteristic Bundle of a Homotopy Equivalence . 80 III. The Hauptvermutung . 93 References . 103 The Princeton Notes on the Hauptvermutung (by M.A.Armstrong, C.P.Rourke, G.E.Cooke) Preface (1996) . 105 I. The Hauptvermutung According to Lashof and Rothenberg (by M.A.Armstrong, 1968) 1. Introduction . 107 2. Splitting theorems . 110 3. Siebenmann's collaring theorems . 112 4. Proof of the splitting theorem . 116 5. Proof of the relative splitting theorem . 121 6. The trivialization problem . 123 References . 125 II. The Hauptvermutung according to Casson and Sullivan (by C.P.Rourke, 1968) 1. Introduction . 129 2. Notation and basic definitions . 132 3. An account of Sullivan theory . 138 4. Surgery obstructions . 143 5. The `canonical Novikov homotopy' . 148 6. Weaker hypotheses . 151 7. Refinements of the Main Theorem . 154 8. Block bundles and homotopy equivalences . 157 References . 162 III. The Hauptvermutung according to Casson and Sullivan (by G.E.Cooke, 1968) 1. Introduction . 165 2. Principal fibrations . 166 3. Postnikov systems . 167 4. Application to G=P L and the Hauptvermutung . 172 References . 186 Coda: Connection with the results of Kirby and Siebenmann (by C.P.Rourke, 1972) . 189 References . 190 Preface The Hauptvermutung is the conjecture that any two triangulations of a poly- hedron are combinatorially equivalent. The conjecture was formulated at the turn of the century, and until its resolution was a central problem of topology. Initially, it was verified for low-dimensional polyhedra, and it might have been expected that further development of high-dimensional topology would lead to a verification in all dimensions. However, in 1961 Milnor constructed high-dimensional polyhedra with combinatorially inequivalent triangulations, disproving the Hauptvermutung in general. These polyhedra were not manifolds, leaving open the Hauptvermu- tung for manifolds. The development of surgery theory led to the disproof of the high-dimensional manifold Hauptvermutung in the late 1960's. Unfortunately, the published record of the manifold Hauptvermutung has been incomplete, as was forcefully pointed out by Novikov in his lecture at the Browder 60th birthday conference held at Princeton in March 1994. This volume brings together the original 1967 papers of Casson and Sulli- van, and the 1968/1972 `Princeton notes on the Hauptvermutung' of Armstrong, Rourke and Cooke, making this work physically accessible. These papers include several other results which have become part of the folklore but of which proofs have never been published. My own contribution is intended to serve as an intro- duction to the Hauptvermutung, and also to give an account of some more recent developments in the area. In preparing the original papers for publication, only minimal changes of punctuation etc. have been made with the exception of the references, which have been completed and updated wherever possible. I should like to thank all who in various ways helped in the preparation of this volume : Tony Armstrong, Tony Bak, Andrew Casson, Cathy Hassell, Bruce Hughes, Sergei Novikov, Jonathan Rosenberg, Colin Rourke, Ron Stern, Dennis Sullivan, Simon Willerton and Victor Wu. A.A.R., Edinburgh, January 1996 1 2 1. introduction 3 On the Hauptvermutung by A. A. Ranicki x1. Introduction. An abstract simplicial complex K determines a topological space, the poly- hedron jKj.A triangulation (K; f) of a topological space X is a simplicial complex K together with a homeomorphism f : jK|−−!X. A topological space X is triangulable if it admits a triangulation (K; f). The topology of a triangulable space X is determined by the combinatorial topology of the simplicial complex K in any triangulation (K; f) of X. Hauptvermutung is short for die Hauptvermutung der kombinator- ischen Topologie, which is German for the main conjecture of combinatorial topology. The conjecture states that the combinatorial topology of a simplicial complex K is determined by the topology of the polyhedron jKj. More technically, the conjecture is that triangulations of homeomorphic spaces are combinatorially equivalent, i.e. become isomorphic after subdivision. A triangulable space would then have a canonical class of triangulations. The problem was formulated by Steinitz [44] and Tietze [48] in 1908, and there are statements in Kneser [20] and Alexandroff and Hopf [2, p.152]. The modern version of combinatorial topology is codified in the PL (piecewise linear) category, for which Rourke and Sanderson [35] is the standard reference. Simplicial Approximation Theorem. Every continuous map f : jK|−−!jLj between polyhedra is homotopic to the topological realization jf 0j : jKj = jK0|−−!jLj of a simplicial map f 0 : K0−−!L, where K0 is a simplicial subdivision of K. Thus every continuous map of polyhedra is homotopic to a PL map. It does not follow that a homeomorphism of polyhedra is homotopic to a PL homeomor- phism. Hauptvermutung. Every homeomorphism f : jK|−−!jLj between polyhedra is homotopic to the topological realization of a simplicial isomorphism f 0 : K0−−!L0, where K0;L0 are simplicial subdivisions of K; L, i.e. every homeomorphism of polyhedra is homotopic to a PL homeomorphism. This will also be called the Polyhedral Hauptvermutung, to distinguish it from the Manifold Hauptvermutung stated below. 4 ranicki The Simplicial Approximation Theorem shows that the homotopy theory of polyhedra is the same as the PL homotopy theory of simplicial complexes. Ever since Seifert and Threlfall [39] standard treatments of algebraic topology have used this correspondence to show that combinatorial homotopy invariants of simplicial complexes (e.g. simplicial homology, the simplicial fundamental group) are in fact homotopy invariants of polyhedra. The Hauptvermutung is not mentioned, allowing the reader to gain the false impression that the topology of polyhedra is the same as the PL topology of simplicial complexes. In fact, the Hauptvermutung has been known for some time to be false, although this knowledge has not yet filtered down to textbook level. A simplicial complex K is finite if and only if the polyhedron jKj is compact. The Hauptvermutung is only considered here for compact polyhedra. However, the resolution of the conjecture in this case requires an understanding of the difference between the PL and continuous topology of open PL manifolds, which is quantified by the Whitehead group. The Polyhedral Hauptvermutung is true in low dimensions : it was verified for 2-dimensional manifolds by the classification of surfaces, for all polyhedra of dimension ≤ 2 by Papakyriakopoulos [32], and by Mo¨ıse [28] for 3-dimensional manifolds. Milnor [25] obtained the first counterexamples to the Polyhedral Hauptver- mutung in 1961, using Reidemeister torsion and some results on non-compact manifolds of Mazur and Stallings to construct a homeomorphism of compact poly- hedra which is not homotopic to a PL homeomorphism. Stallings [43] generalized the construction, showing that any non-trivial h-cobordism determines a coun- terexample to the Polyhedral Hauptvermutung. These counterexamples arise as homeomorphisms of one-point compactifications of open PL manifolds, and so are non-manifold in nature. An m-dimensional combinatorial (or PL) manifold is a simplicial com- plex K such that linkK (σ) is a PL (m − jσj − 1)-sphere for each simplex σ 2 K. Manifold Hauptvermutung. Every homeomorphism f : jK|−−!jLj of the poly- hedra of compact m-dimensional PL manifolds is homotopic to a PL homeomor- phism. Following Milnor's disproof of the Polyhedral Hauptvermutung there was much activity in the 1960's aimed at the Manifold Hauptvermutung { first proving it in special cases, and then disproving it in general. 1. introduction 5 The Manifold Hauptvermutung is the rel @ version of the following conjecture : Combinatorial Triangulation Conjecture. Every compact m-dimensional topological manifold M can be triangulated by a PL manifold. The Manifold Hauptvermutung and Combinatorial Triangulation Conjecture hold in the low dimensions m ≤ 3. The 1963 surgery classification by Kervaire and Milnor of the latter's exotic