The study of triangulations of topological spaces has always been at the root of geometric topology. Among the most studied triangulations are piecewise linear triangulations of high-dimensional topological manifolds. Their study culminated in the late 1960s–early 1970s in a complete classification in the work of Kirby and Siebenmann. It is this classification that we discuss in this book, including the celebrated Hauptvermutung and Triangulation Conjecture.
The goal of this book is to provide a readable and well-organized exposition of the subject, which would be suitable for advanced graduate students in topology. An exposition like this is currently lacking.
Author(s): Yuli Rudyak
Edition: 1
Publisher: World Scientific
Year: 2016
Language: English
Pages: 126
Tags: Topology; Piecewise Linear Topology; Topological Manifolds
Preface vii
Introduction xi
Graph xix
1. Architecture of the Proof 1
1.1 Some Definitions, Notation, and Conventions . . . . . . . 1
1.2 Principal Fibrations ..................... 4
1.3 Preliminaries on Classifying Spaces . . . . . . . . . . . . . 7
1.4 Structures on Manifolds and Bundles . . . . . . . . . . . . 16
1.5 From Manifolds to Bundles ................. 22
1.6 Homotopy PL Structures on T^n×D^k . . . . . . . . . . . 26
1.7 The Product Structure Theorem .............. 28
1.8 Non-contractibility of TOP/PL............... 30
1.9 Homotopy Groups of TOP/PL ............... 32
2. Normal Invariant 37
2.1 Preliminaries on Stable Duality ............... 37
2.2 Thom Spaces. Proof of Theorem 1.5.6 . . . . . . . . . . . 41
2.3 Normal Morphisms and Normal Bordisms . . . . . . . . . 45
2.4 The Sullivan Map s:[M,G/PL]→P_dimM . . . . . . . . . 49
2.5 The Homotopy Type of G/PL[2] .............. 52
2.6 SplittingTheorems...................... 59
2.7 DetectingFamilies ...................... 65
2.8 Normal Invariant of a Homeomorphism V→T^n×S^k . . 67
3. Applications and Consequences of the Main Theorem 69
3.1 The Space G/TOP...................... 69
3.2 The Map a:TOP/PL→G/PL .............. 73
3.3 Normal Invariant of a Homeomorphism. . . . . . . . . . . 76
3.4 Kirby–Siebenmann Class................... 77
3.5 Several Examples....................... 80
3.6 Invariance of Characteristic Classes . . . . . . . . . . . . . 84
Appendix 91
Bibliography 95
List of Symbols 103
Index 105
