Surgery Theory: Foundations (Grundlehren der mathematischen Wissenschaften, 362)

This monograph provides a comprehensive introduction to surgery theory, the main tool in the classification of manifolds. Surgery theory was developed to carry out the so-called Surgery Program, a basic strategy to decide whether two closed manifolds are homeomorphic or diffeomorphic. This book provides a detailed explanation of all the ingredients necessary for carrying out the surgery program, as well as an in-depth discussion of the obstructions that arise. The components include the surgery step, the surgery obstruction groups, surgery obstructions, and the surgery exact sequence. This machinery is applied to homotopy spheres, the classification of certain fake spaces, and topological rigidity. The book also offers a detailed description of Ranicki's chain complex version, complete with a proof of its equivalence to the classical approach developed by Browder, Novikov, Sullivan, and Wall. This book has been written for learning surgery theory and includes numerous exercises. With full proofs and detailed explanations, it also provides an invaluable reference for working mathematicians. Each chapter has been designed to be largely self-contained and includes a guide to help readers navigate the material, making the book highly suitable for lecture courses, seminars, and reading courses. Preface Contents Chapter 1 Introduction 1.1 Some Classical Problems that Can Be Attacked by Surgery Theory 1.2 Overview of the Contents of this Book 1.3 Outlook 1.4 How to Use this Book 1.5 Prerequisites 1.6 Acknowledgement Chapter 2 The s-Cobordism Theorem 2.1 Introduction 2.2 Handlebody Decompositions 2.3 Handlebody Decompositions and CW-Structures 2.4 Reducing the Handlebody Decomposition 2.5 Handlebody Decompositions and Whitehead Groups 2.6 Notes Chapter 3 Whitehead Torsion 3.1 Introduction 3.2 Whitehead Groups 3.3 Algebraic Approach to Whitehead Torsion 3.4 Geometric Approach to Whitehead Torsion 3.5 Reidemeister Torsion and (Generalised) Lens Spaces 3.5.1 Lens Spaces 3.5.2 Generalised Lens Spaces 3.5.3 Homotopy Classification of Generalised Lens Spaces 3.5.4 Reidemeister Torsion 3.5.5 Simple Homotopy Classification of Generalised Lens Spaces 3.5.6 A Variant of Reidemeister Torsion 3.5.7 The Classification of Lens Spaces 3.5.8 The Homeomorphism Classification of Generalised Lens Spaces 3.6 The Spherical Space Form Problem 3.7 Notes Chapter 4 The Surgery Step and ξ-Bordism 4.1 Introduction 4.2 The The CW-Version of the Surgery Step 4.3 Motivation for the Surgery Step 4.4 Motivation for the Bundle Data 4.5 Immersions and Embeddings 4.6 The Surgery Step 4.7 The Pontrjagin–Thom Construction 4.8 Notes Chapter 5 Poincaré Duality 5.1 Introduction 5.2 Local Coefficients 5.3 Passage to Group Rings 5.4 The Determinant Line Bundle 5.5 The Intrinsic Fundamental Class 5.6 Poincaré Duality 5.6.1 Rings with Involutions 5.6.2 Poincaré Complexes 5.6.3 The Signature 5.6.4 The Degree of a Map 5.7 Notes Chapter 6 The Spivak Normal Structure 6.1 Introduction 6.2 Spherical Fibrations 6.3 The Thom Isomorphism 6.4 The Normal Fibration of a Connected Finite CW-Complex 6.5 The Spivak Normal Structure and Its Main Properties 6.6 Existence of Poincaré Complexes without Vector Bundle Reduction 6.7 Characterisation of Poincaré Duality in Terms of the Normal Fibration 6.8 The Existence of the Spivak Normal Fibration 6.9 Spanier–Whitehead Duality 6.10 The Uniqueness of the Spivak Normal Fibration 6.11 Notes Chapter 7 Normal Maps and the Surgery Problem 7.1 Introduction 7.2 An Informal Preview 7.3 Rank k Normal Invariants 7.4 Normal Maps 7.5 Notes Chapter 8 The Even-Dimensional Surgery Obstruction 8.1 Introduction 8.1.1 The Nature of the Surgery Obstruction 8.1.2 The Intrinsic Versions 8.1.3 The Simply Connected Case 8.1.4 The Surgery Problem Relative Boundary andWall’s Realisation Theorem 8.2 Normal Γ-Maps 8.3 Intersection and Self-intersection Pairings 8.3.1 Intersections of Immersions 8.3.2 Self-intersections of Immersions 8.4 Surgery Kernels and Forms 8.4.1 Surgery Kernels 8.4.2 Symmetric Forms and Surgery Kernels 8.4.3 Quadratic Forms and Surgery Kernels 8.5 Even-Dimensional L-Groups 8.5.1 The Definition of L2k (R) via Forms 8.5.2 The L-Groups of Z in Dimensions n = 4k 8.5.3 The L-Groups of Z in Dimensions n = 4k + 2 8.6 The Even-Dimensional Surgery Obstruction in the Universal Covering Case 8.6.1 Normal Maps of Pairs 8.6.2 Surgery Kernels for Maps of Pairs 8.6.3 Proof of Theorem 8.112 8.7 The Intrinsic L-Group and Surgery Obstruction in Even Dimensions 8.7.1 Conjugation Invariant Functors 8.7.2 Half-Conjugation Invariant Functors 8.7.3 The Definition of Intrinsic L-Groups in Even Dimensions 8.7.4 The Intrinsic Even-Dimensional Surgery Obstruction 8.7.5 Bordism Invariance in Even Dimensions for a Not Necessarily Cylindrical Target 8.7.6 The Simply Connected Case in Even Dimensions 8.7.7 The Role of the Bundle Data in the Highly Connected Case 8.8 The Even-Dimensional Surgery Obstruction Relative Boundary 8.8.1 Normal Maps of Pairs 8.8.2 Normal Bordism for Normal Maps of Pairs 8.8.3 The Intrinsic Even-Dimensional Surgery Obstruction Relative Boundary 8.8.4 Kernels for Maps of Triads 8.9 Realisation of Even-Dimensional Surgery Obstructions 8.10 Notes Chapter 9 The Odd-Dimensional Surgery Obstruction 9.1 Introduction 9.2 Odd-Dimensional L-Groups 9.2.1 The Definition of L2k+1(R) via Formations 9.2.2 Presentations of Formations 9.2.3 The Definition of L2k+1(R) via Automorphisms 9.2.4 The Odd-Dimensional L-Groups of Z 9.3 The Odd-Dimensional Surgery Obstruction in the Universal Covering Case 9.3.1 The Kernel Formation 9.3.2 The Bordism Formation 9.3.3 Proof of Theorem 9.60 9.4 The Intrinsic Surgery Obstruction in Odd Dimensions 9.4.1 The Definition of Intrinsic L-Group in Odd Dimensions 9.4.2 The Intrinsic Odd-Dimensional Surgery Obstruction 9.4.3 Bordism Invariance in Odd Dimensions for Not Necessarily Cylindrical Target 9.4.4 The Simply Connected Case in Odd Dimensions 9.5 The Odd-Dimensional Surgery Obstruction Relative Boundary 9.6 Realisation of Odd-Dimensional Surgery Obstructions 9.7 Notes Chapter 10 Decorations and the Simple Surgery Obstruction 10.1 Introduction 10.2 Decorated L-Groups 10.3 Reidemeister U-Torsion 10.4 The Simple Surgery Obstruction 10.4.1 The Even-Dimensional Case 10.4.2 The Odd-Dimensional Case 10.4.3 Main Properties of the Simple Surgery Obstruction 10.5 Notes Chapter 11 The Geometric Surgery Exact Sequence 11.1 Introduction 11.2 The Geometric Structure Set 11.3 The Set of Normal Maps 11.4 The Surgery Obstruction Groups 11.5 The Geometric Surgery Exact Sequence 11.6 The Piecewise Linear and Topological Categories 11.7 Rigidity 11.8 Group Structures in the Surgery Exact Sequences 11.9 Some Information about G/O, G/PL and G/TOP 11.10 Functorial Properties of the Geometric Surgery Exact Sequence 11.11 Notes Chapter 12 Homotopy Spheres 12.1 Introduction 12.2 The Group of Homotopy Spheres 12.3 The Surgery Exact Sequence for Homotopy Spheres 12.4 The J-Homomorphism and Stably Framed Bordism 12.5 The Computation of bPn+1 12.6 The Group Θn/bPn+1 12.7 The Kervaire–Milnor Braid 12.8 Notes Chapter 13 The Geometric Surgery Obstruction Group and Surgery Obstruction 13.1 Introduction 13.2 Surgery on and Normal Bordisms for Pairs 13.2.1 Short Review of Normal Bordism for Manifolds Relative Boundary 13.2.2 Surgery on the Boundary 13.3 The π-π-Theorem 13.3.1 Algebraic Proof of the π-π-Theorem 13.4 13.3.2 Preliminaries for the Proof of the π-π-Theorem 13.4 13.3.3 Proof of the π-π-Theorem 13.4 in the Even-Dimensional Case 13.3.4 Proof of the π-π-Theorem 13.4 in the Odd-Dimensional Case 13.4 The Geometric Surgery Obstruction Group and Surgery Obstruction 13.4.1 The Construction of the Geometric Surgery Groups 13.4.2 Making the Reference Map a π0- and π1-Isomorphism 13.4.3 The Geometric Surgery Obstruction 13.4.4 Identifying the Geometric Surgery Obstruction Groups with the Algebraic L-Groups 13.5 The Geometric Rothenberg Sequence 13.6 The Geometric Shaneson Splitting 13.7 Notes Chapter 14 Chain Complexes 14.1 Introduction 14.2 Modules over Associative Rings 14.3 Modules over Rings with Involution 14.4 Some Basic Chain Complex Constructions 14.5 Further Chain Complex Constructions over Rings with Involution 14.6 Homotopy Theory of Chain Complexes 14.6.1 Chain Homotopy 14.6.2 Pushouts 14.6.3 Mapping Cylinders, Mapping Cones and Suspensions 14.6.4 Cofibrations and Homotopy Cofibrations 14.6.5 Homotopy Pushouts 14.6.6 Homotopy Cocartesian Squares 14.6.7 Pullbacks 14.6.8 Homotopy Fibres 14.6.9 Relating the Homotopy Cofibre and the Homotopy Fibre 14.6.10 Homotopy Pullbacks 14.6.11 Homotopy Colimits of Sequences 14.6.12 Homotopy Limits of Inverse Systems of Maps 14.6.13 Chain Complexes of Projective Modules 14.7 Notes Chapter 15 Algebraic Surgery 15.1 Introduction 15.2 Overview 15.2.1 Statement of the Main Results 15.2.2 Some Basic Notions and Explanations 15.2.3 Outline of the Proofs 15.3 Structured Chain Complexes 15.3.1 Basic Definitions 15.3.2 The Symmetric Construction 15.3.3 Products 15.3.4 Homotopy Invariance 15.3.5 The Suspension Map 15.3.6 The Quadratic Construction 15.3.7 Algebraic Poincaré Complexes 15.3.8 Equivariant S-Duality and Umkehr Maps 15.3.9 Wu Classes 15.3.10 Homotopy Z[Z/2]-Chain Maps 15.4 Forms and Formations 15.4.1 Homotopy Theory of Highly Connected Structured Chain Complexes 15.4.2 Revision of Forms and Formations 15.4.3 Complexes versus Forms and Formations 15.5 L-Groups in Terms of Chain Complexes 15.5.1 Pairs 15.5.2 Cobordisms 15.5.3 The Algebraic Thom Construction 15.5.4 The Algebraic Boundary 15.5.5 Algebraic Surgery 15.6 The Identification of the L-Groups 15.7 The Identification of the Surgery Obstructions 15.7.1 Euler Classes 15.7.2 Self-Intersections 15.7.3 Putting it Together 15.8 Simple L-groups and Simple Surgery Obstruction in Terms of Chain Complexes 15.9 Applications of Algebraic Surgery 15.9.1 Product formulas 15.9.2 The Algebraic Surgery Transfers for Fibrations 15.9.3 The Algebraic Rothenberg Sequence 15.9.4 The Algebraic Shaneson Splitting 15.9.5 The Algebraic Surgery Exact Sequence 15.9.6 The Total Surgery Obstruction 15.10 Notes Chapter 16 Brief Survey of Computations of L-Groups 16.1 Introduction 16.2 More Decorated L-Groups 16.3 Finite Groups 16.4 Torsionfree Groups 16.5 The Farrell–Jones Conjecture 16.5.1 The K-theoretic Farrell–Jones Conjecture with Coefficients in Additive G-Categories 16.5.2 The L-theoretic Farrell–Jones Conjecture in Additive G-Categories with Involution 16.5.3 Reducing the Family 16.5.4 The Full Farrell–Jones Conjecture 16.6 Notes Chapter 17 The Homotopy Type of G/TOP, G/PL and G/O 17.1 Introduction 17.2 Localisation 17.3 G/TOP 17.4 G/PL 17.5 G/O 17.6 H-Space Structures on G/TOP and G/PL 17.7 Splitting Invariants 17.8 Milnor Manifolds and Kervaire Manifolds 17.9 Notes Chapter 18 Computations of Topological Structure Sets of some Prominent Closed Manifolds 18.1 Introduction 18.2 Fake Spaces 18.3 Products of Spheres 18.4 Complex Projective Spaces 18.5 Quaternionic Projective Spaces 18.6 The Join and the Transfer 18.7 The ρ-Invariant 18.8 Real Projective Spaces 18.9 Lens Spaces 18.10 Aspherical Manifolds 18.11 Some Torus Bundles over Lens Spaces 18.12 Notes Chapter 19 Topological Rigidity 19.1 Introduction 19.2 Topological Rigidity and the Surgery Exact Sequence 19.3 Aspherical Closed Manifolds 19.3.1 Homotopy Classification of Aspherical CW-Complexes 19.3.2 Low Dimensions 19.3.3 Non-Positive Curvature 19.3.4 Torsionfree Discrete Subgroups of Almost Connected Lie Groups 19.3.5 Hyperbolisation 19.3.6 Exotic Aspherical Closed Manifolds 19.3.7 Non-Aspherical Closed Manifolds 19.4 The Borel Conjecture 19.5 Non-Aspherical Topologically Rigid Closed Manifolds 19.5.1 Spheres 19.5.2 3-Manifolds 19.5.3 Products of Two Spheres 19.5.4 Homology Spheres 19.5.5 Connected Sums 19.5.6 Homology Isomorphisms and Topological Rigidity 19.6 Smooth Rigidity 19.7 Notes Chapter 20 Modified Surgery 20.1 Introduction 20.2 Stable Diffeomorphisms 20.3 Bordism Groups Associated to Spaces over BO 20.4 Modified Surgery 20.5 Normal k-Type 20.5.1 Embedding Spaces and Gauss Maps up to Contractible Choice 20.5.2 The Moore–Postnikov Factorisation 20.5.3 Normal k-Type 20.6 Classification up to Stable Diffeomorphisms 20.7 Tangential Approach 20.7.1 Passing to the Tangent Bundle 20.7.2 Connected Sum 20.8 Applications 20.8.1 Complete Intersections 20.8.2 Homogeneous Spaces 20.8.3 4-Manifolds 20.9 Notes Chapter 21 Solutions of the Exercises Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7 Chapter 8 Chapter 9 Chapter 10 Chapter 11 Chapter 12 Chapter 13 Chapter 14 Chapter 15 Chapter 16 Chapter 17 Chapter 18 Chapter 19 Chapter 20 References Notation Index