The Novikov Conjecture

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ISBN: 9783764371418 | 3764371412
Cover: Paperback
Copyright: 3/30/2005

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These lecture notes contain a guided tour to the Novikov Conjecture and related conjectures due to Baum-Connes, Borel and Farrell-Jones. They begin with basics about higher signatures, Whitehead torsion and the s-Cobordism Theorem. Then an introduction to surgery theory and a version of the assembly map is presented. Using the solution of the Novikov conjecture for special groups some applications to the classification of low dimensional manifolds are given. Finally, the most recent developments concerning these conjectures are surveyed, including a detailed status report. The prerequisites consist of a solid knowledge of the basics about manifolds, vector bundles, (co-) homology and characteristic classes.

Introduction

A Motivating Problem (K.)

(4)

Dimensions ≤ 4

(1)

Dimension 6

(2)

Dimension 5

(1)

Introduction to the Novikov and the Borel Conjecture (L.)

(6)

The Original Formulation of the Novikov Conjecture

(1)

Invariance Properties of the L-Class

(2)

The Borel Conjecture

(3)

Normal Bordism Groups (K.)

(6)

Normal Bordism Groups

(1)

Rational Computation of Normal Bordism Groups

(2)

Rational Computation of Oriented Bordism Groups

(3)

The Signature (K.)

(8)

The Definition of the Signature

(1)

The Bordism Invariance of the Signature

(2)

Multiplicativity and other Properties of the Signature

(1)

Geometric Interpretation of Cohomology and the Intersection Form

(4)

The Signature Theorem and the Novikov Conjecture (K.)

(8)

The Signature Theorem

(2)

Higher Signatures

(1)

The Novikov Conjecture

(1)

The Pontrjagin Classes are not Homeomorphism Invariants

(4)

The Projective Class Group and the Whitehead Group (L.)

(10)

The Projective Class Group

(2)

The First Algebraic K-Group

(4)

The Whitehead Group

(1)

The Bass Heller Swan Decomposition

(3)

Whitehead Torsion (L.)

(12)

Whitehead Torsion of a Chain Map

(4)

The Cellular Chain Complex of the Universal Covering

(2)

The Whitehead Torsion of a Cellular Map

(3)

Simple Homotopy Equivalences

(3)

The Statement and Consequences of the s-Cobordism Theorem (L.)

(4)

Sketch of the Proof of the s-Cobordism Theorem (L.)

(10)

Handlebody Decompositions

(2)

Handlebody Decompositions and CW-Structures

(2)

Reducing the Handlebody Decomposition

(2)

Handlebody Decompositions and Whitehead Torsion

(4)

From the Novikov Conjecture to Surgery (K.)

(6)

The Structure Set

(1)

The Assembly Idea

(5)

Surgery Below the Middle Dimension I: An Example (K.)

(4)

Surgery and its Trace

(1)

The Effect on the Fundamental Group and Homology Groups

(1)

Application to Knottings

(2)

Surgery Below the Middle Dimension II: Systematically (K.)

(8)

The Effect of Surgery in Homology and Homotopy

(2)

Surgery below the Middle Dimension

(3)

Construction of Certain 6-Manifolds

(3)

Surgery in the Middle Dimension I (K.)

(6)

Motivation for the Surgery Obstruction Groups

(1)

Unimodular Hermitian Forms

(1)

The L-Groups in Dimensions 4m

(1)

The L-Groups in Other Dimensions

(3)

Surgery in the Middle Dimension II (K.)

(6)

Equivariant Intersection Numbers

(1)

Stably Free Modules

(1)

The Quadratic Refinement

(2)

The Surgery Obstruction

(2)

Surgery in the Middle Dimension III (K.)

(10)

Stable Diffeomorphism Classification

(2)

The Surgery Obstruction is a Bordism Invariant

101

(1)

The Main Result

101

(2)

Proof of the Main Theorem

103

(3)

The Exact Surgery Sequence

106

(1)

Stable Classification of Certain 6-Manifolds

107

(2)

An Assembly Map (K.)

109

(4)

More on the Definition of the Assembly Map

109

(3)

The Surgery Version of the Novikov Conjecture

112

(1)

The Novikov Conjecture for Zn (K.)

113

(6)

The Idea of the Proof

113

(1)

Reduction to Mapping Tori

113

(2)

The Proof for Rank 1

115

(2)

The Generalization to Higher Rank

117

(2)

Poincare Duality and Algebraic L-Groups (L. and Varisco)

119

(14)

Poincare duality

119

(5)

Algebraic L-groups

124

(9)

Spectra (L.)

133

(20)

Basic Notions about Spectra

133

(2)

Homotopy Pushouts and Homotopy Pullbacks for Spaces

135

(3)

Homotopy Pushouts and Homotopy Pullbacks for Spectra

138

(1)

(Co-)Homology Theories Associated to Spectra

139

(2)

K-Theory and L-Theory Spectra

141

(2)

The Chern Character for Homology Theories

143

(1)

The Bordism Group Associated to a Vector Bundle

144

(1)

The Thorn Space of a Vector Bundle

145

(1)

The Pontrjagin--Thom Construction

145

(1)

The Stable Version of the Pontrjagin--Thom Construction

146

(2)

The Oriented Bordism Ring

148

(1)

Stable Homotopy

149

(1)

The Thom Isomorphism

150

(1)

The Rationalized Oriented Bordism Ring

150

(1)

The Integral Oriented Bordism Ring

151

(2)

Classifying Spaces of Families (L.)

153

(10)

Basics about G-CW-Complexes

153

(3)

The Classifying Space for a Family

156

(1)

Special Models

157

(6)

The Family of All Subgroups and the Trivial Family

157

(1)

Operator Theoretic Model

157

(1)

Discrete Subgroups of Almost Connected Lie Groups

157

(1)

Simply Connected Non-Positively Curved Manifolds

158

(1)

CAT(0)-Spaces

158

(1)

Trees with Finite Isotropy Groups

158

(1)

Amalgamated Products and HNN-Extensions

158

(1)

Arithmetic Groups

159

(1)

Outer Automorphism Groups of Free groups

160

(1)

Mapping Class groups

160

(1)

One-Relator Groups

160

(1)

Special Linear Groups of (2,2)-Matrices

161

(2)

Equivariant Homology Theories and the Meta-Conjecture (L.)

163

(10)

The Meta-Conjecture

163

(1)

Formulation of the Farrell-Jones and the Baum-Connes Conjecture

164

(1)

Equivariant Homology Theories

165

(3)

The Construction of Equivariant Homology Theories from Spectra

168

(5)

The Farrell--Jones Conjecture (L.)

173

(12)

The Bass--Heller--Swan Decomposition in Arbitrary Dimensions

173

(1)

Decorations in L--Theory and the Shaneson Splitting

174

(2)

Changing the Family

176

(1)

The Farrell--Jones Conjecture for Torsionfree Groups

177

(3)

The Farrell--Jones Conjecture and the Borel Conjecture

180

(1)

The Passage from FLN to UX

181

(4)

The Baum--Connes Conjecture (L.)

185

(6)

Index Theoretic Interpretation of the Baum--Connes Assembly Map

185

(1)

The Baum--Connes Conjecture for Torsionfree Groups

186

(1)

The Trace Conjecture in the Torsionfree Case

187

(1)

The Kadison Conjecture

187

(1)

The Stable Gromov--Lawson--Rosenberg Conjecture

188

(1)

The Choice of the Family FIN in the Baum--Connes Conjecture

189

(2)

Relating the Novikov, the Farrell-Jones and the Baum--Connes Conjectures (L.)

191

(10)

The Farrell--Jones Conjecture and the Novikov Conjecture

191

(4)

Relating Topological K--Theory and L--Theory

195

(2)

The Baum--Connes Conjecture and the Novikov Conjecture

197

(4)

Miscellaneous (L.)

201

(14)

Status of the Conjectures

201

(5)

Methods of Proof

206

(1)

Computations for Finite Groups

206

(2)

Topological K-Theory for Finite Groups

206

(1)

Algebraic K-Theory for Finite Groups

207

(1)

Algebraic L-Theory for Finite Groups

208

(1)

Rational Computations

208

(3)

Rationalized Topological K-Theory for Infinite Groups

209

(1)

Rationalized Algebraic K-Theory for Infinite Groups

210

(1)

Rationalized Algebraic L-Theory for Infinite Groups

211

(1)

Integral Computations

211

(4)

Exercises

215

(8)

Hints to the Solutions of the Exercises

223

(13)

References

236

(18)

Index

254

(7)

Notation

261

(2)

Schedules

263

(4)

List of Participants

267

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The Novikov Conjecture

Summary

Table of Contents

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