New Topological Invariants For Real- And Angle-valued Maps: An Alternative To Morse-novikov Theory

by Dan Burghelea

Overview

This book is about new topological invariants of real- and angle-valued maps inspired by Morse-Novikov theory, a chapter of topology, which has recently raised interest outside of mathematics; for example, in data analysis, shape recognition, computer science and physics. They are the backbone of what the author proposes as a computational alternative to Morse-Novikov theory, referred to in this book as AMN-theory.These invariants are on one side analogues of rest points, instantons and closed trajectories of vector fields and on the other side, refine basic topological invariants like homology and monodromy. They are associated to tame maps, considerably more general than Morse maps, that are defined on spaces which are considerably more general than manifolds. They are computable by computer implementable algorithms and have strong robustness properties. They relate the dynamics of flows that admit the map as 'Lyapunov map' to the topology of the underlying space, in a similar manner as Morse-Novikov theory does.

Product Details

ISBN-13:	9789814618243
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	10/13/2017
Pages:	260
Product dimensions:	6.10(w) x 9.10(h) x 0.80(d)

Table of Contents

Acknowledgments vii

Preface ix

List of Figures xv

1 Preview 1

1.1 Introduction 1

1.2 Morse-Novikov theory 2

1.3 The AMN theory 4

1.4 Contents of the book 8

2 Preparatory Material 13

2.1 Linear algebra 13

2.1.1 Matrices 13

2.1.2 Fredholm maps and Fredholm cross-ratio 22

2.1.3 Appendix 29

2.2 Linear relations 30

2.2.1 General considerations 30

2.2.2 Calculation of R(A, B)reg (an algorithm) 39

2.3 Topology 44

2.3.1 ANRs, tameness, regular and critical values 44

2.3.2 Compact Hilbert cube manifolds 47

2.3.3 Infinite cyclic covers 49

2.3.4 Simplicial complexes, cell complexes and incidence matrices 50

2.3.5 Configurations 54

2.3.6 Algebraic topology of a pair (X, ξ ∈ H¹(X; Z)) 58

3 Graph Representations 65

3.1 Generalities on graph representations 65

3.2 The indecomposable representations 68

3.2.1 Two basic constructions 74

3.2.2 The k[t^-1, t]-module associated to a G_2m-representation 75

3.2.3 The matrix M(p) and the representation p_u 76

3.3 Calculation of indecomposables (an algorithm) 81

3.3.1 Elementary transformations 82

3.3.2 Algorithm for deriving barcodes from M(p) 86

3.3.3 Implementation of T₁(i), T₂(i), T₃(i), T4(i) 88

3.4 Exercises 93

4 Barcodes and Jordan Blocks via Graph Representations 95

4.1 The graph representations associated to a map 95

4.2 Barcodes and Jordan blocks of a tame map 99

4.2.1 The configurations δ^f_r 101

4.2.2 The AM and AN complexes 103

4.2.3 The relevant exact sequences 104

4.3 Barcodes, Jordan cells, and homology 110

4.3.1 Two examples 117

4.4 Barcodes and Borel-Moore homology 120

4.5 Calculations of barcodes and Jordan cells 125

4.6 Exercises 127

5 Configurations δ^f_r and δ^f_r (Alternative Approach) 131

5.1 General considerations 132

5.2 The case of real-valued maps f : X → R 147

5.2.1 The main results 147

5.2.2 Proof of Theorem 5.1 149

5.2.3 Proof of Theorem 5.2 (stability property) 151

5.2.4 Proof of Theorem 5.3 (Pomearé Duality property) 153

5.3 The case of angle-valued map, f : X → S¹ 157

5.3.1 The main results 163

5.3.2 Proofs of Theorems 5.4 and 5.5 (topological results and stability) 165

5.3.3 Proof of Theorem 5.6 (Poincaré Duality property) 167

6 Configurations γ^f_r 177

6.1 General considerations 178

6.2 The case of real-valued maps 189

6.3 The case of angle-valued maps 192

6.4 Exercises 194

7 Monodromy and Jordan Cells 195

7.1 General considerations 195

7.2 Geometric r-monodromy via linear relations 199

7.3 The calculation of Jordan cells of a simplicial angle-valued map: an algorithm 205

8 Applications 209

8.1 Relations with the classical Morse and Morse-Novikov theories 209

8.1.1 The Morse complex 210

8.1.2 The Novikov complex 213

8.1.3 Chain complexes of vector spaces 215

8.1.4 The AM and AN complexes for a Morse map 217

8.2 A few computational applications 219

8.2.1 Novikov-Betti numbers, standard and twisted Betti numbers in relation with Jordan cells 219

8.2.2 Alexander polynomial of a knot, generalizations 223

8.3 Exercises 224

9 Comments 227

9.1 Relation to Persistence Theory 227

9.1.1 Persistence Theory, a summary 227

9.1.2 A few observations about δ^f_r and γ^f_r 231

9.2 A measure-theoretic aspect of the configurations δ^f_r, γ^f_r 233

9.3 An invitation 235

Bibliography 237

Index 241