Lectures on Chern-Weil theory and Witten deformations / Weiping Zhang.

Format:

Book

Author/Creator:

Zhang, Weiping, 1964-

Series:

Nankai tracts in mathematics ; v. 4.

Nankai tracts in mathematics ; vol. 4

Language:

English

Subjects (All):

Chern classes.

Index theorems.

Complexes.

Physical Description:

xi, 117 pages ; 22 cm.

Place of Publication:

River Edge, N.J. : World Scientific, [2001]

Contents:

Chapter 1 Chern-Weil Theory for Characteristic Classes 1

1.1 Review of the de Rham Cohomology Theory 1

1.2 Connections on Vector Bundles 3

1.3 The Curvature of a Connection 4

1.4 Chern-Weil Theorem 6

1.5 Characteristic Forms, Classes and Numbers 8

1.6.1 Chern Forms and Classes 10

1.6.2 Pontrjagin Classes for Real Vector Bundles 11

1.6.3 Hirzebruch's L-class and A-class 12

1.6.4 K-groups and the Chern Character 14

1.6.5 The Chern-Simons Transgressed Form 16

1.7 Bott Vanishing Theorem for Foliations 17

1.7.1 Foliations and the Bott Vanishing Theorem 18

1.7.2 Adiabatic Limit and the Bott Connection 20

1.8 Chern-Weil Theory in Odd Dimension 22

Chapter 2 Bott and Duistermaat-Heckman Formulas 29

2.1 Berline-Vergne Localization Formula 29

2.2 Bott Residue Formula 35

2.3 Duistermaat-Heckman Formula 37

2.4 Bott's Original Idea 38

Chapter 3 Gauss-Bonnet-Chern Theorem 41

3.1 A Toy Model and the Berezin Integral 41

3.2 Mathai-Quillen's Thom Form 43

3.3 A Transgression Formula 46

3.4 Proof of the Gauss-Bonnet-Chern Theorem 47

3.6 Chern's Original Proof 51

Chapter 4 Poincare-Hopf Index Formula: an Analytic Proof 57

4.1 Review of Hodge Theorem 57

4.2 Poincare-Hopf Index Formula 60

4.3 Clifford Actions and the Witten Deformation 61

4.4 An Estimate Outside of U[subscript p[set membership]zero(V)]U[subscript p] 63

4.5 Harmonic Oscillators on Euclidean Spaces 64

4.6 A Proof of the Poincare-Hopf Index Formula 67

4.7 Some Estimates for D[subscript T,i]'s, 2 [less than or equal] i [less than or equal] 4 69

4.8 An Alternate Analytic Proof 73

Chapter 5 Morse Inequalities: an Analytic Proof 75

5.1 Review of Morse Inequalities 75

5.2 Witten Deformation 77

5.3 Hodge Theorem for ([Omega]* (M), d[subscript Tf]) 78

5.4 Behaviour of [square subscript Tf] Near the Critical Points of f 79

5.5 Proof of Morse Inequalities 81

5.6 Proof of Proposition 5.5 83

Chapter 6 Thom-Smale and Witten Complexes 93

6.1 The Thom-Smale Complex 93

6.2 The de Rham Map for Thom-Smale Complexes 95

6.3 Witten's Instanton Complex and the Map e[subscript T] 97

6.4 The Map P[subscript [infinity],T]e[subscript T] 100

6.5 An Analytic Proof of Theorem 6.4 102

Chapter 7 Atiyah Theorem on Kervaire Semi-characteristic 105

7.1 Kervaire Semi-characteristic 106

7.2 Atiyah's Original Proof 107

7.3 A proof via Witten Deformation 108

7.4 A Generic Counting Formula for k(M) 112

7.5 Non-multiplicativity of k(M) 113.

Notes:

Includes bibliographical references and index.

ISBN:

9810246862

9810246854

OCLC:

47863244