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Tables of Contents for Lectures on Chern-Weil Theory and Witten Deformations
Chapter/Section Title
Page #
Page Count
Preface
vii
Chern-Weil Theory for Characteristic Classes
1
28
Review of the de Rham Cohomology Theory
1
2
Connections on Vector Bundles
3
1
The Curvature of a Connection
4
2
Chern-Weil Theorem
6
2
Characteristic Forms, Classes and Numbers
8
2
Some Examples
10
7
Chern Forms and Classes
10
1
Pontrjagin Classes for Real Vector Bundles
11
1
Hirzebruch's L-class and A-class
12
2
K-groups and the Chern Character
14
2
The Chern-Simons Transgressed Form
16
1
Bott Vanishing Theorem for Foliations
17
5
Foliations and the Bott Vanishing Theorem
18
2
Adiabatic Limit and the Bott Connection
20
2
Chern-Weil Theory in Odd Dimension
22
4
References
26
3
Bott and Duistermaat-Heckman Formulas
29
12
Berline-Vergne Localization Formula
29
6
Bott Residue Formula
35
2
Duistermaat-Heckman Formula
37
1
Bott's Original Idea
38
1
References
39
2
Gauss-Bonnet-Chern Theorem
41
16
A Toy Model and the Berezin Integral
41
2
Mathai-Quillen's Thom Form
43
3
A Transgression Formula
46
1
Proof of the Gauss-Bonnet-Chern Theorem
47
3
Some Remarks
50
1
Chern's Original Proof
51
3
References
54
3
Poincare-Hopf Index Formula: an Analytic Proof
57
18
Review of Hodge Theorem
57
3
Poincare-Hopf Index Formula
60
1
Clifford Actions and the Witten Deformation
61
2
An Estimate Outside of Up∈zero(V)Up
63
1
Harmonic Oscillators on Euclidean Spaces
64
3
A Proof of the Poincare-Hopf Index Formula
67
2
Some Estimates for DT,i's, 2 ≤ i ≤ 4
69
4
An Alternate Analytic Proof
73
1
References
74
1
Morse Inequalities: an Analytic Proof
75
18
Review of Morse Inequalities
75
2
Witten Deformation
77
1
Hodge Theorem for (Ω* (M),dTf)
78
1
Behaviour of &sku;Tf Near the Critical Points of f
79
2
Proof of Morse Inequalities
81
2
Proof of Proposition 5.5
83
5
Some Remarks and Comments
88
1
References
89
4
Thom-Smale and Witten Complexes
93
12
The Thom-Smale Complex
93
2
The de Rham Map for Thom-Smale Complexes
95
2
Witten's Instanton Complex and the Map eT
97
3
The Map P∞,TeT
100
2
An Analytic Proof of Theorem 6.4
102
1
References
102
3
Atiyah Theorem on Kervaire Semi-characteristic
105
12
Kervaire Semi-characteristic
106
1
Atiyah's Original Proof
107
1
A proof via Witten Deformation
108
4
A Generic Counting Formula for k(M)
112
1
Non-multiplicativity of k(M)
113
2
References
115
2
Index
117