Preface p. vii
Chern-Weil Theory for Characteristic Classes p. 1
Review of the de Rham Cohomology Theory p. 1
Connections on Vector Bundles p. 3
The Curvature of a Connection p. 4
Chern-Weil Theorem p. 6
Characteristic Forms, Classes and Numbers p. 8
Some Examples p. 10
Chern Forms and Classes p. 10
Pontrjagin Classes for Real Vector Bundles p. 11
Hirzebruch's L-class and A-class p. 12
K-groups and the Chern Character p. 14
The Chern-Simons Transgressed Form p. 16
Bott Vanishing Theorem for Foliations p. 17
Foliations and the Bott Vanishing Theorem p. 18
Adiabatic Limit and the Bott Connection p. 20
Chern-Weil Theory in Odd Dimension p. 22
References p. 26
Bott and Duistermaat-Heckman Formulas p. 29
Berline-Vergne Localization Formula p. 29
Bott Residue Formula p. 35
Duistermaat-Heckman Formula p. 37
Bott's Original Idea p. 38
References p. 39
Gauss-Bonnet-Chern Theorem p. 41
A Toy Model and the Berezin Integral p. 41
Mathai-Quillen's Thom Form p. 43
A Transgression Formula p. 46
Proof of the Gauss-Bonnet-Chern Theorem p. 47
Some Remarks p. 50
Chern's Original Proof p. 51
References p. 54
Poincare-Hopf Index Formula: an Analytic Proof p. 57
Review of Hodge Theorem p. 57
Poincare-Hopf Index Formula p. 60
Clifford Actions and the Witten Deformation p. 61
An Estimate Outside of U[subscript p[set membership]zero(V)]U[subscript p] p. 63
Harmonic Oscillators on Euclidean Spaces p. 64
A Proof of the Poincare-Hopf Index Formula p. 67
Some Estimates for D[subscript T,i]'s, 2 [less than or equal] i [less than or equal] 4 p. 69
An Alternate Analytic Proof p. 73
References p. 74
Morse Inequalities: an Analytic Proof p. 75
Review of Morse Inequalities p. 75
Witten Deformation p. 77
Hodge Theorem for ([Omega]* (M), d[subscript Tf]) p. 78
Behaviour of [square subscript Tf] Near the Critical Points of f p. 79
Proof of Morse Inequalities p. 81
Proof of Proposition 5.5 p. 83
Some Remarks and Comments p. 88
References p. 89
Thom-Smale and Witten Complexes p. 93
The Thom-Smale Complex p. 93
The de Rham Map for Thom-Smale Complexes p. 95
Witten's Instanton Complex and the Map e[subscript T] p. 97
The Map P[subscript [infinity],T]e[subscript T] p. 100
An Analytic Proof of Theorem 6.4 p. 102
References p. 102
Atiyah Theorem on Kervaire Semi-characteristic p. 105
Kervaire Semi-characteristic p. 106
Atiyah's Original Proof p. 107
A proof via Witten Deformation p. 108
A Generic Counting Formula for k(M) p. 112
Non-multiplicativity of k(M) p. 113
References p. 115
Index p. 117