| 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 |
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