Contents -- Preface -- Chapter 1 Chern-Weil Theory for Characteristic Classes -- 1.1 Review of the de Rham Cohomology Theory -- 1.2 Connections on Vector Bundles -- 1.3 The Curvature of a Connection -- 1.4 Chern-Weil Theorem -- 1.5 Characteristic Forms, Classes and Numbers -- 1.6 Some Examples -- 1.6.1 Chern Forms and Classes -- 1.6.2 Pontrjagin Classes for Real Vector Bundles -- 1.6.3 Hirzebruch's L-class and A-class -- 1.6.4 K-groups and the Chern Character -- 1.6.5 The Chern-Simons Transgressed Form -- 1.7 Bott Vanishing Theorem for Foliations -- 1.7.1 Foliations and the Bott Vanishing Theorem -- 1.7.2 Adiabatic Limit and the Bott Connection -- 1.8 Chern-Weil Theory in Odd Dimension -- 1.9 References -- Chapter 2 Bott and Duistermaat-Heckman Formulas -- 2.1 Berline-Vergne Localization Formula -- 2.2 Bott Residue Formula -- 2.3 Duistermaat-Heckman Formula -- 2.4 Bott's Original Idea -- 2.5 References -- Chapter 3 Gauss-Bonnet-Chern Theorem -- 3.1 A Toy Model and the Berezin Integral -- 3.2 Mathai-Quillen's Thom Form -- 3.3 A Transgression Formula -- 3.4 Proof of the Gauss-Bonnet-Chern Theorem -- 3.5 Some Remarks -- 3.6 Chern's Original Proof -- 3.7 References -- Chapter 4 Poincare-Hopf Index Formula: an Analytic Proof -- 4.1 Review of Hodge Theorem -- 4.2 Poincare-Hopf Index Formula -- 4.3 Clifford Actions and the Witten Deformation -- 4.4 An Estimate Outside of Up zero(V) Up -- 4.5 Harmonic Oscillators on Euclidean Spaces -- 4.6 A Proof of the Poincare-Hopf Index Formula -- 4.7 Some Estimates for DT,i's, 2 i 4 -- 4.8 An Alternate Analytic Proof -- 4.9 References -- Chapter 5 Morse Inequalities: an Analytic Proof -- 5.1 Review of Morse Inequalities -- 5.2 Witten Deformation -- 5.3 Hodge Theorem for ( * (M), dTf -- 5.4 Behaviour of rf Near the Critical Points of f -- 5.5 Proof of Morse Inequalities -- 5.6 Proof of Proposition 5.5.

5.7 Some Remarks and Comments -- 5.8 References -- Chapter 6 Thom-Smale and Witten Complexes -- 6.1 The Thom-Smale Complex -- 6.2 The de Rham Map for Thom-Smale Complexes -- 6.3 Witten's Instanton Complex and the Map eT -- 6.4 The Map P, TeT -- 6.5 An Analytic Proof of Theorem 6.4 -- 6.6 References -- Chapter 7 Atiyah Theorem on Kervaire Semi-characteristic -- 7.1 Kervaire Semi-characteristic -- 7.2 Atiyah's Original Proof -- 7.3 A proof via Witten Deformation -- 7.4 A Generic Counting Formula for k(M ) -- 7.5 Non-multiplicativity of k(M) -- 7.6 References -- Index.