Preface p. vii
Chapter 1. The Novikov Numbers p. 1
1.1. Homological algebra of Morse inequalities p. 1
1.2. The Novikov ring Nov([Gamma]) p. 6
1.3. The rational subring R([Gamma]) p. 9
1.4. Homology of local coefficient systems p. 12
1.5. The Novikov numbers p. 17
1.6. Further properties of the Novikov numbers p. 21
1.7. Novikov numbers and Betti numbers of flat line bundles p. 30
Chapter 2. The Novikov Inequalities p. 35
2.1. Closed 1-forms p. 35
2.2. Geometry of Novikov theory p. 38
2.3. The Novikov inequalities p. 45
Chapter 3. The Universal Complex p. 49
3.1. The Main Theorem p. 49
3.2. Line bundles and algebraic integers p. 54
3.3. Generic flat vector bundles p. 56
3.4. Examples p. 58
Chapter 4. Construction of the Universal Complex p. 61
4.1. Chain collapse p. 61
4.2. Proof of Theorem 3.1 in the rank 1 case p. 63
4.3. Proof of Theorem 3.1 in the general case p. 66
4.4. Refined universal complex and deformation complex p. 75
Chapter 5. Bott-type Inequalities p. 81
5.1. Topology of the set of zeros p. 81
5.2. Proofs of Theorems 5.1 and 5.5 p. 86
Chapter 6. Inequalities with Von Neumann Betti Numbers p. 91
Chapter 7. Equivariant Theory p. 99
7.1. Basic 1-forms p. 100
7.2. Equivariant Novikov inequalities p. 101
7.3. Application: Fixed points of a symplectic circle action p. 104
7.4. Signature via Novikov numbers p. 108
Chapter 8. Exactness of the Novikov Inequalities p. 113
8.1. Exactness Theorem p. 113
8.2. Finiteness theorem for codimension two knots p. 114
8.3. Surgery on codimension one submanifolds p. 115
8.4. Algebra of minimal lattices p. 119
8.5. Proof of The Exactness Theorem p. 122
Chapter 9. Morse Theory of Harmonic Forms p. 125
9.1. Topology of singular foliations of closed 1-forms p. 125
9.2. Intrinsically harmonic 1-forms p. 131
9.3. Examples of singular foliations p. 137
9.4. Proof of Calabi's Theorem p. 140
9.5. Morse numbers of harmonic 1-forms p. 147
Chapter 10. Lusternik-Schnirelman Theory, Closed 1-Forms, and Dynamics p. 159
10.1. Colliding the critical points p. 160
10.2. Closed 1-forms on topological spaces p. 162
10.3. Category of a space with respect to a cohomology class p. 165
10.4. Estimate of the number of zeros p. 170
10.5. Gradient-convex neighborhoods p. 177
10.6. Movable homology classes p. 179
10.7. Cohomological lower bound for cat(X, [xi]) p. 181
10.8. Deformations and their spectral sequences p. 184
10.9. Families of flat bundles and higher Massey products p. 190
10.10. Estimate for cat(X, [xi]) in terms of [xi]-survivors p. 194
10.11. Flows, Lyapunov 1-forms and asymptotic cycles p. 197
Appendix A. Manifolds with Corners p. 205
Appendix B. Morse-Bott Functions on Manifolds with Corners p. 213
Appendix C. Morse-Bott Inequalities p. 227
Appendix D. Relative Morse Theory p. 233
Bibliography p. 239
Index p. 245