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| Preface 1. Introduction 1.1 Overview 1.2 Algebraic topology 1.3 Basic Morse theory 1.4 Stable and unstable manifolds 1.5 Basic differential topology 1.6 Morse-Smale functions 1.7 The Morse Homology Theorem 1.8 Morse theory on Grassmann manifolds 1.9 Floer homology theories 1.10 Guide to the book 2. The CW-Homology Theorem 2.1 Singular homology 2.2 Singular cohomology 2.3 CW-complexes 2.4 CW-homology 2.5 Some homotopy theory 3. Basic Morse Theory 3.1 Morse functions 3.2 The gradient flow of a Morse function 3.3 The CW-complex associated to a Morse function 3.4 The Morse Inequalities 3.5 Morse-Bott functions 4. The Stable/Unstable Manifold Theorem 4.1 The Stable/Unstable Manifold Theorem for a Morse function 4.2 The Local Stable Manifold Theorem 4.3 The Global Stable/Unstable Manifold Theorem 4.4 Examples of stable/unstable manifolds 5. Basic Differential Topology 5.1 Immersions and submersions 5.2 Transversality 5.3 Stability 5.4 General position 5.5 Stability and density for Morse functions 5.6 Orientations and intersection numbers 5.7 The Lefschetz Fixed Point Theorem 6. Morse-Smale Functions 6.1 The Morse-Smale transversality condition 6.2 The A-Lemma 6.3 Consequences of the A-Lemma 6.4 The CW-complex associated to a Morse-Smale function 7. The Morse Homology Theorem 7.1 The Morse-Smale-Witten boundary operator 7.2 Examples using the Morse Homology Theorem 7.3 The Conley index 7.4 Proof of the Morse Homology Theorem 7.5 Independence of the choice of the index pairs 8. Morse Theory On Grassmann Manifolds 8.1 Morse theory on the adjoint orbit of a Lie group 8.2 A Morse function on an adjoint orbit of the unitary group 8.3 An almost complex structure on the adjoint orbit 8.4 The critical points and indices of fA : U(n+k)·xo→R 8.5 A Morse function on the complex Grassmann manifold 8.6 The gradient flow lines of fA : Gn,n,+k(C)→R 8.7 The homology of Gn, n+k(C) 8.8 Further generalizations and applications 9. An Overview of Floer Homology Theories 9.1 Introduction to Floer homology theories 9.2 Symplectic Floerhomology 9.3 Floer homology for Lagrangian intersections 9.4 Instanton Floer homology 9.5 A symplectic flavor of the instanton homology Hints and References for Selected Problems Bibliography Symbol Index Index |
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