Morse Homology

1 Introduction.- 1.1 Background.- 1.1.1 Classical Morse Theory.- 1.1.2 Relative Morse Theory.- 1.1.3 The Continuation Principle.- 1.2 Overview.- 1.2.1 The Construction of the Morse Homology.- 1.2.2 The Axiomatic Approach.- 1.3 Remarks on the Methods.- 1.4 Table of Contents.- 1.5 Acknowledgments.- 2 The Trajectory Spaces.- 2.1 The Construction of the Trajectory Spaces.- 2.2 Fredholm Theory.- 2.2.1 The Fredholm Operator on the Trivial Bundle.- 2.2.2 The Fredholm Operator on Non-Trivial Bundles.- 2.2.3 Generalization to Fredholm Maps.- 2.3 Transversality.- 2.3.1 The Regularity Conditions.- 2.3.2 The Regularity Results.- 2.4 Compactness.- 2.4.1 The Space of Unparametrized Trajectories.- 2.4.2 The Compactness Result for Unparametrized Trajectories.- 2.4.3 The Compactness Result for Homotopy Trajectories.- 2.4.4 The Compactness Result for ?-Parametrized Trajectories.- 2.5 Gluing.- 2.5.1 Gluing for the Time-Independent Trajectory Spaces.- 2.5.2 Gluing of Trajectories of the Time-Dependent Gradient Flow.- 2.5.3 Gluing for ?-Parametrized Trajectories.- 3 Orientation.- 3.1 Orientation and Gluing in the Trivial Case.- 3.1.1 The Determinant Bundle.- 3.1.2 Gluing and Orientation for Fredholm Operators.- 3.2 Coherent Orientation.- 3.2.1 Orientation and Gluing on the Manifold M.- 4 Morse Homology Theory.- 4.1 The Main Theorems of Morse Homology.- 4.1.1 Canonical Orientations.- 4.1.2 The Morse Complex.- 4.1.3 The Canonical Isomorphism.- 4.1.4 Topology and Coherent Orientation.- 4.2 The Eilenberg-Steenrod Axioms.- 4.2.1 Extension of Morse Functions and Induced Morse Functions on Vector Bundles.- 4.2.2 The Homology Functor and Homotopy Invariance.- 4.2.3 Relative Morse Homology.- 4.2.4 Summary.- 4.3 The Uniqueness Result.- 5 Extensions.- 5.1 Morse Cohomology.- 5.2 Poincaré Duality.- 5.3 Products.- A Curve Spaces and Banach Bundles.- B The Geometric Boundary Operator.