1.1 Background The subject of this book is Morse homology as a combination of relative Morse theory and Conley's continuation principle. The latter will be useda s an instrument to express the homology encoded in a Morse complex associated to a fixed Morse function independent of this function. Originally, this type of Morse-theoretical tool was developed by Andreas Floer in order to find a proof of the famous Arnold conjecture, whereas classical Morse theory turned out to fail in the infinite-dimensional setting. In this framework, the homological variant of Morse theory is also known as Floer homology. This kind of homology theory is the central topic of this book. But first, it seems worthwhile to outline the standard Morse theory. 1.1.1 Classical Morse Theory The fact that Morse theory can be formulated in a homological way is by no means a new idea. The reader is referred to the excellent survey paper by Raoul Bott [Bol.

Inhalt

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.