This book presents a link between modern analysis and topology. Based upon classical Morse theory it develops the finite dimensional analogue of Floer homology which, in the recent years, has come to play a significant role in geometry. Morse homology naturally arises from the gradient dynamical system associated with a Morse function. The underlying chain complex, already considered by Thom, Smale, Milnor and Witten, analogously forms the basic ingredient of Floer's homology theory. This concept of relative Morse theory in combination with Conley's continuation principle lends itself to an axiomatic homology functor. The present approach consistenly employs analytic methods in strict analogy with the construction of Floers homology groups. That is a calculus for certain nonlinear Fredholm operators on Banach manifolds which here are curve spaces and within which the solution sets form the focal moduli spaces. The book offers a systematic and comprehensive presentation of the analysis of these moduli spaces. All theorems within this analytic schedule comprising Fredholm theory, regularity and compactness results, gluing and orientation analysis, together with their proofs and pre-requisite material, are examined here in detail. This exposition thus brings a methodological insight into present-day analysis.
"The proofs are written with great care, and Schwarz motivates all ideas with great skill...This is an excellent book." - Bulletin of the AMS
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 Poincare Duality.- 5.3 Products.- A Curve Spaces and Banach Bundles.- B The Geometric Boundary Operator.