Morse homology

by Schwarz, Matthias

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Title Statements

Full title :Morse homology / Matthias Schwarz.

Summary

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.

Edition Statements

Publication Date:©1993.

Publication Statements

Publisher(s):Birkhäuser Verlag
Place of Publication:Basel
Language:eng
Publication Date:©1993.

The Physical Object

Description:1 online resource (ix, 235 pages) : : illustrations

Edition Identifiers

ISBN:9783034885775, 3034885776
DOI:10.1007/978-3-0348-8577-5
oclc:ocn681875921
LC Classification:QA331
Dewey Decimal:515/.73

1 Introduction -- 1.1 Background -- 1.2 Overview -- 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 -- 2.3 Transversality -- 2.4 Compactness -- 2.5 Gluing -- 3 Orientation -- 3.1 Orientation and Gluing in the Trivial Case -- 3.2 Coherent Orientation -- 4 Morse Homology Theory -- 4.1 The Main Theorems of Morse Homology -- 4.2 The Eilenberg-Steenrod Axioms -- 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.