Overview

This book presents a link between modem 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 non-linear 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 prerequisite material, are examined here in detail. This exposition thus brings a methodological insight into present-day analysis.

Full Product Details

Author:   Schwarz
Publisher:   Birkhauser Verlag AG
Imprint:   Birkhauser Verlag AG
Edition:   1993 ed.
Volume:   111
Dimensions:   Width: 15.50cm , Height: 1.50cm , Length: 23.50cm
Weight:   1.180kg
ISBN:  

9783764329044

ISBN 10:   3764329041
Pages:   236
Publication Date:   01 October 1993
Audience:   College/higher education ,  Professional and scholarly ,  Postgraduate, Research & Scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   In Print  
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Table of Contents

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.

Reviews

The proofs are written with great care, and Schwarz motivates all ideas with great skill...This is an excellent book. <br> - Bulletin of the AMS

The proofs are written with great care, and Schwarz motivates all ideas with great skill...This is an excellent book. - Bulletin of the AMS

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