Morse Homology
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 Mors
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Series Editors J. Oesterle A. Weinstein
Matthias Schwarz
Morse Homology
Springer Basel AG
Author: Matthias Schwarz ETHZentrum Mathematik 8092 Ziirich Switzerland
Library of Congress Cataloging-in-Publication Data Schwarz, Matthias, 1967Morse homology / Matthias Schwarz. p. cm. - (Progress in mathematics ; voI. 111) Inc1udes bibliographical references and index. ISBN 978-3-0348-9688-7 ISBN 978-3-0348-8577-5 (eBook) DOI 10.1007/978-3-0348-8577-5 1. Morse theory. 2. Homology theory. 1. TItle. II. Series: Progress in mathematics (Boston, Mass.) ; voI. 111. QA331.S43 1993 515'.73 - dc20 Deutsche Bibliothek Cataloging-in-Publication Data
Schwarz, Matthias: Morse homology / Matthias Schwarz. - Basel ; Boston ; Berlin Birkhăuser, 1993 (Progress in mathematics ; VoI. 111) ISBN 978-3-0348-9688-7 NE:GT
This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © 1993 Springer Basel AG Originally published by Birkhăuser Verlag Basel in 1993 Softcover reprint ofthe hardcover Ist edition 1993 Camera-ready copy prepared by the author Printed on acid-free paper produced of chlorine-free pulp
ISBN 978-3-0348-9688-7 987654321
Contents List of symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
viii
1 Introduction 1.1
1.2
Background...........
1
1.1.1
Classical Morse Theory
1
1.1.2
Relative Morse Theory.
5
1.1.3
The Continuation Principle
7
Overview . . . . . . . . . . . . . .
8
1.2.1
The Construction of the Morse Homology
1.2.2
The Axiomatic Approach
8 10
1.3
Remarks on the Methods
14
1.4
Table of Contents .
17
1.5
Acknowledgments.
18
2 The Trajectory Spaces 2.1
The Construction of the Trajectory Spaces.
21
2.2
Fredholm Theory . . . . . . . . . . . . . . .
29
2.2.1
The Fredholm Operator on the Trivial Bundle
29
2.2.2
The Fredholm Operator on Non-Trivial Bundles
39
2.2.3
Generalization to Fredholm Maps.
41
2.3
Transversality . . . . . . . . . . . .
41
2.3.1
The Regularity Conditions
42
2.3.2
The Regularity Results
49
..
CONTENTS
vi 2.4
2.5
Compactness . . . . . . . . . . . . . . . . . . . . .
51
2.4.1
The Space of Unparametrized Trajectories.
52
2.4.2
The Compactness Result for Unparametrized Trajectories 55
2.4.3
The Compactness Result for Homotopy Trajectories ..
63
2.4.4
The Compactness Result for A-Parametrized Trajectories
67
Gluing..............................
68
2.5.1
Gluing for the Time-Independent Trajectory Spaces
68
2.5.2
Gluing of Trajectories of the Time-Dependent Gradient Flow. . . . . . . . . . . . . . . . . . . .
96
Gluing for A-Parametrized Trajectories. . . . . . . . ..
99
2.5.3
3 Orientation
3.1
3.2
Orientation and Gluing in the Trivial Case 3.1.1
The Determinant Bundle .. . . . .
104
3.1.2
Gluing and Orientat
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