Bordism of Diffeomorphisms and Related Topics
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1069
Matthias Kreck
Bordism of Diffeomorphisms and Related Topics With an Appendix by Neal W. Stoltzfus
Springer-Verlag Berlin Heidelberg New York Tokyo 1984
Author
Matthias Kreck Fachbereich Mathematik der Universitat Mainz Saarstr. 21, 6500 Mainz, Federal Republic of Germany
AMS Subject Classification (1980): 57R50, 57R65, 57R90; lOC05, 57R 15 ISBN 3-540-13362-3 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13362-3 Springer-Verlag New York Heidelberg Berlin Tokyo ThIS work ts subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
CONTENTS
Introduction Bordism groups of orientation preserving diffeomorphisms § Report about equivariant Witt groups § The isometric structure of a diffeomorphism § The mapping torus of a diffeomorphism. § Fibrations over sl within their bordism class and the computation of 6* § 6 Addition and subtraction of handles § 7 Proof of Theorem 5.5 in the odd-dimensional case § 8 Proof of Theorem 5.5 in the even-dimensional case § 9 Bordism of diffeomorphisms on manifolds with additional normal structures like Spin-, unitary structures or framings; orientation reversing diffeomorphisms and the unoriented case § 10 Applications to SK-groups § 11 Miscellaneous results: Ring structure, generators, relation to the inertia group
101
References
110
Appendix by Neal W. Stoltzfus The algebraic relationship between Quinn's invariant for open book decomposition bordism and the isometric structure
115
Subject index
142
§
1 2 3 4 5
1 12 17 22 27
33 41 51 54
72 95
Introduction The main theme of this work is the computation of the bordism group of diffeomorphisms. The problem is the following: Given two diffeomorphisms f i: Mi---. Mi on closed manifolds does there exist a diffeomorphism F: W----+-W on a manifold Wwith boundary Mo + (- M1) such that Flaw
=
fo+f 1.
If we collect all diffeomorphism on m-dimensional manifolds and introduce the bordism relation as above we obtain the bordism group of diffeomorphisms. One can consider this group for manifolds with various structures but we concentrate for the moment on oriented manifolds. The bordism group of orientation preserving diffeomorphisms is denoted by To describe the computation of the bordism group we introduce the following invariants. The most obvious invariants are the bordism class of the underlying manifolds and the bordism class of the mapping torus Mf wherethe 7l-action on Mis given by the diffeomorphism f.
=
lR x71 M
For diffeomorphisms on even-dimensional manifolds there is a very interesting invariant cal
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