Homotopy Equivalences of 3-Manifolds with Boundaries
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761 Klaus Johannson
Homotopy Equivalences of 3-Manifolds vvith Boundaries
Springer-Verlag Berlin Heidelberg New York 1979
Author
Klaus Johannson Fakultat fUr Mathematik der Universitat Universitatsstr. 1 D-4800 Bielefeld 1
AMS Subject Classifications (1970): 55A99 ISBN 3-540-09714-7 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-09714-7 Springer-Verlag New York Heidelberg Berlin Library of Congress Cataloging in Publication Data Johannsen, Klaus, 1948Homotopy equivalence of 3-manifolds with boundaries. (Lecture notes in mathematics; 761) Bibliography: p. Includes index. 1. Manifolds (Mathematics) 2. Homotopy equivalences. I. Title. II. Series: Lecture notes in mathematics (Berlin); 761. OA3.L28 no. 761 [OA613] [514'.2]510'.8s 79-23603 ISBN 0-387-09714-7
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© by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany
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Contents. 3
Introduction THE CONCEPTS OF CHARACTERISTIC SUBMANIFOLDS AND MANIFOLDS WITH BOUNDARY-PATTERNS.
Part I.
Chapter I:
General theory.
§1.
Definitions
19
§2.
Useful boundary-patterns
22
§3.
Essential maps
27
§4.
Essential surfaces and useful boundary-patterns
32
Chapter II:
Essential singular surfaces in some special 3-manifolds.
§5.
I-bundles and Seifert fibre spaces
47
§6.
Stallings manifolds
65
§7.
Generalized Seifert fibre spaces
80
Chapter III:
Characteristic submanifolds.
§8.
Definition of a characteristic submanifold
83
§9.
Existence of a characteristic submanifold
86
§10. Uniqueness of the characteristic submanifold Part II.
90
THE ENCLOSING THEOREM.
Chapter IV:
Singular surfaces and characteristic submanifolds.
§11. A lemma on essential intersections
104
§12. Proof of the enclosing theorem
109
Chapter V:
Singular submanifolds and characteristic submanifolds.
§13. An extension of the enclosing theorem
120
§14. Homotopy equivalences between 3-manifolds with torus boundaries
123
Part III.
THE SPLITTING THEOREMS.
Chapter VI:
Invariance of the characteristic submanifolds under homotopy equivalences.
§15. The preimage of an essential F-manifold
135
§16. Singular characteristic submanifolds
147
2
§17. The preimage of the characteristic submanifold
151
§18. Splitting a homotopy at the characteristic submanifold
155
Chapter VII:
Simple 3-manifolds.
§19. Isotopic surfaces in simple 3-manifolds
159
§20. Splitting a homotopy equivalence at a surface
165
§21. Splitting a homotopy at a surface
170
Part IV.
THE CONCLUSION OF THE PROOF OF THE CLASSIFICATION THEOREM.
Ch
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