Topology and Combinatorics of 3-Manifolds
This book is a study of combinatorial structures of 3-mani- folds, especially Haken 3-manifolds. Specifically, it is concerned with Heegard graphs in Haken 3-manifolds, i.e., with graphs whose complements have a free fundamental group. These graphs always
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Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
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Klaus Johannson
Topology and Combinatorics of 3-Manifolds
Springer
Author Klaus Johannson Department of Mathematics University of Tennessee Knoxville, TN 37996, USA E-mail: [email protected]
Mathematics Subject Classification (1991): 57M99
ISBN 3-540-59063-3 Springer-Verlag Berlin Heidelberg New York CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer- Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1995 Printed in Germany
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INTRODUCTION
This book is concerned with the combinatorial rigidity of 3-manifolds. Indeed, the main result of this book says that combinatorial structures of all Haken 3-manifolds without non-trivial Stallings fibrations are virtually rigid, i.e., rigid up to finitely many choices. It follows in particular that the combinatorial structures of all hyperbolic Haken 3-manifolds with infinite volume are virtually rigid. Recall that a Haken 3-manifold is a compact piecewise linear 3-manifold which is irreducible, in the sense that every PL-embedded disc or 2-sphere separates a 3-ball, and which is sufficiently large, in the sense that it contains an orientable surface whose fundamental group is non-trivial and whose embedding induces an injection on the fundamental groups. For instance, all non-trivial knot spaces are Haken 3-manifolds. At the very basis of 3-manifold theory there are the classical theorems that every compact 3-manifold has a piecewise linear structure, i.e., an atlas of charts whose transition maps are piecewise linear, and that any topological homeomorphism between piecewise linear 3-manifolds can be isotoped into a piecewise linear homeomorphism. In short every compact 3-manifold has a rigid piecewise linear structure. This theorem has been proved by Moise [Moi I, 2] (for other proofs see [Bin], [Sh] and [Ham]). It is common to realize the piecewise linear structure of a 3-manifold by a combinatorial structure, e.g., by a triangulation. But note that in doing so one loses information which is reflected by the fact that there are various different triangulations which realize the underlying piecewise linear structure of a 3-manifold. Thus the combinatorics of a 3-manifold is no longer rigid. From the rigidity of piecewise linear structures one can only deduce that any two triangulations of a 3-manifold have a common sub-division
