Seifert Manifolds
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Peter Orlik University of Wisconsin, Madison, WI/USA
Seifert Manifolds
Springer-Verlag Berlin· Heidelbera . New York 1972
AMS Subject Classifications (1970): Primary: 57-02, 55F55, 57AlO, 57E15 Secondary: 14J15, 55A05, 57D85
ISBN 3-540-06014-6 Springer-Verlag Verlin . Heidelberg . New York ISBN 0-387-06014-6 Springer-Verlag New York . Heidelberg . Berlin
This work is 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 the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1972. Library of Congress Catalog Card Number 72-90184. Printed in Germany. Offsetdruck: Julius Beltz, HemsbachiBergstr.
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Art i e
Introduction
These are notes for a lecture series given at the University of Oslo in 1971 - 1972.
Although the manifolds of the title were
constructed by Seifert [1J in 1933, considerable interest has been devoted to them recently.
The principal aim here is to sur-
vey the new results and to emphasize the variety of areas and techniques involved. The equivariant theory comprising the first four chapters was initiated by Raymond [1J. who discovered that two classes of Seifert manifolds coincide with certain fixed point free 3-dimensional
S1-manifolds.
Chapter 1 contains Raymond's classifica-
tion of
S1-actions on 3-manifolds. Chapter 2 describes equivariant plumbing of n2 - bundl e s over 2-manifolds and identifies the boundary 3-manifolds.
This is used in chapter 3 to resolve sin-
gularities of complex algebraic surfaces with
C*-action.
The
technique is to compute the Seifert invariants of a suitable neighborhood boundary of the singular point and use these to construct an equivariant resolution following Orlik-Wagreich [1,2J. The equivariant fixed point free cobordism classification of Seifert manjfolds due to Ossa [1J is given in chapter 4. The remaining chapters contain topological results.
The
homeomorphism classification by Orlik-Vogt-Zieschang [1J using fundamental groups is obtained in chapter 5. actions of finite groups on Seifert-Threlfall [1J. manifolds fiber over
The known free
S3 are given in chapter 6 following
In chapter 7 we determine which Seifert S1 •
The important results of Waldhausen
[1,2J are outlined in the last chapter together with a number of
VI other topics that we could not discuss in detail in the frame of the lectures. I would like to thank my friends Frank Raymond and Philip Wagreich for teaching me directly or through collaboration much of the contents of these notes; the mathematicians in Oslo in general and Per Holm and Jon Reed in particular for their hospitality; and Professor F. Hirzebruch for inviting me to Bonn and for recommending the pUblication
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