Multiaxial Actions on Manifolds
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643 Michael Davis
Multiaxial Actions on Manifolds
Springer-Verlag Berlin Heidelberg New York 1978
Author Michael Davis Mathematics Department Columbia University New York, NY 10027/U.SA
Library of Congrpss Cataloging in Publication Data
Davis, Nichael, 1949Hultiaxial actions on maroifolds_ (Lecture notes in mathematics ; 643) Includes bibliographical references and index. 1. Topological transformation groups. 2. Lie groups. 3. Manifolds (Hathematics) I. Title. II. Series: LectUre notes in mathematics (Berlin) ; 643. QA3.L28 no. 643 [QA613.7] 510'.5s [522'.55) 78-3765
AMS Subject Classifications (1970): 57E15 ISBN 3-540-08667-6 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-08667-6 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 1978 Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
PREFACE These are the notes for a series of five lectures which I gave in the Transformation Groups Seminar at the Institute for Advanced Study during February of 1977 .
They concern the study of smooth
actions of the compact classical groups (O(n) ,U(n) or Sp(n»
which
resemble or are "modeled on" the linear representation k Pn '
In the
literature
such actions have generally been called "regular" 0 (n) ,
U(n) or Sp(n)-actions; however, following a suggestion of Bredon, I have adopted the terminology "k-axial actions." My interest in these actions was ignited by the beautiful theory of biaxial actions on homotopy spheres discovered by the Hsiangs , JHnich, and Bredon.
Perhaps the most striking result in area is the
theorem , due to the Hsiangs and independently to J§nich , which essentially identifies the study of biaxial O(n)-actions on homotopy spheres (with fixed p oints) with k not theory.
Also of interest is
Hirzebruch's observation, that many Brieskorn varieties support canonical biax i al a c tions. In my thes i s ,
Such mater i al is discussed in Chapter I .
I studied the theory of k-axial O(n) , U(n) and
Sp(n) actions for arbitrary
k
such that n
~
k.
The main result,
here called the Structure Theorem, i s proved (in outline) in Chapter IV.
This theorem implies that (assuming a certain obviously necessary
condition) any k - axial action is a pullback of its linear model.
In
Chapter VI, I indicate how this result can be combined with Smith theory and surgery theory in order to classify all such actions on homotopy spheres up to concordance . Many proofs are omitted and some are only sketched .
In the case
IV
of U(n) or Sp(n)-actions the classification up to
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