Proceedings of the Conference on Transformation Groups: New Orleans, 1967

469 Pages·1968·10.736 MB·English

by  William Browder (auth.), Paul S. Mostert (eds.)

Proceedings
of the Conference on
Transformation Groups
New Orleans, 1967
Edited by Paul S. Mostert
Springer-Verlag New York Inc. 1968
PAUL S. MOSTERT
Tulane University
Department of Mathematics
New Orleans, Louisiana/USA
ISBN· 13: 978·3·642-46 I 43·9 e·1SBN· J3: 978·3·642-46 I 4 1·5
DOl: 10.1007/978·3·642·46141·5
All rights reserved. No part of this book may be translated or reproduced in any fonn without written
permission from the Publishers. © by Springer-Verlag Berlin' Heidelberg 1968. Library of
Congress Catalog Card Number 68-27313. Title No. D14
Softcover reprint of the hardcover 1st edition 1968
Preface
These Proceedings contain articles based on the lectures and in
formal discussions at the Conference on Transformation Groups held
at Tulane University, May 8 to June 2, 1967 under the sponsorship of
the Advanced Science Seminar Projects of the National Science Foun
dation (Contract No. GZ 400). They differ, however, from many such
Conference proceedings in that particular emphasis has been given to
the review and exposition of the state of the theory in its various mani
festations, and the suggestion of direction to further research, rather
than purely on the publication of research papers. That is not to say
that there is no new material contained herein. On the contrary, there
is an abundance of new material, many new ideas, new questions, and
new conjectures~arefully incorporated within the framework of the
theory as the various authors see it.
An original objective of the Conference and of this report was
to supply a much needed review of and supplement to the theory
since the publication of the three standard works, MONTGOMERY
and ZIPPIN, Topological Transformation Groups, Interscience Pub
lishers, 1955, BOREL et aI., Seminar on Transformation Groups,
Annals of Math. Surveys, 1960, and CONNER and FLOYD, Differen
tial Periodic Maps, Springer-Verlag, 1964. Considering this objective
ambitious enough, it was decided to limit the survey to that part of
Transformation Group Theory derived from the Montgomery School.
Thus, that part of the theory generally referred to as Topological Dynam
ics has been purposely excluded, in order to concentrate on the narrower
objektive.
The survey articles contain few proofs. However, they do give
adequate bibliographical data to enable the reader to check with the
original sources. Since the objective of the Conference (as reflected in
these Proceedings) was to review, consolidate, and give new direction
to the research in Transformation Groups, such omission is not in
appropriate. However, not all articles are of a primarily expository
nature. But, in the ordering of the papers, precedence is given to the
expository papers, so that the first three or four papers in each part
are of this nature, and indeed these papers form the nucleus around
which the part is organized, and give it its character. With very few
exceptions, all papers, whether expository or not, were either reported
IV Preface
on during the Conference or were inspired by work during the Con
ference.
The most dynamic movement lately in the Theory of Transformation
Groups, and at the same time the subject almost totally lacking in
coverage by the standard works referred to above, is the theory of
compact connected Lie groups acting differentiably on a manifold.
Main tools in this branch of the theory have been representation theory
and differential topology-particularly the Browder-N ovikov theory. We
have prevailed upon W. BROWDER to write an exposition of this theory
as it applies or is applicable to the Theory of Transformation Groups.
This article immediately follows the Introductory Remarks by MONT
GOMERY and precedes the division of the material into parts. The four
parts are then:
Part I Differentiable Transformation Groups
Part II Algebraic Topological and Other Techniques
Part III Compact Non-Lie Transformation Groups
Part IV Non-Compact Transformation Groups.
These divisions must at some point become arbitrary. Thus, for example,
Parts II and III have considerable overlap, with BREDON'S review of
cohomological methods in II rather than in III, though much of the
work there applies equally well to compact non-Lie group actions.
Each part contains a section for short notes and observations,
examples, etc., and a section on problems. It should be pointed out,
however, that many problems are contained within the various papers
and are not repeated in the sections on problems. There is one paper
by the HSIANG brothers devoted entirely to problems.
To the contributors of this volume, I wish to extend my sincere
thanks for the enthusiasm and energy which they have given to this
task, and for the cooperation they have shown in meeting the deadlines.
Although it is perhaps unjust to single out any of the individual authors,
still the enormous contributions of G. BREDON and W.-Y. HSIANG de
serve special mention. Also, special thanks are due H.-T. and M.-C. Ku
for their help in reading some of the manuscripts for mathematical and
typographical errors. To C.-T. YANG for suggesting the Conference, to
K. H. HOFMANN for his help in organizing the Conference, and to the
Mathematics Department of Tulane University for making available
unusually comfortable facilities at the expense of some invonvenience
to faculty and graduate students, there is further thanks due. For
comments and suggestions as to the proper ordering of problems and
about other matters, I wish to express my appreciation to PETER ORLIK.
Preface v
Finally, to Dr. KLAUS PETERS and Springer-Verlag for the skillful
and expeditious handling of the printing and publication, the entire
community of transformation group theorists owes a debt of gratitude.
Princeton, May 15, 1968 PAUL S. MOSTERT
Contents
Preface . . . . . . . . . . . . . III
List of Participants at the Conference X
List of Contributors. . . . . . . . XI
MONTGOMERY, D.: Introductory Remarks XIII
BROWDER, W.: Surgery and the Theory of Differentiable Trans-
formation Groups . . . . . . . . . . . . . . . . . . .
PAR T I Differentiable Transformation Groups
BREDON, G. E.: Exotic Actions on Spheres . . . . . . . . 47
HSIANG, W.-Y. : A Survey on Regularity Theorems in Differentiable
Compact Transformation Groups . . . . . . . . . . .. 77
MONTGOMERY, D. and C. T. YANG: Differentiable Actions on
Homotopy Seven Spheres, II . . . . . . . . . . . . . . 125
lANICH, K.: On the Classification of Regular O(n)-Manifolds in
Terms of their Orbit Bundles . . . . . . . . . . . . . . 135
LIVESAY, G. R. and C. B. THOMAS: Involutions on Homotopy
Spheres. . . . . . . . . . . . . . . . . . . . . . . . 143
HIRZEBRUCH, F.: Involutionen auf Mannigfaltigkeiten . . . . . 148
L6PEZ DE MEDRANO, S.: Some Results on Involutions of Homo-
topy Spheres . . . . . . . . . . . . . . . . . . . . . 167
MONTGOMERY, D. and C. T. YANG: Free Differentiable Actions
on Homotopy Spheres . . . . . . . . . . . . . . . . . 175
Su, 1. c.: Some Results on Cyclic Transformation Groups on
Homotopy Spheres. . . . . . . . . . . . . . . . 193
ORLIK, P.: Examples of Free Involutions on 3-Manifolds. . . . 205
SHORT NOTES
LEE, C. N.: Cyclic Group Actions on Homotopy Spheres. 207
LEE, R.: Non-Existence of Free Differentiable Actions of Sl and
7L2 on Homotopy Spheres. . . . . . . . . . . . . . . . 208
CALABI, E.: On Differentiable Actions of Compact Lie Groups on
Compact Manifolds . . . . . . . . . . . . 210
Ku, H.-T.: Examples of Differentiable Involutions. . . . . . . 214
VIII Contents
NEUMANN, W. D.: 3-Dimensional G-Manifolds with 2-Dimensio-
nal Orbits . . . . . . . . . . . . . . . . . . . . .. 220
HSIANG, W.-C. and W.-Y. HSIANG: Some Problems in Differenti-
able Transformation Groups 223
Problems . . . . . . . . . . . . . . . . . . . . . . .. 235
PART II Algebraic Topological and Other Techniques
LEE, C. N.: Equivariant Homology Theories . . . . . . .. 237~
BREDON, G. E.: Cohomological Aspects of Transformation·
Groups. . . . . . . . . . . . . . . . . . . . . . .. 247
BREDON, G. E.: Equivariant Homotopy . . . . . . . . . .. 281
MANN, L. N.: Gaps in the Dimensions of Compact Transfor
mation Groups . . . . . . . . . . . . . . . . . . .. 293
ORLIK, P. and F. RAYMOND: Actions of 80(2) on 3-Manifolds . 297
Ku, H.-T.: The Fixed Point Set of Zp in a C* (m1,m2, ••• ,mq- 1; Zp)-
Space . . . . . . . . . . . . . . . . . . 319
FARY, I.: Group Actions and a Spectral Sequence.. . . . . .. 325
SHORT NOTES
RA YMOND, F.: Exotic PL Actions which are Topologically Linear 339
Ku, H.-T. and M.-C. Ku: The Lefschetz Fixed Point Theorem
for Involutions . . . . . . . . . . . . . . . . . . .. 341
KWUN, K. W.: Involutions on the n-Cell. . . . . . . . . .. 343
RAYMOND, F.: Topological Actions do not Necessarily have
Reasonable Slices . . . . . . . . . . . . . . . . . " 345
Ku, H.-T. and M.-C. Ku: A Simple Proof of the Maximal Tori
Theorem ofE. CARTAN. . . . . . . . . . . . . . .. 346
SCHAFER, J. A. : A Relation between Group Actions and Index 349
Su, J. c.: An Example. 351
Problems . . . . . . . . . . . . . . . . . . . . . .. 352
PART III Compact (Non-Lie) Transformation Groups
RAYMOND, F.: Cohomological and Dimension Theoretical Pro
perties of Orbit Spaces of p-Adic Actions . . . . . . . . . 354
WILLIAMS, R:F.: Compact Non-Lie Groups . . . . . . . .. 366
HOFMANN, K. H. and P. S. MOSTERT: Applications of Transfor-
mation Groups to Problems in Topological Semi groups . 370
Ku, M.-C.: On the Action of Compact Groups . . . . 381
Ku, H.-T.: A Generalization of the Conner Inequalities . . 401
Contents IX
SHORT NOTES
MOSTERT, Po So: Some Simple Observations about Ap-Actions 415
0
Problems 418
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
PART IV Non-Compact Transformation Groups
FARY, I.: On Hilbert's Theory of Closed Group Actions 419
0 0
RICHARDSON, Ro: On the Variation of Isotropy Subalgebras 429
0
WILLIAMS, Ro Fo: Non-Compact Lie Group Actions 441
0 0 0 0
WEST, Jo Eo: Fixed Point Sets of Transformation Groups on
Separable Infinite-Dimensional Frechet Spaces 446
0 0 0 0 0 0
KINOSHITA, So: On a Kind of Discrete Transformation Group 451
0
Problems 457
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
List of Participants at the Conference
BREDON, G. E. LIVESAY, G. R.
BROWDER, W. LOPEZ DE MEDRANO, S.
CONNELL, E. MANN, L. N.
FARY, I. MONTGOMERY, D.
FELDMAN, L. A. MOSTERT, P. S.
FLOYD, E. E. NOVIKOV, S. P.
GOTO, M. ORLIK, P.
HOFMANN, K. H. PAK, J.
Hoo, C.-S. RAYMOND, F.
HSIANG, W.-c. RICHARDSON, R. W.
HSIANG, W.-Y. ROSEMAN, D.
JANICH, K. SAMELSON, H.
KINOSHITA, S. SU, J. C.
Ku, H.-T. THOMAS, C. B.
Ku, M.-C. WANG, H.-C.
KWUN, K. W. WASSERMAN, A.
LANDWEBER, P. S. WEST, J. E.
LEE, C. N. WILLIAMS, R. F.
LEE, R. YANG, C. T.
LININGER, L.
List of Contributors
BREDON, G. E., Rutgers University, Department of Mathematics, New
Brunswick, New Jersey/USA.
BROWDER, W., Princeton University, Department of Mathematics,
Princeton, New Jersey/USA.
CALABI, E., Institut des Hautes Etudes Scientifiques, Bures-sur-Yvette/
France.
FARY, I., University of California at Berkeley, Department of Mathe
matics, Berkeley, California/USA.
HIRZEBRUCH, F., Mathematisches Institut der Universitat, Bonn/
Germany.
HOFMANN, K. H., Tulane University, Department of Mathematics,
New Orleans, Lousiana/USA.
HSIANG, W.-C., Yale University, Department of Mathematics, New
Haven, Connecticut/USA.
HSIANG, W.-Y., University of Chicago, Department of Mathematics,
Chicago, Illinois/USA.
JANIeH, K., Mathematisches Institut der Universitat, Bonn/Germany.
KINOSHITA, S., Florida State University, Department of Mathematics,
Tallahassee, Florida/USA.
Ku, H.-T., University of Massachusetts, Department of Mathematics,
Amherst, Massachusetts/USA.
Ku, M.-C., University of Massachusetts, Department of Mathematics,
Amherst, Massachusetts/USA.
KWUN, K. W., Michigan State University, Department of Mathematics,
East Lansing, Michigan/USA.
LEE, C. N., University of Michigan, Department of Mathematics, Ann
Arbor, Michigan/USA.
LEE, R., University of Michigan, Department of Mathematics, Ann
Arbor, Michigan/USA.
LIVESAY, G. R., Cornell University, Department of Mathematics,
Ithaca, New York/USA.
LOPEZ DE MEDRANO, S., Princeton University, Department of Mathe
matics, Princeton, New Jersey/USA, and Universidad National
Autonoma de Mexico, INIC, Mexico City/Mexico.
MANN, D. N., University of Massachusetts, Department of Mathe
matics, Amherst, Massachusetts/USA.