Volume 2, issue 2 (2002)

Download this article
For printing
Recent Issues

Volume 20
Issue 7, 3219–3760
Issue 6, 2687–3218
Issue 5, 2145–2685
Issue 4, 1601–2143
Issue 3, 1073–1600
Issue 2, 531–1072
Issue 1, 1–529

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
Author Index
To Appear
 
Other MSP Journals
Smith equivalence and finite Oliver groups with Laitinen number 0 or 1

Krzysztof Pawałowski and Ronald Solomon

Algebraic & Geometric Topology 2 (2002) 843–895

arXiv: math.AT/0210373

Abstract

In 1960, Paul A. Smith asked the following question. If a finite group G acts smoothly on a sphere with exactly two fixed points, is it true that the tangent G–modules at the two points are always isomorphic? We focus on the case G is an Oliver group and we present a classification of finite Oliver groups G with Laitinen number aG = 0 or 1. Then we show that the Smith Isomorphism Question has a negative answer and aG 2 for any finite Oliver group G of odd order, and for any finite Oliver group G with a cyclic quotient of order pq for two distinct odd primes p and q. We also show that with just one unknown case, this question has a negative answer for any finite nonsolvable gap group G with aG 2. Moreover, we deduce that for a finite nonabelian simple group G, the answer to the Smith Isomorphism Question is affirmative if and only if aG = 0 or 1.

Keywords
finite group, Oliver group, Laitinen number, smooth action, sphere, tangent module, Smith equivalence, Laitinen–Smith equivalence
Mathematical Subject Classification 2000
Primary: 57S17, 57S25, 20D05
Secondary: 55M35, 57R65.
References
Forward citations
Publication
Received: 15 September 2001
Accepted: 17 June 2002
Published: 15 October 2002
Authors
Krzysztof Pawałowski
Faculty of Mathematics and Computer Scienc
Adam Mickiewicz University
ul. Umultowska 87
61-614 Poznań
Poland
Ronald Solomon
Department of Mathematics
The Ohio State University
231 West 18th Avenue
Columbus, OH 43210–1174
USA