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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


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.

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.
Forward citations
Received: 15 September 2001
Accepted: 17 June 2002
Published: 15 October 2002
Krzysztof Pawałowski
Faculty of Mathematics and Computer Scienc
Adam Mickiewicz University
ul. Umultowska 87
61-614 Poznań
Ronald Solomon
Department of Mathematics
The Ohio State University
231 West 18th Avenue
Columbus, OH 43210–1174