Algebraic and Geometric Topology 2 (2002), paper no. 35, pages 843-895.

Smith equivalence and finite Oliver groups with Laitinen number 0 or 1

Krzysztof Pawalowski Ronald Solomon

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 a_G = 0 or 1. Then we show that the Smith Isomorphism Question has a negative answer and a_G > 1 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 a_G > 1. Moreover, we deduce that for a finite nonabelian simple group G, the answer to the Smith Isomorphism Question is affirmative if and only if a_G = 0 or 1.

Keywords. Finite group, Oliver group, Laitinen number, smooth action, sphere, tangent module, Smith equivalence, Laitinen-Smith equivalence.

AMS subject classification. Primary: 57S17, 57S25, 20D05. Secondary: 55M35, 57R65..

DOI: 10.2140/agt.2002.2.843

E-print: arXiv:math.AT/0210373

Submitted: 15 September 2001. Accepted: 17 June 2002. Published: 15 October 2002.

Notes on file formats

Krzysztof Pawalowski Ronald Solomon
Faculty of Mathematics and Computer Science, Adam Mickiewicz University
ul. Umultowska 87, 61-614 Poznan, Poland
Department of Mathematics, The Ohio State University
231 West 18th Avenue, Columbus, OH 43210--1174, USA


AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to