#### 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

and

Department of Mathematics, The Ohio State
University

231 West 18th Avenue, Columbus, OH 43210--1174, USA

Email: kpa@main.amu.edu.pl, solomon@math.ohio-state.edu

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