## Tohoku Mathematical Journal

### Richness of Smith equivalent modules for finite gap Oliver groups

Toshio Sumi

#### Abstract

Let $G$ be a finite group not of prime power order. Two real $G$-modules $U$ and $V$ are $\mathcal{P}(G)$-connectively Smith equivalent if there exists a homotopy sphere with smooth $G$-action such that the fixed point set by $P$ is connected for all Sylow subgroups $P$ of $G$, it has just two fixed points, and $U$ and $V$ are isomorphic to the tangential representations as real $G$-modules respectively. We study the $\mathcal{P}(G)$-connective Smith set for a finite Oliver group $G$ of the real representation ring consisting of all differences of $\mathcal{P}(G)$-connectively Smith equivalent $G$-modules, and determine this set for certain nonsolvable groups $G$.

#### Article information

Source
Tohoku Math. J. (2), Volume 68, Number 3 (2016), 457-469.

Dates
Revised: 29 January 2015
First available in Project Euclid: 23 September 2016

https://projecteuclid.org/euclid.tmj/1474652268

Digital Object Identifier
doi:10.2748/tmj/1474652268

Mathematical Reviews number (MathSciNet)
MR3550928

Zentralblatt MATH identifier
1361.57040

Subjects
Primary: 57S17: Finite transformation groups
Secondary: 20C15: Ordinary representations and characters

#### Citation

Sumi, Toshio. Richness of Smith equivalent modules for finite gap Oliver groups. Tohoku Math. J. (2) 68 (2016), no. 3, 457--469. doi:10.2748/tmj/1474652268. https://projecteuclid.org/euclid.tmj/1474652268

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