Tohoku Mathematical Journal 2016 September SECOND SERIES VOL. 68, NO. 3

 Tohoku Math. J. 68 (2016), 457-469

Title RICHNESS OF SMITH EQUIVALENT MODULES FOR FINITE GAP OLIVER GROUPS

Dedicated to Professor Mikiya Masuda on his sixtieth birthday

Author Toshio Sumi

(Received February 25, 2014, revised January 29, 2015)
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 $G$.

Mathematics Subject Classification. Primary 57S17; Secondary 20C15.

Key words and phrases. Smith problem, tangential representation, gap group, Oliver groups.