A new family of finite Oliver groups satisfying the Laitinen Conjecture

TL;DR

This paper introduces a new infinite family of finite Oliver groups for which the Laitinen Conjecture holds, advancing understanding of group actions on spheres with two fixed points.

Contribution

It identifies a novel infinite family of Oliver groups satisfying the Laitinen Conjecture using induction of group representations.

Findings

Confirmed the Laitinen Conjecture for the new family of Oliver groups.

Provided a representation-theoretic approach to the conjecture.

Expanded the class of groups known to satisfy the conjecture.

Abstract

This paper is concerned with the Laitinen Conjecture. The conjecture predicts an answer to the Smith question which reads as follows. Is it true that for a finite group acting smoothly on a sphere with exactly two fixed points, the tangent spaces at the fixed points have always isomorphic group module structures defined by differentiation of the action? Using the technique of induction of group representations, we indicate a new infinite family of finite Oliver groups for which the Laitinen Conjecture holds.