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Inducing of exotic smooth two fixed point actions on spheres

Piotr Mizerka

Abstract

This paper is concerned with 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? We show that one can answer this question negatively by using the technique of induction of group representations. We apply our results to indicate new dimensions of spheres admitting actions of specific Oliver groups, which give the negative answer to the Smith question. In particular, for the first time, we indicate some solvable non-nilpotent Oliver groups which yield negative answers to the Smith question.

Inducing of exotic smooth two fixed point actions on spheres

Abstract

This paper is concerned with 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? We show that one can answer this question negatively by using the technique of induction of group representations. We apply our results to indicate new dimensions of spheres admitting actions of specific Oliver groups, which give the negative answer to the Smith question. In particular, for the first time, we indicate some solvable non-nilpotent Oliver groups which yield negative answers to the Smith question.

Paper Structure

This paper contains 8 sections, 17 theorems, 28 equations.

Table of Contents

  1. Induction homomorphisms
  2. Proof of Theorem \ref{['theorem:A']}
  3. Application to the case $\operatorname{SL}(2,q)\trianglelefteq\operatorname{GL}(2,q)$
  4. Existence of Smith exotic actions
  5. Sufficient conditions for an exotic action
  6. Ensuring that the induced modules are Smith equivalent
  7. Application to direct products with $\operatorname{SL}(2,5)$ as a direct summand
  8. Examples for which Conjecture \ref{['conjecture:C']} holds

Key Result

Theorem A

Let $G$ be a finite group and $H$ its a subgroup. Then the following holds.

Theorems & Definitions (36)

  • Theorem A
  • Theorem B
  • Conjecture C
  • Theorem 1.1
  • Lemma 1.2
  • proof
  • Lemma 1.3
  • proof
  • proof : Proof of Theorem \ref{['theorem:A']}
  • Corollary 1.4
  • ...and 26 more