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.
Cite
@article{arxiv.2002.04735,
title = {Inducing of exotic smooth two fixed point actions on spheres},
author = {Piotr Mizerka},
journal= {arXiv preprint arXiv:2002.04735},
year = {2026}
}
Comments
After posting this manuscript, the author became aware of potential flaws in some proofs and conclusions (especially the statements after Definition 2.2). At present the author lacks the time to verify all details rigorously. Readers should treat the results with caution