Smooth finite group actions on homology six-spheres with odd euler chracteristic fixed point sets

TL;DR

This paper classifies finite groups acting smoothly on homology six-spheres with fixed points having odd Euler characteristic, showing they are limited to specific well-known groups with a single fixed point.

Contribution

It establishes a classification of finite group actions on homology six-spheres with odd Euler characteristic fixed points, identifying the exact groups involved.

Findings

The acting group is isomorphic to A5, S5, or A5 × Z2.

The fixed point set consists of exactly one point.

The group actions are constrained to these specific groups.

Abstract

In this paper, we prove that if a finite group acts smoothly and effectively on an integral homology six-sphere and the fixed point set has an odd Euler characteristic, then the acting group is isomorphic to either the alternating group on five letters, the symmetric group on five letters, or the Cartesian product of the alternating group on five letters and a group of order 2 and the fixed point set consists of precisely one point.