Finite group actions on manifolds without odd cohomology

TL;DR

This paper proves a uniform bound on the structure of finite group actions on certain smooth manifolds with torsion-free, even-degree cohomology, confirming a conjecture of Ghys using classification of finite simple groups.

Contribution

It establishes a universal bound on the abelian subgroups of finite groups acting smoothly on manifolds with even cohomology, confirming Ghys's conjecture.

Findings

Existence of a constant C bounding the index of abelian subgroups

Such subgroups can be generated by a number of elements related to manifold dimension

The fixed point set's Euler characteristic matches that of the manifold components

Abstract

Let $X$ be a compact smooth manifold, possibly with boundary. Denote by $X_1,\dots,X_r$ the connected components of $X$. Assume that the integral cohomology of $X$ is torsion free and supported in even degrees. We prove that there exists a constant $C$ such that any finite group $G$ acting smoothly and effectively on $X$ has an abelian subgroup $A$ of index at most $C$, which can be generated by at most $\sum_i[\dim X_i/2]$ elements, and which satisfies $\chi(X_i^A)=\chi(X_i)$ for every $i$. This proves, for all such manifolds $X$, a conjecture of \'Etienne Ghys. An essential ingredient of the proof is a result on finite groups by Alexandre Turull and the author which uses the classification of finite simple groups.