Almost fixed points of finite group actions on manifolds without odd cohomology

TL;DR

The paper proves that smooth manifolds without odd cohomology have the almost fixed point property under the action of their diffeomorphism groups, extending to certain algebraic varieties.

Contribution

It establishes the almost fixed point property for manifolds with no odd cohomology and applies this to real affine varieties with the same cohomological condition.

Findings

Manifolds with no odd cohomology have the almost fixed point property.

The property extends to real affine varieties with no odd cohomology.

Results connect topological properties with automorphism groups.

Abstract

If $X$ is a smooth manifold and ${\mathcal{G}}$ is a subgroup of $Diff(X)$ we say that $(X,{\mathcal{G}})$ has the almost fixed point property if there exists a number $C$ such that for any finite subgroup $G\leq{\mathcal{G}}$ there is some $x\in X$ whose stabilizer $G_x\leq G$ satisfies $[G:G_x]\leq C$. We say that $X$ has no odd cohomology if its integral cohomology is torsion free and supported in even degrees. We prove that if $X$ is compact and possibly with boundary and has no odd cohomology then $(X,Diff(X))$ has the almost fixed point property. Combining this with a result of Petrie and Randall we conclude that if $Z$ is a non necessarily compact smooth real affine variety, and $Z$ has no odd cohomology, then $(Z,Aut(Z))$ has the almost fixed point property, where $Aut(Z)$ is the group of algebraic automorphisms of $Z$ lifting the identity on $Spec\,{\mathbb{R}}$.