ALMOST FIXED POINTS OF FINITE GROUP ACTIONS ON MANIFOLDS WITHOUT ODD COHOMOLOGY

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 ≤ $$ \mathcal{G} $$ there is some x ∈ X whose stabilizer Gx ≤ $$ \mathcal{G} $$ satisfies [G : Gx] ≤ 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 ℝ. The main ingredients in the proof are: (1) the Jordan property for diffeomorphism groups of compact manifolds with nonzero Euler characteristic, and (2) the study of λ-stability, a condition on actions of finite abelian groups on manifolds that we introduce in this paper.