Finite group actions on manifolds without odd cohomology

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 Étienne 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.

34 pages. v4: Corollary 1.2 of v3 (which in v4 is Theorem 1.3) is proved for arbitrary compact manifolds independently of Ghys' conjecture, using a group theoretical result of Guralnick and Lucchini; some other minor changes in the introduction

References in corpus (5)

• Groups acting on manifolds: around the Zimmer program
• Finite group actions on homology spheres and manifolds with nonzero Euler characteristic
• Finite subgroups of diffeomorphism groups
• Finite group actions on 4-manifolds with nonzero Euler characteristic
• Boosting an analogue of Jordan's theorem for finite groups

Cited by in corpus (10)

• Finite group actions on homology spheres and manifolds with nonzero Euler characteristic
• Diffeomorphism Groups of Compact 4-manifolds are not always Jordan
• Finite subgroups of diffeomorphism groups
• Non Jordan groups of diffeomorphisms and actions of compact Lie groups on manifolds
• Finite group actions on 4-manifolds with nonzero Euler characteristic
• Boosting an analogue of Jordan's theorem for finite groups
• Automorphism groups of $P^1$-bundles over a non-uniruled base
• Finite groups acting symplectically on $T^2\times S^2$
• Finite groups of bimeromorphic selfmaps of non-uniruled Kähler threefolds
• Automorphism groups of Moishezon threefolds