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Finite group actions on manifolds without odd cohomology

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 $χ(X_i^A)=χ(X_i)$ 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.