Jordan’s theorem for the diffeomorphism group of some manifolds

Abstract:

Let $M$ be a compact connected $n$-dimensional smooth manifold admitting an unramified covering $\widetilde {M}\to M$ with cohomology classes $\alpha _1,\dots ,\alpha _n \in H^1(\widetilde {M};\mathbb {Z})$ satisfying $\alpha _1\cup \dots \cup \alpha _n\neq 0$. We prove that there exists some number $c$ such that: (1) any finite group of diffeomorphisms of $M$ contains an abelian subgroup of index at most $c$; (2) if $\chi (M)\neq 0$, then any finite group of diffeomorphisms of $M$ has at most $c$ elements. We also give a new and short proof of Jordan’s classical theorem for finite subgroups of $\mathrm {GL}(n,\mathbb {C})$, of which our result is an analogue for $\mathrm {Diff}(M)$.