A Jordan 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)$.