Jordan’s theorem for the diffeomorphism group of some manifolds

Let M M be a compact connected n n -dimensional smooth manifold admitting an unramified covering M ~ → M M M with cohomology classes α 1, …, α n ∈ H 1 (M ~ ; Z) ₁, , ₙ H¹ (M; Z) satisfying α 1 ∪ ⋯ ∪ α n ≠ 0 ₁ ₙ 0. We prove that there exists some number c c such that: (1) any finite group of diffeomorphisms of M M contains an abelian subgroup of index at most c c ; (2) if χ (M) ≠ 0 (M) 0, then any finite group of diffeomorphisms of M M has at most c <mml: annotation encoding