A compact manifold with infinite-dimensional co-invariant cohomology

Abstract

Let $M$ be a smooth manifold. When $Γ$ is a group acting on the manifold $M$ by diffeomorphisms one can define the $Γ$-co-invariant cohomology of $M$ to be the cohomology of the differential complex $Ω_c(M)_Γ=\mathrm{span}\{ω-γ^*ω,\;ω\inΩ_c(M),\;γ\inΓ\}.$ For a Lie algebra $\mathcal{G}$ acting on the manifold $M$, one defines the cohomology of $\mathcal{G}$-divergence forms to be the cohomology of the complex $\mathcal{C}_{\mathcal{G}}(M)=\mathrm{span}\{L_Xω,\;ω\inΩ_c(M),\;X\in\mathcal{G}\}.$ In this short paper we present a situation where these two cohomologies are infinite dimensional.