A compact manifold with infinite-dimensional co-invariant cohomology

Abstract

Let $M$ be a smooth manifold. When $\Gamma$ is a group acting on $M$ by diffeomorphisms, one can define the $\Gamma$-co-invariant cohomology of $M$ to be the cohomology of the complex $\Omega_c(M)_\Gamma=\operatorname{span}\{\omega-\gamma^*\omega : \omega\in\Omega_c(M),\,\gamma\in\Gamma\}$. 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 $\mathcal{C}_{\mathcal{G}}(M)=\operatorname{span}\{L_X\omega : \omega\in\Omega_c(M),\,X\in\mathcal{G}\}$. In this short paper we present a situation where these two cohomologies are infinite dimensional on a compact manifold.