A compact manifold with infinite-dimensional co-invariant cohomology

Abstract

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