A compact manifold with infinite-dimensional co-invariant cohomology

TL;DR

This paper constructs examples of smooth manifolds with group or Lie algebra actions where the associated co-invariant and divergence form cohomologies are infinite-dimensional, highlighting complex topological structures.

Contribution

It introduces specific scenarios demonstrating infinite-dimensional cohomologies for group and Lie algebra actions on manifolds, expanding understanding of their topological complexity.

Findings

Co-invariant cohomology can be infinite-dimensional.

Cohomology of divergence forms can be infinite-dimensional.

Provides explicit examples of such phenomena.

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 $\Omega_c(M)_\Gamma=\mathrm{span}\{\omega-\gamma^*\omega,\;\omega\in\Omega_c(M),\;\gamma\in\Gamma\}.$ 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.