Revista de la
Unión Matemática Argentina
A compact manifold with infinite-dimensional co-invariant cohomology
Mehdi Nabil
Volume 63, no. 2 (2022), pp. 317–325

Download PDF


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.


  1. A. Abouqateb, M. Boucetta and M. Nabil, Cohomology of coinvariant differential forms, J. Lie Theory 28 (2018), no. 3, 829–841. MR 3808895.
  2. A. Abouqateb, Cohomologie des formes divergences et actions propres d'algèbres de Lie, J. Lie Theory 17 (2007), no. 2, 317–335. MR 2325702.
  3. A. El Kacimi-Alaoui, Invariants de certaines actions de Lie. Instabilité du caractère Fredholm, Manuscripta Math. 74 (1992), no. 2, 143–160. MR 1147559.
  4. W. Greub, S. Halperin and R. Vanstone, Connections, Curvature, and Cohomology. Vol. II, Pure and Applied Mathematics, Vol. 47-II, Academic Press, New York, 1973. MR 0336651.