Journal of Lie Theory

no. 3
Cohomology of Coinvariant Differential Forms
Abdelhak Abouqateb1; Mohamed Boucetta1; Mehdi Nabil2
1Dept. of Mathematics, Faculty of Sciences and Technologies, Cadi Ayyad University, Gueliz Marrakesh, Morocco
2Dept. of Mathematics, Faculty of Sciences and Technologies, Cadi Ayyad University, Gueliz Marrkesh, Morocco
Journal of Lie Theory, Volume 28 (2018) no. 3, pp. 829-841
\newcommand{\diff}{\mathrm{Diff}} Let $M$ be a smooth manifold and $\Gamma$ a group acting on $M$ by diffeomorphisms; which means that there is a group morphism $\rho\colon \Gamma\rightarrow \diff(M)$ from $\Gamma$ to the group of diffeomorphisms of $M$. For any such action we associate a cohomology ${\rm H}(\Omega(M)_\Gamma)$ which we call the cohomology of $\Gamma$-coinvariant forms. This is the cohomology of the graded vector space generated by the differentiable forms $\omega - \rho(\gamma)^*\omega$ where $\omega$ is a differential form with compact support and $\gamma\in \Gamma$. The present paper is an introduction to the study of this cohomology. More precisely, we study the relations between this cohomology, the de Rham cohomology and the cohomology of invariant forms ${\rm H}(\Omega(M)^\Gamma)$ in the case of isometric actions on compact Riemannian oriented manifolds and in the case of properly discontinuous actions on manifolds.
DOI: 10.5802/jolt.1028
Classification: 57S15, 14F40, 14C30
Keywords: Cohomology, transformation groups, Hodge theory

Abdelhak Abouqateb  1 ; Mohamed Boucetta  1 ; Mehdi Nabil  2

1 Dept. of Mathematics, Faculty of Sciences and Technologies, Cadi Ayyad University, Gueliz Marrakesh, Morocco
2 Dept. of Mathematics, Faculty of Sciences and Technologies, Cadi Ayyad University, Gueliz Marrkesh, Morocco 
Abdelhak Abouqateb; Mohamed Boucetta; Mehdi Nabil. Cohomology of Coinvariant Differential Forms. Journal of Lie Theory, Volume 28 (2018) no. 3, pp. 829-841. doi: 10.5802/jolt.1028
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     title = {Cohomology of {Coinvariant} {Differential} {Forms}},
     journal = {Journal of Lie Theory},
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