A compact manifold with infinite-dimensional co-invariant cohomology

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),...

Full description

Saved in:
Bibliographic Details
Published inarXiv.org
Main Author Mehdi Nabil
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 04.01.2021
Subjects
Homology
Invariants
Isomorphism
Lie groups
Online AccessGet full text

Cover

More Information
Summary: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.
ISSN:2331-8422