The de Rham cohomology of a Lie group modulo a dense subgroup

Brant Clark; Francois Ziegler

doi:10.1090/proc/17595

The de Rham cohomology of a Lie group modulo a dense subgroup

Differential Geometry 2026-04-22 v2 Algebraic Topology

Authors: Brant Clark , Francois Ziegler

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Abstract

Let $H$ be a dense subgroup of a Lie group $G$ with Lie algebra $\mathfrak g$ . We show that the (diffeological) de Rham cohomology of $G/H$ equals the Lie algebra cohomology of $\mathfrak g/\mathfrak h$ , where $\mathfrak h$ is the ideal $\{Z\in\mathfrak g:\exp(tZ)\in H \text{ for all } t\in\mathbf R\}$ .

Keywords

de Rham cohomology Lie group dense subgroup Lie algebra cohomology diffeological cohomology

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@article{arxiv.2407.07381,
  title  = {The de Rham cohomology of a Lie group modulo a dense subgroup},
  author = {Brant Clark and Francois Ziegler},
  journal= {arXiv preprint arXiv:2407.07381},
  year   = {2026}
}

Comments

Genuine introduction (motivation / history) and appendix A (little known subgroup properties) added per referee request. Accepted version, to appear in Proceedings of the AMS

The de Rham cohomology of a Lie group modulo a dense subgroup

Abstract

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