# De Rham cohomology of Lie groupoid

Let $$G$$ be a Lie group acting on a manifold $$M$$.

Consider the transformation groupoid $$\mathcal{G}=(G\times M\rightrightarrows M)$$. We have the notion of de Rham cohomology of a Lie groupoid by considering de Rham cohomology of simplicial manifold associated to it, denoted by $$\mathcal{G}_\bullet$$.

Is there any relation between de Rham cohomology of Lie groupoid $$\mathcal{G}$$ (eg page 11 of Laurent-Gengoux–Tu–Xu, Chern-Weil map for principal bundles over groupoids, Math. Z. 255 (2007) pp451–491, arXiv:math/0401420) and the de Rham (??) cohomology of the Lie group $$G$$ and the de Rham cohomology of the manifold $$M$$?

• What is DeRham Cohomology of a Lie groupoid ? Do you mean groupoid cohomology defined the same way as group cohomology but restricting to composable chains ? – InfiniteLooper Jul 11 '19 at 22:11
• Please see edit @InfiniteLooper – Praphulla Koushik Jul 11 '19 at 22:17
• In this special case your de Rham cohomology associated to the Lie groupoid is classically known as equivariant (Borel) cohomology. There is a Serre spectral sequence that has the flavour of what you want and is often used to compute equivariant cohomology; see (1.2.1) of math.ias.edu/~goresky/pdf/equivariant.jour.pdf. Note that first page of the Serre spectral sequence involves cohomology over $BG$, in general with twisted coefficient group. The relation to $H^*(G)$ is more subtle (this is the "Koszul duality" in the title of the linked paper). – Daniel Pomerleano Jul 12 '19 at 0:00
• @DanielPomerleano Thanks for the link. Can you give a reference for “Lie groupoid cohomology same thing as Equivariant cohomology”. Thank you :) – Praphulla Koushik Jul 12 '19 at 8:07

Proposition $$13$$ and Remark $$16$$ in page $$10$$ of Cohomology or Stacks says that, there is a natural isomorphism
$$H^i_G(X)\rightarrow H^i(X\times G\rightrightarrows X)$$ where, $$H^i_G(X)$$ is the $$i^{\text{th}}$$ equivariant cohomology of $$X$$ with respect to action of $$G$$ and $$H^i(X\times G\rightrightarrows X)$$ is the $$i^{\text{th}}$$ deRham cohomology of the transformation groupoid $$(X\times G\rightrightarrows X)$$.
• The existence of such an isomorphism does not depend on $G$ compact, as stated in Remark 16 of the paper you link. – Mike Miller Eismeier Jul 12 '19 at 11:58