Continuous and smooth Lie groupoid cohomology

Question

In the paper by Weinstein and Xu: Extensions of symplectic groupoids and quantization, J. Reine Angew. Math. 417 (1991), there are two versions of Lie groupoid cohomology. The same differential $\delta$ can be in fact applied to globally smooth cochains or only to cochains which are globally continous and smooth in a neighbourhood of the diagonal.

If $\Gamma$ denotes the total space of the groupoid and $\Gamma^{(n)}$ the set of composable $n$-tuples then one can consider:

$C^n(\Gamma:\mathbb R)={\cal C}^\infty(\Gamma^{(n)},\mathbb R)$,

$C^n_{es}(\Gamma:\mathbb R)=\{\sigma:\Gamma^{(n)}\to\mathbb R, \sigma\quad \mathrm{smooth}\quad \mathrm{around}\quad \Delta^{(n)}\}$

and of course one could consider just continuous cochains

$C^n_0(\Gamma:\mathbb R)={\cal C}(\Gamma^{(n)},\mathbb R)$

In the same paper it is proven that the first two cohomologies are different by giving an example in which globally smooth 2-cohomology is 0 while $H^2_{es}(\Gamma;\mathbb R)$ is non zero (in fact in the example coefficients are in $\mathbb S^1$ but this should make no big difference).

Does anyone knows:

1. If also continuous cohomology differs from the two previous ones

2. Other examples in which the cohomologies are different and/or equal

3. General conditions under which the cohomologies are known to be equal.

Answer 1

Sorry for answering so late, I just was pointed to this question. Before answering your questions in the case of a Lie group (i.e., one object) let me point out that the difference between $\mathbb{R}$-coefficients and $\mathbb{S}^1$-coefficients it not negligible but is somehow the whole heart of the story.

1. The cohomology of the continuous cochains is equal to the cohomology of the smooth cochains. Moreover, there is yet another complex $C^n_{ec}$ of cochains that are continuous around $\Delta^{(n)}$. Then $H^n_{ec}$ equals $H^n_{es}$ (all this for coefficients in $\mathbb{R}$, $\mathbb{S}^1$ or more general tori).

2. The universal cover $\mathbb{Z}\to\mathbb{R}\to\mathbb{S}^1$ is described by a cocycle in $Z^2_{es}$ that cannot be smooth (otherwise $ \mathbb{R}\to\mathbb{S}^1$ would have a smooth global section), see also 3.

3. $H^n_{es}=H^n$ (in your notation form above) if either the Lie group $G$ or the coefficients are contractible (in particular the difference between $\mathbb{R}$ and $\mathbb{S}^1$ plays a significant role).

A reference for the equality of smooth and continuous cohomology is Hochschild, Mostow, "Cohomology of Lie groups" and for the more general case this paper, together with F. Wagemann.

I expect that the case of general Lie groupoids can be treated similarly, for instance I would expect that $H^n=H^n_{es}$ if the Lie groupoid $\Gamma$ is topologically trivial (i.e., its simplicial manifold $N\Gamma$ is contractible). However, the generalisation of our approach is not entirely straight forward.