# Compactly supported cohomology of a topological DM stack

Consider a separated topological Deligne-Mumford stack $$\mathfrak X$$, i.e., a topological stack which is presentable by a proper etale topological groupoid (equivalently, $$\mathfrak X$$ is locally isomorphic to the stack quotient $$[X/G]$$ for $$X$$ a Hausdorff space and $$G$$ a finite group). Let's assume that $$\mathfrak X$$ is locally compact Hausdorff, i.e., for every etale map (or equivalently, one etale atlas) $$X\to\mathfrak X$$, where $$X$$ is a topological space, $$X$$ has an open cover by locally compact Hausdorff spaces.

Is there a natural way to define the compactly supported cohomology groups $$H^i_c(\mathfrak X,\mathbb Q)$$?

I would like the definition to agree with $$H^i_c(X,\mathbb Q)^G$$ for the simple case of a global quotient stack $$\mathfrak X = [X/G]$$, where $$G$$ is a finite group. Another property I'd like to have is a Thom isomorphism like statement, i.e., if we have an oriented rank $$r$$ vector bundle $$\pi:\mathfrak E\to\mathfrak X$$, then we have isomorphisms $$\pi_!:H^{*+r}_c(\mathfrak E,\mathbb Q)\to H^*_c(\mathfrak X,\mathbb Q)$$.

• The standard approach in AG (see Laszlo-Olsson papers) is as follows: take the nerve $X^\bullet$ of the atlas, and set $R\Gamma_c(\mathfrak{X}, \mathbf{Q}) = \mathrm{colim} R\Gamma_c(X^\bullet, \mathbf{Q})$, where the colimit is over the simplex category (and derived...); it makes sense because compactly supported cohomology is covariantly functorial for etale maps. Does this not apply here? – witt-voodoo Dec 1 '19 at 16:15
• Thanks for the reference. I'll take a look at the papers you mention and see if I'm able to get what I need from there. – Mohan Swaminathan Dec 1 '19 at 16:32
• I just took a look at the Laszlo-Olsson papers myself. It seems they do not follow the approach above, but instead take an approach via duality. (Their comment about a dual approach suggested by Gabber might be related to my previous comment.) Putting aside the question of whether or not such an approach is worked out in the literature, can you adopt their approach via duality to your setting? – witt-voodoo Dec 1 '19 at 17:51
• The spaces/stacks I want to apply this to might be singular (i.e. not topological orbifolds), so I don't know if any version of Poincare duality can apply here. – Mohan Swaminathan Dec 1 '19 at 18:01
• Smoothness is not relevant here (at least in the AG context). But it is important to have a dualizing complex and Verdier duality to run this construction. I was under the impression that these notions are available for locally compact Hausdorff spaces, but perhaps I'm wrong... – witt-voodoo Dec 1 '19 at 18:11