Group cohomology version of Deligne-Beilinson cohomology

Question

I appreciate Deligne-Beilinson cohomology as a topological cohomology generalization of de Rham cohomology, which concerns the topological structure of manifolds.

On the other hand, we know that there is Group Cohomology theory, suitable for describing the classifying spaces $BG$ of a group $G$, for example, see ncatlab: group+cohomology and ncatlab: Dijkgraaf-Witten gauge theory.

Is there a group cohomology version of Deligne-Beilinson cohomology, concerning classifying spaces $B^n G$ of a group $G$? What are some key intro References? Thank you in advance.

Answer 1

Note that Deligne-Beilinson cohomology is not really a "topological cohomology" generalization of de Rham cohomology, it depends on additional analytic structure. So one would expect that some additional analytical structure would be required on the classifying space of a group $G$ to have some Deligne-Beilinson group cohomology.

A version of Deligne-Beilinson cohomology for affine groups schemes equipped with a mixed Hodge structure has been studied recently (it applies in particular to completions of fundamental groups of smooth complex varieties w.r.t monodromy representations):

• R. Hain. Deligne-Beilinson cohomology of affine groups. arXiv:1507.03144, link to paper

Answer 2

The nth Deligne cohomology is defined as cohomology with coefficients in a truncated chain complex of sheaves of U(1)-valued differential forms: U(1)→Ω^1→Ω^2→⋯→Ω^n for some n≥0.

Thus starting with an arbitrary Lie group G one can take the truncated simplicial object of sheaves of groups of G-valued differential forms (defined using crystals, for example), and then take the sheaf cohomology with values in this simplicial presheaf.

This defines a nonabelian analog of the Deligne cohomology for any choice of the underlying site: smooth, holomorphic, algebraic.

Answer 3

Sorry, I just saw this question.

One can consider Deligne cohomology for simplicial manifolds in the rather obvious way, namely by adding an additional "simplicial manifold" direction to the ususal double complex, and then take the total cohomology. As resolutions one can use certain hypercovers.

For the simplicial manifold $BG$, I have described and done this is Section 2.2. of my paper

Waldorf, Konrad, Multiplicative bundle gerbes with connection, Differ. Geom. Appl. 28, No. 3, 313-340 (2010). ZBL1191.53022.

As explained there, the underlying version of "smooth group cohomology" is Brylinski's "Differentiable cohomology of gauge groups", which remained unpublished. It is equivalent to the Segal-Mitchison smooth group cohomology.

References on general simplicial Deligne cohomology are:

Brylinski, J.-L.; McLaughlin, D. A., The geometry of degree-four characteristic classes and of line bundles on loop spaces. I, Duke Math. J. 75, No. 3, 603-638 (1994). ZBL0844.57025.

Gomi, Kiyonori, Equivariant smooth Deligne cohomology, Osaka J. Math. 42, No. 2, 309-337 (2005). ZBL1081.14030.