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Consider some oriented surface $S$ with fundamental group $\pi_1(S)$. The group cohomology of $\pi_1(S)$ with coefficients in $\mathbb{R}$ is isomorphic to the de Rham cohomology of $S$. In degree 2, integration gives a map $\int_S\colon H^2_{dR}(S;\mathbb{R})\to \mathbb{R}$.

In terms of group cohomology, with cochains being (inhomogeneous) maps $\pi_1(S)\times \pi_1(S)\to \mathbb{R}$ instead of 2-forms, what is the analogue of integration ? Say differently, is there a natural map $H^2(\pi_1(S);\mathbb{R})\to \mathbb{R}$ that corresponds to integration of differential forms ?

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    $\begingroup$ I think you would have to use cyclic homology/hochschild homology, and the map you are talking about is essentially the residue map. $\endgroup$
    – Bombyx mori
    Commented Aug 5, 2019 at 21:02
  • $\begingroup$ Thanks for your answer! Can you maybe elaborate a little bit (or share a link if you have a reference to suggest). Cheers ;) $\endgroup$
    – Arnaud Maret
    Commented Aug 6, 2019 at 14:03
  • $\begingroup$ It is not an answer, just some vague idea. If I know a proof, I would have written down an answer already. The question is a good one. One thing I want to point out is that by taking the "integral" technically you are evaluating the residue. And you do not need de Rham cohomology to define the residue. Since formally hochschild homology is an analog of group cohomology, there is a similar residue formula in literature. But this is all I know. $\endgroup$
    – Bombyx mori
    Commented Aug 6, 2019 at 18:32
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    $\begingroup$ There is such a thing as group homology. There is a perfect pairing $H^2(G;\mathbb R)\otimes H_2(G;\mathbb R)\to \mathbb R$. Thus integration is an element of group homology, just as it is an element of the homology of a surface. See also Poincaré duality groups and Bieri-Eckmann duality groups. $\endgroup$
    – Ben Wieland
    Commented Aug 7, 2019 at 20:32

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We always have a pairing $H_k(X, \mathbb{R}) \otimes H^k(X, \mathbb{R}) \to \mathbb{R}$ between chains and cochains. If $X$ is a closed oriented $n$-manifold, the orientation equips it with a class $[X] \in H_n(X, \mathbb{R})$ called the fundamental class, and pairing this class with elements in $H^n(X, \mathbb{R})$ reproduces integration of $n$-forms when $X$ is smooth. When $X$ is furthermore aspherical, so a $K(\pi_1(X), 1)$, then its homology and cohomology can be computed as group homology and cohomology of $\pi_1(X)$ as Ben Wieland says in the comments.

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