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Suppose $M$ is a compact orientable aspherical manifold and $G$ its fundamental group. Is there a nice description of representatives of the fundamental class of $M$ and its dual in the (homogenous) bar complex of $G$? I'm especially interested in the dual fundamental class and I think it might have the more natural description.

If $M$ is negatively curved and one fixes a base point in $\tilde{M}$ one can probably map an $n+1$-tuple of group elements to the volume of their convex hull in $\tilde{M}$ to get a representative of the dual fundamental class, but I don't see how this extends to the non negatively curved case.

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  • $\begingroup$ Something like this was done for the Klein bottle (with mod 2 coefficients) in arxiv.org/abs/1612.03133 using Fox calculus. I suspect the method could work for orientable surfaces and integer coefficients. Not sure about higher dimensions. $\endgroup$
    – Mark Grant
    Oct 15, 2021 at 15:05

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