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I'm trying to compute group cohomology $H^n(G,\mathbb{Z})$ of some crystal groups $G$ which are infinite but finitely generated groups. I succeed in obtaining cohomology groups using projective resolutions, "HomToIntegralModule" and "Cohomology". But the outcomes are just some abstract numbers representing Abelian groups. Is there any way of obtaining explicit cocycles using GAP?

Note: when the group $G$ acts trivially on $\mathbb{Z}$, I know that we have some way of doing it as in http://hamilton.nuigalway.ie/Hap/www/SideLinks/About/aboutCocycles.html. But I don't think this method works for the case where $G$ has non-trivial actions on $\mathbb{Z}$.

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  • $\begingroup$ AFAIR, GAP calculates (co)homology in a way described in G. Ellis, P. Smith, Computing group cohomology rings from the Lyndon–Hochschild–Serre spectral sequence, Journal of Symbolic Calculations 46, and it doesn't involve building explicit resolvents or any calculations in bar resolvent, so, I guess, no cocycles for you. $\endgroup$ – xsnl Mar 26 '18 at 18:25
  • $\begingroup$ Hi @xsnl, I think they actually use explicit resolvents to calculate the cohomology group as described in the following link: link. $\endgroup$ – Xu Yang Apr 6 '18 at 17:57

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