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In the book Cohomology of Groups by Brown it is proved that if $G = F/R$ is a presentation of a group ($G$ is a set, $F$ a set of relations) then we have:

$$H_2(G) = [F, F] / [F, R].$$

I read somewhere that there are similar formulas in higher dimensions. I need some reference for them.

A collateral question could be: are those formulas useful in any sense?

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see the article: https://ac.els-cdn.com/S0001870807003015/1-s2.0-S0001870807003015-main.pdf?_tid=8bde0553-5195-4d59-a77f-6a2f1970e6e2&acdnat=1531763502_2fde10e4c97488cfe98f73d6d8d8a75b

also consider the bbbliography in the introduction (brown and ellis and two other).

key words: hopf formulae in higher dimensions

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