# how can we compute the homology of these groups without using topology?

I'd like to know the homology of a free group and a free abelian group of rank 2. I know that they could be computed topologically, but I'm searching a proof purely algebraic, could you help me please?

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Infinite cyclic groups are easily handled by writing down an explicit resolution: if we write $G$ for the infinite cyclic group generated by $\sigma$, we can use $$\mathbb ZG\xrightarrow{d}\mathbb ZG\stackrel\varepsilon\twoheadrightarrow\mathbb Z$$ with $\varepsilon$ the usual augmentation and $d$ the unique $\mathbb ZG$-linear map such that $d(1)=\sigma-1$.
Alternatively, every textbook includes an explicit description of the functors $H^0(G,\mathord-)$ and $H^1(G,\mathord-)$, which in the case of the infinite cyclic group makes them immediately computable, and then you can check any problems that $H^1(G,\mathord-)$ is right exact. It follows that the higher $H^p$'s are zero, and that we have computed the whole cohomology. One can proceed in a similar way for homology.