What is the homology group $H_k(T^2 x T^2 x T^2, \mathbb{Z})$? The homology group (which I think of as the number of equivalence classes of $k$-cycles, but maybe that is wrong) for the $n$-torus is $\mathbb{Z}=\binom n k$, but I don't know if the same is true here.
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1This is a $6$-torus, so...? – Qiaochu Yuan Oct 06 '23 at 08:52
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Did you learn the Kunneth formula? – Moishe Kohan Oct 06 '23 at 13:19