Why is the second homology of $T^3=S^1\times S^1\times S^1$ generated by $S^1\times S^1\times \{\mathrm{pt}.\}$, $S^1\times \{\mathrm{pt}.\}\times S^1$ and $ \{\mathrm{pt}.\}\times S^1\times S^1$?
$H_2(T^3;\mathbb{Z})=\mathbb{Z}^3$ and $H_2(T^2;\mathbb{Z})=\mathbb{Z}$, so that fits already nicely. But how does one actually prove some collection of homology classes is a generator of a homology group?