The $n$-torus is the product of $n$ circles:

$$T^{n}=\underbrace{S^{1} \times \ldots \times S^{1}}_{n \text { times }} .$$

For all $n \geqslant 1$ and $0 \leqslant k \leqslant n$, compute $H_{k}\left(T^{n}\right)$.

[You may assume that relevant spaces are triangulable, but you should state carefully any version of any theorem that you use.]