text: Q:
  
  Homology of the $n$-torus using the Künneth Formula
  
  I'm trying to apply the Künneth Formula
  $$H_{n}(X \times Y) \simeq \displaystyle \bigoplus_{r+s=n} H_{r}(X) \otimes H_{s}(Y)$$
  to compute the homology groups of the $n$-torus. For the double torus, if I could write $\mathbb{2T}^{2} = \mathcal{U} \times \mathcal{V}$ it would be easier but I don't see how to do it.
  Also, is there a way to relate $H_{\ast}(X \times Y \times Z)$ to $H_{\ast}(X)$, $H_{\ast}(Y)$ and $H_{\ast}(Z)$ and generalize the Künneth Formula?
  I truly appreciate the help :]
  
  A:
  
  Write the two torus as $S^1 \times S^1$. Use that $H_0(S^1) = H_1 (S^1) = \mathbb Z$ and $H_i (S^1) = 0$ for $i > 1$.
  Also use the fact that $\mathbb Z \otimes_{\mathbb Z} \mathbb Z = \mathbb Z$.
  The answers you would get would be $H_0(T^2) = \mathbb Z$.
  $H_1(T^2) = \mathbb Z \oplus \mathbb Z$.
  $H_2(T^2) = \mathbb Z$.
  $H_i(T^2) = 0$ for $i >2$.
  Now I must say that the Kunneth formula for homology has some $\text{Tor}$ terms. It works over here because homology groups of sphere are free.
  That formula won't work for say $\mathbb RP^2 \times \mathbb RP^2$.
  Anyway, that formula does work for coefficients in a field, since all modules over a field are free.
  To relate product of three or more spaces, you can just apply the Kunneth formula repeatedly. 
  So for n torus, use the Kunneth formula repeatedly to obtain
  $$H_k(S^1\times\dots\times S^1) = \bigoplus_{i_1 + \dots + i_r = k}H_{i_1}(S^1)\otimes\dots\otimes H_{i_r}(S^1).$$
  Note that nonzero things appear only when the indices are $0$ or $1$. Use it to get the following :
  $$H_k(T^n) = \mathbb Z ^{\binom{n}{k}}.$$
  
  