I got stuck with this problem:
Let $X_1 , X_2$ be two topological spaces. Prove that $i_j :X_j \rightarrow X_1 \times X_2$ induces a monomorphism $H_* (i_j) :H_* (X_j) \rightarrow H_* (X_1 \times X_2)$.
Conclude from this that $S^{n+m}$ is not homotopy equivalent to $S^n \times S^m$.
I think I got the first part: since the projections $p_j : X_1 \times X_2 \rightarrow X_j$ verify $p_j \circ i_j = \operatorname{Id}_{X_i}$ then $(p_j \circ i_j)_* = (p_j)_* \circ (i_j)_* = \operatorname{Id}_{H(X_i)}$ and in particular $(i_j)_*$ is injective for $j=1,2$. However I don't see how to connect this with the second question. I've tried to use that for homotopy equivalent spaces $X$ and $Y$, $H_n (X)$ is isomorphic to $H_n (Y)$ for all $n$, but with no result.