I was trying to calculate the homology groups $H_1(SO(4))$, $H_2(SO(4))$ and $H_3(SO(4))$, but I have no clue on how to do this.
I calculate other homology groups, for like the sphere $S^n$, projective spaces $\mathbb C\mathbb P^n$ and $\mathbb R\mathbb P^n$, the Klein bottle and so on. All these homology groups I could use some relation, like the Meyer-Vietoris sequence, or use that the inclusion was a cofibration.
The only thing that I know is that $SO(4)\cong SO(3)\times S^3 $, but still I dont know how to compute $H_n(X\times Y)$.
What could I try to do that?
Thank in advance.