I'm working through old topology quals and came across this question:
Let M be a closed, connected, orientable 4-manifold with fundamental group $π_1(M) = \mathbb{Z}_3 ∗ \mathbb{Z}_3$ and Euler characteristic $χ(M) = 5$.
(a) Compute $H_i(M,\mathbb{Z})$ for all $i$.
(b) Prove that M is not homotopy equivalent to any CW complex with no 3-cells.
I know $H_0 = H_4 = \mathbb{Z}$, and $H_1$ is the abelianization of $π_1(M)$. I'm lost on finding $H_2$ and $H_3$. I assume we need to use $\chi(M) = \sum_n (-1)^n \text{ rank}(H_n)$ at some point.
Any help much appreciated