Simply-connected 4-manifolds are classified by data including the second homotopy group (in the guise of the second homology group). In case you complain that one might as well use $H_2$, certain non-simply-connected 4-manifolds are classified by $\pi_1$, $\pi_2$ as a $\pi_1$-module and some extra data involving $\pi_2$.

(apologies for brief and sketchy answer!)