Hurewicz theorem says that for a simply connected space $X$, $\pi_2(X)\cong H_2(X,\mathbb Z)$. So $\pi_2(K3)\cong H_2(K3,\mathbb Z)\cong \mathbb Z^{20}$Z^{22}$. Here is a link: http://en.wikipedia.org/wiki/Hurewicz_theorem 1 Hurewicz theorem says that for a simply connected space$X$,$\pi_2(X)\cong H_2(X,\mathbb Z)$. So$\pi_2(K3)\cong \mathbb Z^{20}\$. Here is a link: