Hurewicz theorem

Remarks

The map $h$ is a homomorphism, because a sum of maps $S^n \to X$ factors through the pinch map $S^n \to S^n \vee S^n$. For $n \ge 2$, one replaces $X$ by a weakly equivalent CW complex with a single $0$-cell and no cells of dimensions $1$ to $n - 1$. Then both groups are computed from the $n$-cells and the attaching maps of the $(n+1)$-cells. Example: $S^n$ is $(n-1)$-connected and $H_n(S^n) \cong \Z$, so $\pi_n(S^n) \cong \Z$, generated by the identity. The isomorphism stops at the first nonzero degree: $S^2$ is $1$-connected, yet $\pi_3(S^2) \cong \Z$ while $H_3(S^2) = 0$.

Going deeper

For a student

The theorem gives the first nonzero homotopy group of a simply connected space from its homology. With the Whitehead theorem, it shows that a homology isomorphism between simply connected CW complexes is a homotopy equivalence. A relative version for pairs of spaces extends it further. It is also the starting point for Serre's mod $\mathcal{C}$ theory, which compares homotopy and homology modulo classes of abelian groups.