Hurewicz theorem

If a path-connected space is $(n-1)$-connected for $n\ge 2$, then its first potentially nonzero homotopy group maps isomorphically to homology:

A sphere map $f:S^n\to X$ is sent to $f_{\ast }[S^n]\in H_n(X;\mathbb{Z})$. After tensoring with $\mathbb{Q}$ this gives the rational Hurewicz homomorphism. It is natural in $X$, sends Whitehead products to zero, and is an isomorphism in the first nonzero homotopy degree of a simply connected space by the Hurewicz theorem.

If $X$ is simply connected and $h_{\ast }:{\pi }_{\ast }(X)\otimes \mathbb{Q}\to {\tilde{H}}_{\ast }(X;\mathbb{Q})$ is onto, choose a homogeneous homology basis and representing sphere maps into $X_{\mathbb{Q}}$. Their wedge sum gives a rational homology equivalence, hence a rational homotopy equivalence by the rational Whitehead theorem. All positive cup products therefore vanish. In a minimal model, the induced map from cohomology to the indecomposable quotient is injective; replacing a set of generators by cocycle representatives produces a quasi-isomorphism to the square-zero cohomology algebra, proving formality of a topological space.

If $X$ and $A$ are simply connected and the pair is $(r-1)$-connected, with $r\ge 3$, its first possible nonzero relative homotopy group maps isomorphically to relative integral homology. This lets a basis of a free relative homology group be realized by attaching disks.

For a simply connected space, if lower homotopy groups lie in an admissible Serre class, the Hurewicz map in the next degree is an isomorphism modulo that class, and the lower homology groups lie in the class. This applies to finitely generated groups and to finite groups supported at prescribed primes.