Homological Whitehead theorem ($H_{\ast }(f;\mathbb{Z})\text{ invertible}\Rightarrow {\pi }_{\ast }(f)\text{ invertible}$)

An integral homology equivalence between simply connected spaces is a weak homotopy equivalence. A first nonzero relative homotopy group would contradict the Relative Hurewicz theorem. Between CW complexes the ordinary Whitehead theorem then gives a homotopy equivalence.