Theorem 1 (J.H.C. Whitehead)   If $\funcdef{f}{X}{Y}$ is a weak homotopy equivalence and $X$ and $Y$ are path-connected and of the homotopy type of CW complexes, then $f$ is a strong homotopy equivalence.

Remark 1   It is essential to the theorem that isomorphisms between $\pi_k(X)$ and $\pi_k(Y)$ for all $k$ are induced by a map $\funcdef{f}{X}{Y};$ if an isomorphism exists which is not induced by a map, it need not be the case that the spaces are homotopy equivalent.

For example, let $X=\rp^m\cross S^n$ and $Y=\rp^n\cross S^m.$ Then the two spaces have isomorphic homotopy groups because they both have a universal covering space homeomorphic to $S^m\cross S^n,$ and it is a double covering in both cases. However, for $m<n,$ $X$ and $Y$ are not homotopy equivalent, as can be seen, for example, by using homology: \begin{eqnarray*} H_m(X;\zmod{2})&\isom&\zmod{2}, \quad \textrm{but} \\ H_m(Y;\zmod{2})&\isom&\zmod{2}\oplus\zmod{2}. \end{eqnarray*} (Here, $\rp^n$ is $n$ -dimensional real projective space, and $S^n$ is the $n$ -sphere.)