Whitehead theorem

Theorem 1 (J.H.C. Whitehead)

If $f:X\to Y$ 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 $f:X\to Y;$ 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 $X=ℝP^m\times S^n$ and $Y=ℝP^n\times S^m.$Then the two spaces have isomorphic homotopy groups because they both have a universal covering space homeomorphic to $S^m\times S^n,$ and it is a double covering in both cases. However, for $X$ and $Y$ are not homotopy equivalent, as can be seen, for example, by using homology:

$H_m(X;\mathbb{Z}/2\mathbb{Z})$$\cong$$\mathbb{Z}/2\mathbb{Z},\text{but}$

$H_m(Y;\mathbb{Z}/2\mathbb{Z})$$\cong$$\mathbb{Z}/2\mathbb{Z}\oplus \mathbb{Z}/2\mathbb{Z}.$

(Here, $ℝP^n$ is $n$-dimensional real projective space, and $S^n$ is the $n$-sphere.)