Theorem 1 (J.H.C. Whitehead)

If $f:X\rightarrow Y$ 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 $f:X\rightarrow Y;$ 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 $X={\mathbb{R}}P^{m}\times S^{n}$ and $Y={\mathbb{R}}P^{n}\times S^{m}.$ 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 $m $X$ and $Y$ are not homotopy equivalent, as can be seen, for example, by using homology   :

 $\displaystyle H_{m}(X;{\mathbb{Z}}/2{\mathbb{Z}})$ $\displaystyle\cong$ $\displaystyle{\mathbb{Z}}/2{\mathbb{Z}},\quad\textrm{but}$ $\displaystyle H_{m}(Y;{\mathbb{Z}}/2{\mathbb{Z}})$ $\displaystyle\cong$ $\displaystyle{\mathbb{Z}}/2{\mathbb{Z}}\oplus{\mathbb{Z}}/2{\mathbb{Z}}.$

(Here, ${\mathbb{R}}P^{n}$ is $n$-dimensional real projective space, and $S^{n}$ is the $n$-sphere.)

Title Whitehead theorem  WhiteheadTheorem 2013-03-22 13:25:48 2013-03-22 13:25:48 antonio (1116) antonio (1116) 10 antonio (1116) Theorem msc 55P10 msc 55P15 msc 55Q05 ConjectureApproximationTheoremHoldsForWhitneyCrMNSpaces WeakHomotopyEquivalence ApproximationTheoremAppliedToWhitneyCrMNSpaces