Whitehead theorem

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 $m<n,$ $X$ 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
Canonical name	WhiteheadTheorem
Date of creation	2013-03-22 13:25:48
Last modified on	2013-03-22 13:25:48
Owner	antonio (1116)
Last modified by	antonio (1116)
Numerical id	10
Author	antonio (1116)
Entry type	Theorem
Classification	msc 55P10
Classification	msc 55P15
Classification	msc 55Q05
Related topic	ConjectureApproximationTheoremHoldsForWhitneyCrMNSpaces
Related topic	WeakHomotopyEquivalence
Related topic	ApproximationTheoremAppliedToWhitneyCrMNSpaces