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Whitehead theorem (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.)



"Whitehead theorem" is owned by antonio.
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See Also: approximation theorem for arbitrary spaces applied to Whitney ${C}^r (M,N)$ spaces
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Cross-references: projective space, real, homology, covering, homeomorphic, universal covering space, homotopy groups, isomorphic, homotopy equivalent, map, induced, isomorphisms, strong homotopy equivalence, CW complexes, homotopy type, path-connected, weak homotopy equivalence
There are 4 references to this entry.

This is version 7 of Whitehead theorem, born on 2003-02-07, modified 2004-02-16.
Object id is 3988, canonical name is WhiteheadTheorem.
Accessed 2349 times total.

Classification:
AMS MSC: 55P10 (Algebraic topology :: Homotopy theory :: Homotopy equivalences)
 55P15 (Algebraic topology :: Homotopy theory :: Classification of homotopy type)
 55Q05 (Algebraic topology :: Homotopy groups :: Homotopy groups, general; sets of homotopy classes)
Pending Errata and Addenda
None.
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CW complex is not defined by Dr_Absentius on 2003-02-07 13:05:03
Hi

 as you have probably noticed there isn't an entry on the definition of CW complex (at least not one I could locate). I was wondering if you plan to write one, since you seem to be doing a lot of basic homotopy theoretic entries.
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