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In [[homotopy theory]] (a branch of [[mathematics]]), the '''Whitehead theorem''' states that if a [[continuous mapping]] ''f'' between [[topological space]]s ''X'' and ''Y'' induces [[isomorphisms]] on all [[homotopy group]]s, then ''f'' is a [[homotopy equivalence]] provided ''X'' and ''Y'' are [[connected space|connected]] and have the homotopy-type of [[CW complex]]es. This result was proved by [[J. H. C. Whitehead]] in two landmark papers from 1949, and provides a justification for working with the [[CW complex]] concept that he introduced there.
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== Statement ==
 
More accurately, we suppose given CW complexes ''X'' and ''Y'', with respective base points ''x'' and ''y''. Given a continuous mapping
 
:<math>f\colon X \to Y</math>
 
such that ''f''(''x'') = ''y'', we consider for ''n'' ≥ 0 the induced homomorphisms
 
:<math>f_*\colon \pi_n(X,x) \to \pi_n(Y,y),</math>
 
where π<sub>''n''</sub> denotes for ''n'' ≥ 1 the ''n''-th homotopy group. For ''n'' = 0 this means the mapping of the path-connected components; if we assume both ''X'' and ''Y'' are connected we can ignore this as containing no information. We say that ''f'' is a '''weak homotopy equivalence''' if the homomorphisms ''f''<sub>*</sup> are all isomorphisms. The Whitehead theorem then states that a weak homotopy equivalence, for connected CW complexes, is a [[homotopy|homotopy equivalence]].
 
== Spaces with isomorphic homotopy groups may not be homotopy equivalent ==
 
A word of caution: it is not enough to assume π<sub>''n''</sub>(''X'') is isomorphic to π<sub>''n''</sub>(''Y'') for each ''n'' ≥ 1 in order to conclude that ''X'' and ''Y'' are homotopy equivalent. One really needs a map  ''f'' : ''X'' → ''Y'' inducing such isomorphisms in homotopy. For instance, take ''X''= [[Hypersphere|''S''<sup>2</sup>]] × [[real projective space|'''RP'''<sup>3</sup>]] and ''Y''= '''RP'''<sup>2</sup> × ''S''<sup>3</sup>. Then ''X'' and ''Y'' have the same fundamental group, namely '''Z'''<sub>2</sub>, and the same universal cover, namely ''S''<sup>2</sup> × ''S''<sup>3</sup>;  thus, they have isomorphic homotopy groups. On the other hand their homology groups are different (as can be seen from the [[Künneth theorem|Künneth formula]]); thus, ''X'' and ''Y'' are not homotopy equivalent.  
 
The Whitehead theorem does not hold for general topological spaces or even for all subspaces of '''R<sup>n</sup>'''. For example, the [[Warsaw circle]], a subset of the plane, has all homotopy groups zero, but the map from the Warsaw circle to a single point is not a homotopy equivalence. The study of possible generalizations of Whitehead's theorem to more general spaces is part of the subject of [[shape theory (mathematics)|shape theory]].
 
== Generalization to model categories ==
In any [[model category]], a weak equivalence between cofibrant-fibrant objects is a homotopy equivalence.
 
==References==
* J. H. C. Whitehead, ''Combinatorial homotopy. I.'', Bull. Amer. Math. Soc., 55 (1949), 213&ndash;245
* J. H. C. Whitehead, ''Combinatorial homotopy. II.'', Bull. Amer. Math. Soc., 55 (1949), 453&ndash;496
* A. Hatcher, [http://www.math.cornell.edu/~hatcher/AT/ATpage.html ''Algebraic topology''], Cambridge University Press, Cambridge, 2002. xii+544 pp.&nbsp;ISBN 0-521-79160-X and ISBN 0-521-79540-0  (see Theorem 4.5)
 
[[Category:Homotopy theory]]
[[Category:Theorems in algebraic topology]]

Latest revision as of 17:46, 11 June 2014

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