Whitehead theorem

Theorem 1 (J.H.C. Whitehead)

If f:XY is a weak homotopy equivalence and X and Y are path-connected and of the homotopy typeMathworldPlanetmath of CW complexes, then f is a strong homotopy equivalence.

Remark 1

It is essential to the theorem that isomorphismsMathworldPlanetmathPlanetmath between πk(X) and πk(Y) for all k are induced by a map f:XY; 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=Pm×Sn and Y=Pn×Sm. Then the two spaces have isomorphic homotopy groupsMathworldPlanetmath because they both have a universal covering space homeomorphicMathworldPlanetmath to Sm×Sn, 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 homologyMathworldPlanetmathPlanetmath:

Hm(X;/2) /2,but
Hm(Y;/2) /2/2.

(Here, Pn is n-dimensional real projective space, and Sn is the n-sphere.)

Title Whitehead theoremMathworldPlanetmath
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