Whitehead theorem

In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping "f" between topological spaces "X" and "Y" induces isomorphisms on all homotopy groups, then "f" is a homotopy equivalence provided "X" and "Y" are connected CW complexes. 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.

Stating it more accurately, we suppose given CW complexes "X" and "Y", with respective base points "x" and "y". Given a continuous mapping

:$fcolon\; X\; o\; Y$

such that "f"("x") = "y", we consider for "n" ≥ 0 the induced homomorphisms

:$f\_*colon\; pi\_n(X,x)\; o\; pi\_n(Y,y),$

where π_"n" 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"_{* are all bijective. The Whitehead theorem then states that a weak homotopy equivalence, for connected CW complexes, is an actual homotopy equivalence.}

A word of caution: it is not enough to assume π_"n"("X") is isomorphic to π_"n"("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"= "S"² × RP³ and "Y"= RP² × "S"³. Then "X" and "Y" have the same fundamental group, namely Z₂, and the same universal cover, namely "S"² × "S"³; thus, they have isomorphic homotopy groups. On the other hand their homology groups are different (as can be seen from the 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 Rⁿ. 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 homotopically equivalence. The study of possible generalizations of Whitehead's theorem to more general spaces is part of the subject of shape theory.

References

* J. H. C. Whitehead, "Combinatorial homotopy. I.", Bull. Amer. Math. Soc., 55 (1949), 213–245
* J. H. C. Whitehead, "Combinatorial homotopy. II.", Bull. Amer. Math. Soc., 55 (1949), 453–496
* A. Hatcher, [http://www.math.cornell.edu/~hatcher/AT/ATpage.html "Algebraic topology"] , Cambridge University Press, Cambridge, 2002. xii+544 pp. ISBN 0-521-79160-X and ISBN 0-521-79540-0 (see Theorem 4.5)