Whitehead theorem

Short description: When a mapping that induces isomorphisms on all homotopy groups is a homotopy equivalence

In homotopy theory (a branch of mathematics), the Whitehead theorem states that if a continuous mapping f between CW complexes X and Y induces isomorphisms on all homotopy groups, then f is a homotopy equivalence. This result was proved by J. H. C. Whitehead in two landmark papers from 1949, and provides a justification for working with the concept of a CW complex that he introduced there. It is a model result of algebraic topology, in which the behavior of certain algebraic invariants (in this case, homotopy groups) determines a topological property of a mapping.

Statement

In more detail, let X and Y be topological spaces. Given a continuous mapping

[math]\displaystyle{ f\colon X \to Y }[/math]

and a point x in X, consider for any n ≥ 1 the induced homomorphism

[math]\displaystyle{ f_*\colon \pi_n(X,x) \to \pi_n(Y,f(x)), }[/math]

where π_n(X,x) denotes the n-th homotopy group of X with base point x. (For n = 0, π₀(X) just means the set of path components of X.) A map f is a weak homotopy equivalence if the function

[math]\displaystyle{ f_*\colon \pi_0(X) \to \pi_0(Y) }[/math]

is bijective, and the homomorphisms f_* are bijective for all x in X and all n ≥ 1. (For X and Y path-connected, the first condition is automatic, and it suffices to state the second condition for a single point x in X.) The Whitehead theorem states that a weak homotopy equivalence from one CW complex to another is a homotopy equivalence. (That is, the map f: X → Y has a homotopy inverse g: Y → X, which is not at all clear from the assumptions.) This implies the same conclusion for spaces X and Y that are homotopy equivalent to CW complexes.

Combining this with the Hurewicz theorem yields a useful corollary: a continuous map [math]\displaystyle{ f\colon X \to Y }[/math] between simply connected CW complexes that induces an isomorphism on all integral homology groups is a homotopy equivalence.

Spaces with isomorphic homotopy groups may not be homotopy equivalent

A word of caution: it is not enough to assume π_n(X) is isomorphic to π_n(Y) for each n in order to conclude that X and Y are homotopy equivalent. One really needs a map f : X → Y inducing an isomorphism on homotopy groups. For instance, take X= S² × RP³ and Y= RP² × S³. Then X and Y have the same fundamental group, namely the cyclic group Z/2, and the same universal cover, namely S² × S³; 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 compact 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.

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), 213–245
• J. H. C. Whitehead, Combinatorial homotopy. II., Bull. Amer. Math. Soc., 55 (1949), 453–496
• A. Hatcher, 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)