Whitehead theorem


In homotopy theory, 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 determines a topological property of a mapping.

Statement

In more detail, let X and Y be topological spaces. Given a continuous mapping
and a point x in X, consider for any n ≥ 1 the induced homomorphism
where πn denotes the n-th homotopy group of X with base point x. A map f is a weak homotopy equivalence if the function
is bijective, and the homomorphisms f* are bijective for all x in X and all n ≥ 1. The Whitehead theorem states that a weak homotopy equivalence from one CW complex to another is a homotopy equivalence. 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 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 πn is isomorphic to πn 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 ; 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.