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

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

where π_{n}(*X*,*x*) denotes the *n*-th homotopy group of *X* with base point *x*. (For *n* = 0, π_{0}(*X*) just means the set of path components of *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. (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 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*= *S*^{2} × **RP**^{3} and *Y*= **RP**^{2} × *S*^{3}. Then *X* and *Y* have the same fundamental group, namely the cyclic group **Z**/2, 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 **R ^{n}**. 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)