# 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 and have the homotopy-type of 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.

## Statement

More accurately, we suppose given CW complexes *X* and *Y*, with respective base points *x* and *y*. Given a continuous mapping

such that *f*(*x*) = *y*, we consider for *n* ≥ 1 the induced homomorphisms

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 isomorphisms. The Whitehead theorem then states that a weak homotopy equivalence, for connected CW complexes, 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* ≥ 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} × **RP**^{3} and *Y*= **RP**^{2} × *S*^{3}. Then *X* and *Y* have the same fundamental group, namely **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 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)