Whitehead Theorem - Spaces With Isomorphic Homotopy Groups May Not Be Homotopy Equivalent

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= S2 × RP3 and Y= RP² × S³. Then X and Y have the same fundamental group, namely Z₂, 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 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.