The dual Whitehead theorems

Summary

For Peter Hilton on his 60^th birthday

Eckmann-Hilton duality has been around for quite some time and is something we now all take for granted. Nevertheless, it is a guiding principle to “the homotopical foundations of algebraic topology” that is still seldom exploited as thoroughly as it ought to be. In 1971, I noticed that the two theorems commonly referred to as Whitehead's theorem are in fact best viewed as dual to one another. I've never published the details. (They were to appear in a book whose title is in quotes above and which I contracted to deliver to the publishers in 1974; 1984, perhaps?) This seems a splendid occasion to advertise the ideas. The reader is referred to Hilton's own paper [2] for a historical survey and bibliography of Eckmann-Hilton duality. We shall take up where he left off.

The theorems in question read as follows.

Theorem A. A weak homotopy equivalence e :Y → Z between CW complexes is a homotopy equivalence.

Theorem B. An integral homology isomorphism e :Y → Z between simple spaces is a weak homotopy equivalence.

In both, we may as well assume that Y and Z are based and (path) connected and that e is a based map. The hypothesis of Theorem A (and conclusion of Theorem B) asserts that e_*:π_*(Y) → π_*(Z) is an isomorphism. The hypothesis of Theorem B asserts that e_*:H_*(X) → H_*(Y) is an isomorphism. A simple space is one whose fundamental group is Abelian and acts trivially on the higher homotopy groups.