Spaces With Isomorphic Homotopy Groups May Not Be Homotopy Equivalent
A word of caution: it is not enough to assume πn(X) is isomorphic to πn(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.
Read more about this topic: Whitehead Theorem
Famous quotes containing the words spaces, groups and/or equivalent:
“Surely, we are provided with senses as well fitted to penetrate the spaces of the real, the substantial, the eternal, as these outward are to penetrate the material universe. Veias, Menu, Zoroaster, Socrates, Christ, Shakespeare, Swedenborg,—these are some of our astronomers.”
—Henry David Thoreau (1817–1862)
“As in political revolutions, so in paradigm choice—there is no standard higher than the assent of the relevant community. To discover how scientific revolutions are effected, we shall therefore have to examine not only the impact of nature and of logic, but also the techniques of persuasive argumentation effective within the quite special groups that constitute the community of scientists.”
—Thomas S. Kuhn (b. 1922)
“But then people don’t read literature in order to understand; they read it because they want to re-live the feelings and sensations which they found exciting in the past. Art can be a lot of things; but in actual practice, most of it is merely the mental equivalent of alcohol and cantharides.”
—Aldous Huxley (1894–1963)