Theorem 1 (J.H.C. Whitehead)
It is essential to the theorem that isomorphisms between and for all are induced by a map if an isomorphism exists which is not induced by a map, it need not be the case that the spaces are homotopy equivalent.
For example, let and Then the two spaces have isomorphic homotopy groups because they both have a universal covering space homeomorphic to and it is a double covering in both cases. However, for and are not homotopy equivalent, as can be seen, for example, by using homology:
(Here, is -dimensional real projective space, and is the -sphere.)
|Date of creation||2013-03-22 13:25:48|
|Last modified on||2013-03-22 13:25:48|
|Last modified by||antonio (1116)|