The Rank of the group of homotopy equivalences

Abstract

In this chapter we completely determine the rank of the group of homotopy equivalences of a large class of spaces, namely those whose rational cohomology is an exterior algebra on odd dimensional generators. In addition to topological groups this class includes products of odd dimensional spheres, quaternionic, complex and some real Stiefel varieties and other homogeneous spaces. Aside from any intrinsic interest that the results in this section may have, they are included to illustrate how one can make statements on the rank of certain groups arising in homotopy theory whose group operation, being derived from composition of maps, is of a very different nature from those considered in the preceding chapter.