On the Group &[x] of Homotopy Equivalence Maps

formed by the homotopy classes of equivalence maps from X into itself with the operation induced by the composition of maps. It is clear to see that this group depends only on the homotopy type of X, hence should be determined by the known homotopy invariants of X. This is the problem which we shall try to study here. In fact, there exists a spectral sequence converging to S[X], whose initial terms are given, roughly speaking, by the cohomology of X and the automorphism group of its homotopy group. Besides the satisfaction of curiosity, the group S[X] seems to have other interests. For example, it operates canonically on the special cohomology group [ l ] of X