Gottlieb groups and subgroups of the group of self-homotopy equivalences

Let E-#(X) be the subgroups of E(X) consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of X and E-* (X) be the subgroup of E(X) that fix homology groups for all dimension. In this paper, we establish some connections between the homotopy group of X and the subgroup E# (X) boolean AND E* (X) of E(X). We also give some relations between pi(n)(W), as well as a generalized Gottlieb group G(n)(f) (W, X), and a subset M-#N(f) (X, W) of [X, W]. Finally we establish a connection between the coGottlieb group of X and the subgroup of E(X) consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups.