GOTTLIEB GROUPS AND SUBGROUPS OF THE GROUP OF SELF-HOMOTOPY EQUIVALENCES

Abstract

Let <TEX>$\varepsilon_#(X)$</TEX> be the subgroups of <TEX>$\varepsilon(X)$</TEX> consisting of homotopy classes of self-homotopy equivalences that fix homotopy groups through the dimension of X and <TEX>$\varepsilon_*(X) $</TEX> be the subgroup of <TEX>$\varepsilon(X)$</TEX> that fix homology groups for all dimension. In this paper, we establish some connections between the homotopy group of X and the subgroup <TEX>$\varepsilon_#(X)\cap\varepsilon_*(X)\;of\;\varepsilon(X)$</TEX>. We also give some relations between <TEX>$\pi_n(W)$</TEX>, as well as a generalized Gottlieb group <TEX>$G_n^f(W,X)$</TEX>, and a subset <TEX>$M_{#N}^f(X,W)$</TEX> of [X, W]. Finally we establish a connection between the coGottlieb group of X and the subgroup of <TEX>$\varepsilon(X)$</TEX> consisting of homotopy classes of self-homotopy equivalences that fix cohomology groups.