Journal of the
Korean Mathematical Society JKMS

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

Keywords: self-homotopy equivalences, Gottlieb groups, coGottlieb groups

MSC numbers: 55P10, 55P62