Generalized Jiang and Gottlieb groups

Abstract

Given a map $f:X\to Y$, we extend a Gottlieb's result to the generalized Gottlieb group $G^f(Y,f(x_0))$ and show that the canonical isomorphism ${\pi }_1(Y,f(x_0))\stackrel{\approx }{\to }\mathcal{D}(Y)$ restricts to an isomorphism $G^f(Y,f(x_0))\stackrel{\approx }{\to }{\mathcal{D}}^{{\tilde{f}}_0}(Y)$, where ${\mathcal{D}}^{{\tilde{f}}_0}(Y)$ is some subset of the group $\mathcal{D}(Y)$ of deck transformations of $Y$ for a fixed lifting ${\tilde{f}}_0$ of $f$ with respect to universal coverings of $X$ and $Y$, respectively.