Generalized Jiang and Gottlieb groups

TL;DR

This paper generalizes Gottlieb groups for maps between spaces, establishing isomorphisms with subgroups of deck transformations, thereby extending classical results in algebraic topology.

Contribution

It extends Gottlieb's classical result to generalized Gottlieb groups associated with maps, linking them to subsets of deck transformation groups.

Findings

Established isomorphism between generalized Gottlieb groups and subsets of deck transformations.

Extended classical Gottlieb results to a broader context involving maps between spaces.

Provided a framework connecting fundamental groups, deck transformations, and generalized Gottlieb groups.

Abstract

Given a map $f\colon X \to Y$, we extend a Gottlieb's result to the generalized Gottlieb group $G^f(Y,f(x_0))$ and show that the canonical isomorphism $\pi_1(Y,f(x_0))\xrightarrow{\approx}\mathcal{D}(Y)$ restricts to an isomorphism $G^f(Y,f(x_0))\xrightarrow{\approx}\mathcal{D}^{\tilde{f}_0}(Y)$, 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.