A rational obstruction to be a Gottlieb map

TL;DR

This paper studies Gottlieb maps within rational homotopy theory, introducing an obstruction group and invariant to analyze when maps induce isomorphisms on Gottlieb groups, with examples illustrating their properties.

Contribution

It defines an obstruction group and invariant for Gottlieb maps, providing new tools to analyze their structure and splitting properties in rational homotopy theory.

Findings

Defined the obstruction group O(f) for Gottlieb maps

Introduced the numerical invariant o(f) to measure obstructions

Provided rational examples distinguishing Gottlieb and non-Gottlieb maps

Abstract

We investigate {\it Gottlieb map}s, which are maps $f:E\to B$ that induce the maps between the Gottlieb groups $\pi_n (f)|_{G_n(E)}:G_n(E)\to G_n(B)$ for all $n$, from a rational homotopy theory point of view.We will define the obstruction group $O(f)$ to be a Gottlieb map and a numerical invariant $o(f)$. It naturally deduces a relative splitting of $E$ in certain cases. We also illustrate several rational examples of Gottlieb maps and non-Gottlieb maps by using derivation arguments in Sullivan models.