A rational obstruction to be a Gottlieb map

Toshihiro Yamaguchi

paper

A rational obstruction to be a Gottlieb map

arXiv:1002.1746

Abstract

We investigate {\it Gottlieb map}s, which are maps $f : E \to B$ that induce the maps between the Gottlieb groups $π_{n} (f) ∣_{G_{n} (E)} : G_{n} (E) \to G_{n} (B)$ 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.