Generalized Gottlieb subgroups and Serre fibrations

Bull. Korean Math. Soc. 2009 Vol. 46, No. 1, 25-33 Printed January 1, 2009

Jae-Ryong Kim Kookmin University

Abstract : Let $\pi : E \to B$ be a Serre fibration with fibre $F$. We prove that if the inclusion map $i: F \to E$ has a left homotopy inverse $r$ and $\pi : E \rightarrow B$ admits a cross section $\rho : B \to E$, then $G_n(E,F) \cong \pi_n(B) \oplus G_n(F)$. This is a generalization of the case of trivial fibration which has been proved by Lee and Woo in [8]. Using this result, we will prove that $\pi_n(X^A) \cong \pi_n(X) \oplus G_n(F)$ for the function space $X^A$ from a space $A$ to a weak $H_*$-space $X$ where the evaluation map $\omega : X^A \to X$ is regarded as a fibration.

Keywords : generalized Gottlieb subgroups, Serre fibrations, $G$-sequence