An estimate in Gottlieb ranks of fibration

Abstract

As an application of the Gottlieb sequence of fibration, we give an upper bound of the rank of Gottlieb group $G(E) =\oplus_{i>0}G_i(E)$ of the total space $E$ of a fibration $\xi :X\to E\to B$ and define the {\it Gottlieb type $(a,b,c;s,t,u)$}, which describes a rational homotopical condition of fibration with $\rank G(E)=s+t+u$. We also note various examples showing the different situations that can occur. Finally we comment about an interaction with a Halperin's conjecture on fibration.