A rational realization problem in Gottlieb group

Abstract

We define the fibre-restricted Gottlieb group with respect to a fibration ξ : X → E → Y in CW complexes. It is a subgroup of the Gottlieb group of X. When X and E are finite simply connected, its rationalized model is given by the arguments of derivations of Sullivan models based on Félix, Lupton and Smith [5]. We consider the realization problem of groups in a Gottlieb group as fibre-restricted Gottlieb groups in rational homotoy theory. Especially we define an invariant named as (Gottlieb) depth of X over Y . In particular, when Y = BS 1 , it is related to the rational toral rank of X.