A fibre-restricted Gottlieb group and its rational realization problem

We define the fibre-restricted Gottlieb group G⁎ξ(X) with respect to a fibration ξ:X→E→Y in CW complexes. It is a subgroup of the Gottlieb group G⁎(X) 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 Y. Félix, G. Lupton and S.B. Smith [6]. Then we consider the problem of realizing subgroups of a Gottlieb group as fibre-restricted Gottlieb groups. Especially we define a numerical invariant named as the Gottlieb depth of X over Y, denoted by depthY(X). It gives a measure for the realizations. We illustrate an example of the poset structures of inclusions of realized subgroups in the rational Gottlieb groups, named as the Gottlieb poset of X over Y.