A rational realization problem in Gottlieb group

We define the fibre-restricted Gottlieb group with respect to a fibration $ξ:X\to E\to Y$ 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 \cite{FLS}. 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 $X$.

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