Rational homotopy type of relative universal fibrations

Abstract

For any group $G$ of self homotopy equivalences of the finite nilpotent complex $X$, acting nilpotently on its homology, and for any nilpotent subcomplex $A$, we prove that the universal fibration $$X⟶B(\ast ,{\mathrm{aut}}_G^A(X),X)⟶B{\mathrm{aut}}_G^A(X),$$ which classifies $A$-fibrations for which the image of the $A$-holonomy action lies in $G$, has a Lie model of the form $$L⟶L\tilde{\times }\mathcal{D}e{r}^{M}L⟶\mathcal{D}e{r}^{M}L$$ in which: $M\hookrightarrow L$ is a Lie model of $A\hookrightarrow X$ and $\mathcal{D}e{r}^{M}L$ is a connected complete differential graded Lie algebra of derivations of $L$ which vanish on $M$. The rational homotopy type of extended relative mapping fibrations is also similarly characterized.