Given a fibration \(\xi\to X\), a \(\xi\)-fibration is defined to be a pair \((\pi,\zeta)\) consisting of a fibration \(\pi: E\to B\) with fiber \(X\) and a fiber bundle \(\zeta\to E\) such that the restriction to the fiber \(\zeta_b\) is equivalent to \(\xi\) “in an appropriate sense” [p. 50]. If \(B=*\), a \(\xi\)-fibration is precisely a fiber bundle equivalent to \(\xi\), so a \(\xi\)-fibration can be thought of as a family of fiber bundles equivalent to \(\xi\) parametrized by a base space \(B\). On the other hand, if the fiber of \(\xi\) is a point, a \(\xi\)-fibration is simply a fibration with fiber \(X\).
The article studies the universal \(\xi\)-fibration \(E_\xi\to \mathrm{Baut}(\xi)\) in the context of rational homotopy theory. The main result [Theorem 3.8] is the construction of a relative Sullivan model for this universal \(\xi\)-fibration. This is then used for a number of sample calculations and applications, among which one can find the computation of the rational cohomology of \(\mathrm{Baut}(\tau_{S^m})\), where \(\tau_M\) shall denote the tangent bundle of \(M\) [Theorems 1.1 & 1.2]. Since any fiber bundle with fiber \(M\) yields a \(\tau_M\)-fibration given by the vertical tangent bundle, one obtains a map \(B\mathrm{Diff}(M)\to \mathrm{Baut}(\tau_M)\) and one can consider the induced map \(H^*(\mathrm{Baut}(\tau_M);\mathbb{Q})\to H^*(B\mathrm{Diff}(M);\mathbb{Q})\). This is shown to be split injective if \(M=S^m\) [Corollary 6]. Furthermore, the rational cohomology of \(\mathrm{Baut}(\xi)\) is computed in the case, where \(\xi\) is a \(\mathbb{CP}^n\)-bundle with some restrictions on its structure group [Theorems 1.7 & 1.8].