Fiber-homotopy self-equivalences and a classification of fibrations in rational homotopy (Q504546)

The introduction of the present paper starts by recalling the following problem by \textit{D. H. Gottlieb} [Ann. Math. (2) 87, 42--55 (1968; Zbl 0173.25901)]: Which homotopy equivalences of \(X\) into itself can be extended to fibre homotopy equivalences of \(E\) into itself? The authors note that this problem can be restated as follows: Which homotopy classes of homotopy equivalences of \(X\) into itself can be extended to the homotopy class of fibre homotopy equivalences of \(E\) into itself? In a rational homotopy theory context, this is equivalent to the Leray-Serre spectral sequence of the fibration degenerating on the \(E_2\)-term (the Halperin conjecture posed for \(F_0\)-spaces). In Section 2, the authors outline some well known results about Sullivan models related to derivations, in particular the result that models the rational homotopy type of \(\text{aut}_1(p)\) for a fibration \(p: E\longrightarrow B\), with \(E\) and \(B\) simply connected and \(E\) finite. Their approach (see Sections 3 and 4) is to consider for any fibration \(\xi:X\to E\to B\) the image of the rationalized homotopy group homomorphism \(\pi_\ast(\mathrm{res }\xi)_{\mathbb Q}:\pi_\ast(\mathrm{aut}_1p)_{\mathbb Q}\to\pi_\ast(\mathrm{aut}_1X)_{\mathbb Q}\) obtained from the fibre-restricting map \(\mathrm{res } \xi:\mathrm{aut}_1p \to\mathrm{aut}_1X\). Theorem 3.4 gives a partial answer to the surjection problem. In particular, the authors measure the size of the finite type classification of \(\xi\) by a rational homotopy invariant depth\(_BX\) for \(X\) certain homogeneous spaces and \(B\) spheres. Lemmas 4.3, 4.4 and 4.5 give some answers. The paper finishes (see Section 5) with many interesting examples and a classification table.