Rational type of classifying spaces for fibrations (Q2734875)

The author identifies the rational type of the classifying space, Baut\(_1(X)\), for orientable fibrations with fibre \(X\) when \(X\) is restricted in natural rational homotopy theoretic ways. For instance, it is shown that \(\text{Baut}_1(X)\) is a product of rational Eilenberg-MacLane spaces when \(X\) is an elliptic space with evenly graded cohomology whose negative degree cohomology derivations vanish. Recall that an elliptic space is one for which the rational cohomology and homotopy are finite dimensional. Halperin conjectured that elliptic spaces \(X\) with evenly graded cohomology have no negative degree derivations on cohomology (leading to the collapsing of the Serre spectral sequence for fibrations with fibre \(X\), for instance), so the last assumption may always hold. Another interesting result is proved when \(X\) is a simply connected rational \(H\)-space (of finite type) [compare \textit{S. Piccarreta}, Prog. Math. 196, 349-355 (2001; Zbl 0983.55008), see above]. Namely, the rational type of the classifying space is that of the spatial realization of the Lie algebra of negative degree derivations of cohomology. In particular, \(\text{Baut}_1(X)\) is then a coformal space; a space whose rational type is determined by its rational homotopy vector spaces and the accompanying Whitehead product structure.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00050].