Rational L. S. category of function space components for \(F_0\)-spaces (Q1301687)

By definition, an \(F_0\)-space is a finite, simply connected complex with finite-dimensional rational homotopy and evenly graded rational cohomology. The following long-standing conjecture is due to S. Halperin: ``The rational Serre spectral sequence collapses for every \(\mathbb{Q}\)-orientable fibration with fibre an \(F_0\)-space \(X\).'' Denote by \(M_1(X,X)\) the component of the space of all continuous functions, containing the identity. From works of \textit{W. Meier} [Math. Ann. 258, 329-340 (1982; Zbl 0466.55012)] and \textit{J.-C. Thomas} [Ann. Inst. Fourier 31, No. 3, 71-90 (1981; Zbl 0446.55009)] we know that the Halperin conjecture is true for \(X\) if and only if the rational LS-category of \(M_1(X,X)\) is finite. The aim of the paper under review is the study of the LS-category of some path-components of function space to find new special cases where the Halperin conjecture is true.