Rational cohomology of classifying spaces and Hilali conjecture

Abstract

Let X be a simply connected space with finite-dimensional rational homotopy groups. Denote by Bau₁(X) the classifying space of fibrations with fiber X and max π_∗(X)=max {i; π_i(X)≠0}. In this paper, we show that the dimensional rational cohomology and the Lusternik-Schnirelmann category of Bau₁(X) are infinite if max π_∗(X) is odd. Our results apply, in particular, when X is elliptic. As a consequence, we prove the Hilali conjecture for classifying space.