Algebraic models for classifying spaces of fibrations

Alexander Berglund, Tomáš Zeman

Published 2022-03-04Version 1

We construct an algebraic model for the rational homotopy type of the classifying space $B\operatorname{aut}(X)$ for fibrations with fiber a simply connected finite CW-complex $X$. The ingredients are a nilpotent dg Lie algebra $\mathfrak{g}(X)$ in the category of algebraic representations of a certain reductive algebraic group $R(X)$ together with an arithmetic subgroup $\Gamma(X)$ of $R(X)$. The algebraic model reduces the computation of the rational cohomology ring of $B\operatorname{aut}(X)$ to the computation of Chevalley-Eilenberg cohomology of dg Lie algebras and cohomology of arithmetic groups with coefficients in algebraic representations. This has strong consequences for the structure of this ring.