Algebraic models for classifying spaces of fibrations

TL;DR

This paper develops algebraic models for the rational homotopy type of classifying spaces of fibrations with simply connected fibers, extending classical theorems and providing explicit cohomology descriptions.

Contribution

It introduces a nilpotent dg Lie algebra model for the equivariant rational homotopy type of classifying spaces of fibrations, generalizing Sullivan--Wilkerson theorem.

Findings

Rational cohomology groups are algebraic representations of transformation groups.

Extension of Sullivan--Wilkerson theorem to higher homotopy and cohomology groups.

Explicit algebraic models for the cohomology ring involving arithmetic groups.

Abstract

We prove new structural results for the rational homotopy type of the classifying space $B\operatorname{aut}(X)$ of fibrations with fiber a simply connected finite CW-complex $X$. We first study nilpotent covers of $B\operatorname{aut}(X)$ and show that their rational cohomology groups are algebraic representations of the associated transformation groups. For the universal cover, this yields an extension of the Sullivan--Wilkerson theorem to higher homotopy and cohomology groups. For the cover corresponding to the kernel of the homology representation, this proves algebraicity of the cohomology of the homotopy Torelli space. For the cover that classifies what we call normal unipotent fibrations, we then prove the stronger result that there exists a nilpotent dg Lie algebra $\mathfrak g(X)$ in algebraic representations that models its equivariant rational homotopy type. This leads to an algebraic model for the space $B\operatorname{aut}(X)$ and to a description of its rational cohomology ring as the cohomology of a certain arithmetic group $\Gamma(X)$ with coefficients in the Chevalley-Eilenberg cohomology of $\mathfrak g(X)$. This has strong structural consequences for the cohomology ring and, in certain cases, allows it to be completely determined using invariant theory and calculations with modular forms. We illustrate these points with concrete examples. As another application, we significantly improve on certain results on self-homotopy equivalences of highly connected even-dimensional manifolds due to Berglund--Madsen, and we prove parallel new results in odd dimensions.