Algebraic models for classifying spaces of fibrations
Alexander Berglund, Tomáš Zeman
TL;DR
The paper develops an algebraic framework for the rational homotopy type of $B\operatorname{aut}(X)$ for simply connected finite CW complexes $X$ by passing to a normal unipotent cover and constructing a nilpotent dg Lie algebra $\mathfrak g(X)$ that encodes the equivariant rational homotopy type. It proves that the deck-quotient action yields an algebraic representation on higher homotopy and cohomology, and provides a space-level model via $\langle\mathfrak g(X)\rangle_{h\Gamma(X)}$, together with a commutative dg algebra model $\Omega^*(B\operatorname{aut}(X)) \simeq \Omega^*(\Gamma(X),C_{CE}^*(\mathfrak g(X)))$. Consequently, $H^*(B\operatorname{aut}(X);\mathbb{Q})$ is computed as the group cohomology $H^*(\Gamma(X),H_{CE}^*(\mathfrak g(X)))$, linking invariants of arithmetic groups with characteristic classes of fibrations. The results yield strong consequences for the cohomology ring structure, enable computations in concrete cases (e.g., products of spheres and highly connected manifolds), and illuminate the relationship between characteristic classes and automorphic forms. A counterexample highlights limitations when formality fails. Overall, the work provides a robust space-level algebraic model for the classifying space of self-homotopy equivalences and significant new structural insights into its cohomology and representations.
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 $Γ(X)$ 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.