We prove new structural results for the rational homotopy type of the classifying space Baut.X / of fibrations with fiber a simply connected finite CW-complex X. We first study nilpotent covers of Baut.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 g.X / in algebraic representations that models its equivariant rational homotopy type. This leads to an algebraic model for the space Baut.X / and to a description of its rational cohomology ring as the cohomology of a certain arithmetic group Τ.X / with coefficients in the Chevalley–Eilenberg cohomology of 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 and Madsen, and we prove parallel new results in odd dimensions.

The purpose of this note is to give a concise account of some fundamental properties of the exponential group and the Maurer–Cartan space associated to a complete dg Lie algebra. In particular, we give a direct elementary proof that the Maurer–Cartan space is a delooping of the exponential group. This leads to a short proof that the Maurer–Cartan space functor is homotopy inverse to Quillen’s functor from simply connected pointed spaces to positively graded dg Lie algebras.

The generalized Miller–Morita–Mumford classes of a manifold bundle with fiber M depend only on the underlying τM-fibration, meaning the family of vector bundles formed by the tangent bundles of the fibers. This motivates a closer study of the classifying space for τM-fibrations, Baut(τM), and its cohomology ring, i.e., the ring of characteristic classes of τM-fibrations. For a bundle ξ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal ξ-fibration with holonomy in a given connected monoid, together with explicit cocycle representatives for the characteristic classes of the canonical bundle over its total space. This yields tools for computing the rational cohomology ring of Baut(ξ) as well as the subring generated by the generalized Miller–Morita–Mumford classes. To illustrate, we carry out sample computations for spheres and complex projective spaces. We discuss applications to tautological rings of simply connected manifolds and to the problem of deciding whether a given τM-fibration comes from a manifold bundle.

We construct a dg" role="presentation" style="display: inline; line-height: normal; font-size: 17.3333px; overflow-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; border: 0px; padding: 0px; margin: 0px; color: rgb(64, 64, 64); font-family: "Times New Roman", Times, serif; position: relative;">dgdg Lie algebra model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a given subspace. We derive the model from a known model for based homotopy automorphisms together with general result on rational models for geometric bar constructions.

We introduce a notion of Koszul A(infinity)-algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A(infinity)-algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.

Given a fiber bundle, we construct a differential graded Lie algebra model, in the sense of Quillen's rational homotopy theory, for the classifying space of the monoid of homotopy equivalences of the base covered by a fiberwise isomorphism of the total space.

We express the coefficients of the Hirzebruch L-polynomials in terms of certain alternating multiple zeta values. In particular, we show that every monomial in the Pontryagin classes appears with a non-zero coefficient, with the expected sign. Similar results hold for the polynomials associated to the Â-genus.

In this article we revisit the theory of homotopic Hopf-Galois extensions introduced in [9], in light of the homotopical Morita theory of comodules established in [3]. We generalize the theory to a relative framework, which we believe is new even in the classical context and which is essential for treating the Hopf-Galois correspondence in [19]. We study in detail homotopic Hopf-Galois extensions of differential graded algebras over a commutative ring, for which we establish a descent-type characterization analogous to the one Rognes provided in the context of ring spectra [26]. An interesting feature in the differential graded setting is the close relationship between homotopic Hopf-Galois theory and Koszul duality theory. We show that nice enough principal fibrations of simplicial sets give rise to homotopic Hopf-Galois extensions in the differential graded setting, for which this Koszul duality has a familiar form.

A coring (A,C) consists of an algebra A in a symmetric monoidal category and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf-Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings (A,C) and (B,D) in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules V (C)(A) and V (D)(B) are Quillen equivalent. As an illustration of the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring.