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Alexander Berglund

Publications and source records attributed to Alexander Berglund.

At least 19 recordsLinked to original sources
arXiv2025

Higher structures in rational homotopy theory

These notes are based on a series of three lectures given (online) by the first named author at the workshop "Higher Structures and Operadic Calculus" at CRM Barcelona in June 2021. The aim is to give a concise introduction to rational homotopy theory through the lens of higher structures. The rational homotopy type of a simply connected space of finite type is modeled by a $C_\infty$-algebra structure on the rational cohomology groups, or alternatively an $L_\infty$-algebra structure on the rational homotopy groups. The first lecture is devoted to explaining these models and their relation to the classical models of Quillen and Sullivan. The second lecture discusses the relation between Koszul algebras, formality and coformality. The main result is that a space is formal if and only if the rational homotopy $L_\infty$-algebra is Koszul and, dually, a space is coformal if and only if the cohomology $C_\infty$-algebra is Koszul. For spaces that are both formal and coformal, this collapses to classical Koszul duality between Lie and commutative algebras. In the third lecture, we discuss certain higher structure in the rational homotopy theory of automorphisms of high dimensional manifolds, discovered by Berglund and Madsen. The higher structure in question is Kontsevich's Lie graph complex and variants of it.

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arXiv2025

Equivariant algebraic models for relative self-equivalences and block diffeomorphisms

We construct rational models for classifying spaces of self-equivalences of bundles over simply connected finite CW-complexes relative to a given simply connected subcomplex. Via work of Berglund-Madsen and Krannich this specializes to rational models for classifying spaces of block diffeomorphism groups of simply connected smooth manifolds of dimension at least 6 with simply connected boundary. The main application is a formula for the rational cohomology of these classifying spaces in terms of the cohomology of arithmetic groups and dg Lie algebras. We furthermore prove that our models are compatible with gluing constructions, and deduce that the model for block diffeomorphisms is compatible with boundary connected sums of manifolds whose boundary is a sphere. As in preceding work of Berglund-Zeman on spaces of self-homotopy equivalences, a key idea is to study equivariant algebraic models for nilpotent coverings of the classifying spaces.

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arXiv2024

Algebraic models for classifying spaces of fibrations

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.

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arXiv2023

On exponential groups and Maurer-Cartan spaces

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.

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arXiv2020

Characteristic classes for families of bundles

The generalized Miller-Morita-Mumford classes of a manifold bundle with fiber $M$ depend only on the underlying $τ_M$-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 $τ_M$-fibrations, $Baut(τ_M)$, and its cohomology ring, i.e., the ring of characteristic classes of $τ_M$-fibrations. For a bundle $ξ$ over a simply connected Poincaré duality space, we construct a relative Sullivan model for the universal orientable $ξ$-fibration 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 $τ_M$-fibration comes from a manifold bundle.

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arXiv2020

Rational homotopy theory of automorphisms of manifolds

We study the rational homotopy types of classifying spaces of automorphism groups of smooth simply connected manifolds of dimension at least five. We give dg Lie algebra models for the homotopy automorphisms and the block diffeomorphisms of such manifolds. Moreover, we use these models to calculate the rational cohomology of the classifying spaces of the homotopy automorphisms and block diffeomorphisms of the manifold #^g S^d x S^d relative to an embedded disk as g tends to infinity. The answer is expressed in terms of stable cohomology of arithmetic groups and invariant Lie algebra cohomology. Through an extension of Kontsevich's work on graph complexes, we relate our results to the (unstable) homology of automorphisms of free groups with boundaries.

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arXiv2017

Koszul A-infinity algebras and free loop space homology

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.

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arXiv2017

Hirzebruch L-polynomials and multiple zeta values

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 A-hat genus.

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arXiv2017

Rational models for automorphisms of fiber bundles

Given a fiber bundle, we construct a differential graded Lie algebra model for the classifying space of the monoid of homotopy equivalences of the base covered by a fiberwise isomorphism of the total space.

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arXiv2016

Homotopical Morita theory for corings

A coring (A,C) consists of an algebra A 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 in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules are Quillen equivalent. The category of comodules over the trivial coring (A,A) is isomorphic to the category of A-modules, so the question above englobes that of when two algebras are homotopically Morita equivalent. We discuss this special case in the first part of the paper, extending previously known results. To approach the general question, we introduce the notion of a 'braided bimodule' and show that adjunctions between A-Mod and B-Mod that lift to adjunctions between (A,C)-Comod and (B,D)-Comod correspond precisely to braided bimodules between (A,C) and (B,D). We then give criteria, in terms of homotopic descent, for when a braided bimodule induces a Quillen equivalence. In particular, we obtain criteria for when a morphism of corings induces a Quillen equivalence, providing a homotopic generalization of results by Hovey and Strickland on Morita equivalences of Hopf algebroids. To illustrate the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring.

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arXiv2016

Homotopic Hopf-Galois extensions revisited

In this article we revisit the theory of homotopic Hopf-Galois extensions introduced in arXiv:0902.3393v2 [math.AT], in light of the homotopical Morita theory of comodules established in arXiv:1411.6517 [math.AT]. 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 forthcoming work of the second author and Karpova. 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. 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.

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arXiv2015

Free Loop Space Homology of Highly Connected Manifolds

We calculate the homology of the free loop space of (n-1)-connected closed manifolds of dimension at most 3n-2 (n > 1), with the Chas-Sullivan loop product and loop bracket. Over a field of characteristic zero, we obtain an expression for the BV-operator. We also give explicit formulas for the Betti numbers, showing they grow exponentially. Our main tool is the connection between formality, coformality and Koszul algebras that was elucidated in earlier work by the first author.

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arXiv2014

Homological perturbation theory for algebras over operads

We extend homological perturbation theory to encompass algebraic structures governed by operads and cooperads. The main difficulty is to find a suitable notion of algebra homotopy that generalizes to algebras over operads O. To solve this problem, we introduce what we call thick maps of O-algebras and special thick maps that we call pseudo-derivations, which serve as appropriate generalizations of algebra homotopies for the purposes of homological perturbation theory. As an application, we derive explicit formulas for transferring Cobar(C)-algebra structures along contractions, where C is any connected cooperad in chain complexes. This specializes to transfer formulas for O-infinity algebras for any Koszul operad O, in particular for A-infinity, C-infinity, L-infinity and G-infinity algebras. A key feature is that our formulas are expressed in terms of the compact description of Cobar(C)-algebras as coderivation differentials on cofree C-coalgebras. Moreover, we get formulas not only for the transferred structure and a structure on the inclusion, but also for structures on the projection and the homotopy.

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arXiv2012

Homological stability of diffeomorphism groups

In this paper we prove a stability theorem for block diffeomorphisms of 2d-dimensional manifolds that are connected sums of S^d x S^d. Combining this with a recent theorem of S. Galatius and O. Randal-Williams and Morlet's lemma of disjunction, we determine the homology of the classifying space of their diffeomorphism groups relative to an embedded disk in a stable range.

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arXiv2011

Koszul spaces

We prove that a nilpotent space is both formal and coformal if and only if it is rationally homotopy equivalent to the derived spatial realization of a graded commutative Koszul algebra. We call such spaces Koszul spaces and we show that the rational homotopy groups and the rational homology of iterated loop spaces of Koszul spaces can be computed by applying certain Koszul duality constructions to the cohomology algebra.

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arXiv2008

Shellability and the strong gcd-condition

Shellability is a well-known combinatorial criterion for verifying that a simplicial complex is Cohen-Macaulay. Another notion familiar to commutative algebraists, but which has not received as much attention from combinatorialists as the Cohen-Macaulay property, is the notion of a Golod ring. Recently, a criterion on simplicial complexes reminiscent of shellability, called the strong gcd-condition, was shown to imply Golodness of the associated Stanley-Reisner ring. The two algebraic notions were tied together by Herzog, Reiner and Welker who showed that if the Alexander dual of a complex is sequentially Cohen-Macaulay then the complex itself is Golod. In this paper, we present a combinatorial companion of this result, namely that if the Alexander dual of a complex is (non-pure) shellable then the complex itself satisfies the strong gcd-condition. Moreover, we show that all implications just mentioned are strict in general but that they are equivalences for flag complexes.

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