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  • 1.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics. Matematik.
    Cofinite Hochschild cohomology2007Report (Other academic)
  • 2.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Homological perturbation theory for algebras over operads2014In: Algebraic and Geometric Topology, ISSN 1472-2747, E-ISSN 1472-2739, Vol. 14, no 5, p. 2511-2548Article in journal (Refereed)
    Abstract [en]

    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 thick maps of  O –algebras and special thick maps that we call pseudo-derivations that serve as appropriate generalizations of algebra homotopies for the purposes of homological perturbation theory. 

       As an application, we derive explicit formulas for transferring  Ω(C) –algebra structures along contractions, where C  is any connected cooperad in chain complexes. This specializes to transfer formulas for  O ∞  –algebras for any Koszul operad O , in particular for A ∞  –,  C ∞  –,  L ∞  – and  G ∞  –algebras. A key feature is that our formulas are expressed in terms of the compact description of  Ω(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

  • 3.
    Berglund, Alexander
    University of Copenhagen, Denmark.
    Koszul spaces2014In: Transactions of the American Mathematical Society, ISSN 0002-9947, E-ISSN 1088-6850, Vol. 366, no 9, p. 4551-4569Article in journal (Refereed)
    Abstract [en]

    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 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.

  • 4.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics. Matematik.
    Poincaré series and homotopy Lie algebras of monomial rings2005Licentiate thesis, monograph (Other academic)
  • 5.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics. Matematik.
    Poincaré series of monomial rings2006In: Journal of Algebra, Vol. 295, no 1, p. 211-230Article in journal (Refereed)
  • 6.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebras2015In: Homology, Homotopy and Applications, ISSN 1532-0073, E-ISSN 1532-0081, Vol. 17, no 2, p. 343-369Article in journal (Refereed)
    Abstract [en]

    We calculate the higher homotopy groups of the Deligne–Getzler ∞-groupoid associated to a nilpotent L∞-algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.

  • 7.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Shellability and the strong gcd-condition2009In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed)
    Abstract [en]

    Shellability is a well-known combinatorial criterion on a simplicial complex for verifying that the associated Stanley-Reisner ring k[] is Cohen-Macaulay. Anotion familiar to commutative algebraists, but which has not received as muchattention from combinatorialists as the Cohen-Macaulay property, is the notion ofa Golod ring. Recently, J¨ollenbeck introduced a criterion on simplicial complexesreminiscent of shellability, called the strong gcd-condition, and he together with theauthor proved that it implies Golodness of the associated Stanley-Reisner ring. Thetwo algebraic notions were earlier tied together by Herzog, Reiner and Welker, whoshowed that if k[∨] is sequentially Cohen-Macaulay, where ∨ is the Alexanderdual of , then k[] is Golod. In this paper, we present a combinatorial companionof this result, namely that if ∨ is (non-pure) shellable then satisfies the stronggcd-condition. Moreover, we show that all implications just mentioned are strict ingeneral but that they are equivalences if is a flag complex.

  • 8.
    Berglund, Alexander
    Stockholm University, Faculty of Science, Department of Mathematics.
    Shellability and the strong gcd-condition2009In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, no 2Article in journal (Refereed)
    Abstract [en]

    Shellability is a well-known combinatorial criterion on a simplicial complex for verifying that the associated Stanley-Reisner ring k[] is Cohen-Macaulay. Anotion familiar to commutative algebraists, but which has not received as muchattention from combinatorialists as the Cohen-Macaulay property, is the notion ofa Golod ring. Recently, J¨ollenbeck introduced a criterion on simplicial complexesreminiscent of shellability, called the strong gcd-condition, and he together with theauthor proved that it implies Golodness of the associated Stanley-Reisner ring. Thetwo algebraic notions were earlier tied together by Herzog, Reiner and Welker, whoshowed that if k[∨] is sequentially Cohen-Macaulay, where ∨ is the Alexanderdual of , then k[] is Golod. In this paper, we present a combinatorial companionof this result, namely that if ∨ is (non-pure) shellable then satisfies the stronggcd-condition. Moreover, we show that all implications just mentioned are strict ingeneral but that they are equivalences if is a flag complex.

  • 9.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Bergström, Jonas
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hirzebruch L-polynomials and multiple zeta values2018In: Mathematische Annalen, ISSN 0025-5831, E-ISSN 1432-1807, Vol. 372, no 1-2, p. 125-137Article in journal (Refereed)
    Abstract [en]

    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.

  • 10.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics. Matematik.
    Blasiak, Jonah
    Hersh, Patricia
    Combinatorics of multigraded Poincaré series for monomial rings2007In: Journal of Algebra, Vol. 308, no 1, p. 73-90Article in journal (Refereed)
  • 11.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Börjeson, Kaj
    Stockholm University, Faculty of Science, Department of Mathematics.
    Free loop space homology of highly connected manifolds2017In: Forum mathematicum, ISSN 0933-7741, E-ISSN 1435-5337, Vol. 29, no 1, p. 201-228Article in journal (Refereed)
    Abstract [en]

    We calculate the homology of the free loop space of (n - 1)-connected closed manifolds of dimension at most 3 n - 2 (n >= 2), 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 by the first author [6].

  • 12.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Börjeson, Kaj
    Stockholm University, Faculty of Science, Department of Mathematics.
    Koszul A-algebras and free loop space homologyManuscript (preprint) (Other academic)
    Abstract [en]

    We introduce a notion of Koszul A-algebra that generalizes Priddy’s notion of a Koszul algebra and we use it to construct small A- 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. 

  • 13.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hess, Kathryn
    Homotopic Hopf-Galois extensions revisitedManuscript (preprint) (Other academic)
    Abstract [en]

    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.

  • 14.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hess, Kathryn
    Homotopic Hopf-Galois extensions revisited2018In: Journal of Noncommutative Geometry, ISSN 1661-6952, E-ISSN 1661-6960, Vol. 12, no 1, p. 107-155Article in journal (Refereed)
    Abstract [en]

    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.

  • 15.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hess, Kathryn
    Homotopical Morita theory for coringsManuscript (preprint) (Other academic)
    Abstract [en]

    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.

  • 16.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Hess, Kathryn
    Homotopical Morita theory for corings2018In: Israel Journal of Mathematics, ISSN 0021-2172, E-ISSN 1565-8511, Vol. 227, no 1, p. 239-287Article in journal (Refereed)
    Abstract [en]

    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.

  • 17.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics. Matematik.
    Jöllenbeck, Michael
    On the Golod property of Stanley-Reisner rings2007In: Journal of Algebra, Vol. 315, no 1, p. 249-273Article in journal (Refereed)
  • 18.
    Berglund, Alexander
    et al.
    Köpenhamns universitet, Danmark.
    Madsen, Ib
    University of Copenhagen.
    Homological stability of diffeomorphism groups2013In: Pure and Applied Mathematics Quarterly, ISSN 1558-8599, E-ISSN 1558-8602, Vol. 9, no 1, p. 1-48Article in journal (Refereed)
  • 19.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Madsen, Ib
    Rational homotopy theory of automorphisms of highly connected manifoldsManuscript (preprint) (Other academic)
    Abstract [en]

    We study the rational homotopy types of classifying spaces of automorphism groups of 2d-dimensional (d-1)-connected manifolds (d > 2). We prove that the rational homology groups of the homotopy automorphisms and the block diffeomorphisms of the manifold #^g S^d x S^d relative to a disk stabilize as g increases. Via a theorem of Kontsevich, we obtain the striking result that the stable rational cohomology of the homotopy automorphisms comprises all unstable rational homology groups of all outer automorphism groups of free groups.

  • 20.
    Berglund, Alexander
    et al.
    Stockholm University, Faculty of Science, Department of Mathematics.
    Saleh, Bashar
    Stockholm University, Faculty of Science, Department of Mathematics.
    A dg Lie model for relative homotopy automorphismsManuscript (preprint) (Other academic)
    Abstract [en]

    We construct a dg Lie model for the universal cover of the classifying space of the grouplike monoid of homotopy automorphisms of a space that fix a given subspace.

1 - 20 of 20
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