Let $C$ be a differential graded coalgebra, $
\barΩC$ the Adams cobar construction and $C^\vee$ the dual algebra. We prove that for a large class of coalgebras $C$ there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies $HH^\ast (C^\vee, C ^\vee)$ and $HH^\ast (\barΩC ; \barΩC)$. This result permits to describe a Hodge decomposition of the loop space homology of a closed oriented manifold, in the sense of Chas-Sullivan, when the field of coefficients is of characteristic zero.
We show that the diagonal complex computing the Gerstenhaber-Schack cohomology of a bialgebra (that is, the cohomology theory governing bialgebra deformations) can be given the structure of an operad with multiplication if the bialgebra is a (not necessarily finite dimensional) Hopf algebra with invertible antipode; if the antipode is involutive, the operad is even cyclic. Therefore, the Gerstenhaber-Schack cohomology of any such Hopf algebra carries a Gerstenhaber resp. Batalin-Vilkovisky algebra structure; in particular, one obtains a cup product and a cyclic boundary B that generate the Ger…
On the tensor product of two homotopy Gerstenhaber algebras we construct a Hirsch algebra structure which extends the canonical dg algebra structure. Our result applies more generally to tensor products of "level 3 Hirsch algebras" and also to the Mayer-Vietoris double complex.
We show that any homotopy Gerstenhaber algebra is naturally a strongly homotopy commutative (shc) algebra in the sense of Stasheff-Halperin with a homotopy associative structure map. In the presence of certain additional operations corresponding to a cup-1 product on the bar construction, the structure map becomes homotopy commutative, so that one obtains an shc algebra in the sense of Munkholm.
The goal of this paper is to complete Getzler-Jones' proof of Deligne's Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the $B_\infty$-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: one of them is a $B_\infty$-algebra, another, called a hom…
A method for establishing a Gerstenhaber algebra structure on the cohomology of Loday-type algebras is presented. This method is then applied to dendriform dialgebras and three types of trialgebras introduced by Loday and Ronco. Along the way, our results are combined with a result of McClure-Smith to prove an analogue of Deligne's conjecture for Loday algebras.
A homotopy Gerstenhaber structure on a differential graded algebra is essentially a family of operations defining a multiplication on its bar construction. We prove that the normalized singular cochain algebra of a Davis-Januszkiewicz space is formal as a homotopy Gerstenhaber algebra, for any coefficient ring. This generalizes a recent result by the author about classifying spaces of tori and also strengthens the well-known dga formality result for Davis-Januszkiewicz spaces due to the author and Notbohm-Ray. As an application, we determine the cohomology rings of free and based loop spaces o…
Using non-commutative differential forms, we construct a complex called singular Hochschild cochain complex for any associative algebra over a field. The cohomology of this complex is isomorphic to the Tate-Hochschild cohomology in the sense of Buchweitz. By a natural action of the cellular chain operad of the spineless cacti operad, introduced by R. Kaufmann, on the singular Hochschild cochain complex, we provide a proof of the Deligne's conjecture for this complex. More concretely, the complex is an algebra over the (dg) operad of chains of the little $2$-discs operad. By this action, we als…
We show that the operadic cohomology for any type of algebras over a non-symmetric operad A can be computed as Ext in the category of operadic A-modules. We use this principle to prove that the Gerstenhaber-Schack diagram cohomology is operadic cohomology.
We study the notion of twisting elements $da=a\cup_1a$ with respect to $\cup_1$ product when it is a part of homotopy Gerstenhaber algebra structure. This allows to bring to one context the two classical concepts, the theory of deformation of algebras of M. Gerstenhaber, and $A(\infty)$-algebras of J. Stasheff.
The goal of this paper is to prove a Koszul duality result for E_n-operads in differential graded modules over a ring. The case of an E_1-operad, which is equivalent to the associative operad, is classical. For n>1, the homology of an E_n-operad is identified with the n-Gerstenhaber operad and forms another well known Koszul operad. Our main theorem asserts that an operadic cobar construction on the dual cooperad of an E_n-operad defines a cofibrant model of E_n. This cofibrant model gives a realization at the chain level of the minimal model of the n-Gerstenhaber operad arising from Koszul du…
Sinha has constructed a cosimplicial space X for a fixed integer N. One of his main result states that for N > 3, X is a cosimplicial model for the space of long knots (modulo immersions). On the other hand, Lambrechts, Turchin and Volic showed that for N > 3 the homology Bousfield-Kan spectral sequence associated to Sinha's cosimplicial space collapses at the page 2 rationally. Their approach consists first to prove the formality of some other diagrams approximating X and next deduce the collapsing result. In this paper, we prove directly the formality of Sinha's cosimplicial space and this a…
Using Kadeishvili's formulas with appropriate signs, we show that the classical cobar construction from coalgebras to algebras Ω: CoAlg -> Alg can be enhanced to a functor from Hopf algebras to E_2 algebras (for a certain choice of E_2 operad) Ω: HopfAlg -> E_2 Alg, which, unlike its classical counterpart, is not strictly adjoint, but homotopically equivalent to the left adjoint of the enhanced bar construction B: E_2 Alg -> HopfAlg studied by Gerstenhaber--Voronov and Fresse.
The double cobar construction of a double suspension comes with a Connes-Moscovici structure, that is a homotopy G-algebra (or Gerstenhaber-Voronov algebra) structure together with a particular BV-operator up to a homotopy. We show that the homology of the double cobar construction of a double suspension is a free BV-algebra. In characteristic two, a similar result holds for the underlying $2$-restricted Gerstenhaber algebra. These facts rely on a formality theorem for the double cobar construction of a double suspension.
Consider the space of `long knots' in R^n, K_{n,1}. This is the space of knots as studied by V. Vassiliev. Based on previous work of the authors, it follows that the rational homology of K_{3,1} is free Gerstenhaber-Poisson algebra. A partial description of a basis is given here. In addition, the mod-p homology of this space is a `free, restricted Gerstenhaber-Poisson algebra'. Recursive application of this theorem allows us to deduce that there is p-torsion of all orders in the integral homology of K_{3,1}. This leads to some natural questions about the homotopy type of the space of long knot…
In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin-Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalised Lie derivative and a generalised (cyclic) cap product that obey a Cartan-Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Ho…
Let $G$ be a compact connected Lie group and $K$ a closed connected subgroup. Assume that the order of any torsion element in the integral cohomology of $G$ and $K$ is invertible in a given principal ideal domain $k$. It is known that in this case the cohomology of the homogeneous space $G/K$ with coefficients in $k$ and the torsion product of $H^{*}(BK)$ and $k$ over $H^{*}(BG)$ are isomorphic as $k$-modules. We show that this isomorphism is multiplicative and natural in the pair $(G,K)$ provided that 2 is invertible in $k$. The proof uses homotopy Gerstenhaber algebras in an essential way. I…
We propose a simple method for constructing formal deformations of differential graded algebras in the category of minimal $A_\infty$-algebras. The basis for our approach is provided by the Gerstenhaber algebra structure on the $A_\infty$-cohomology, which we define in terms of the brace operations. As an example, we construct a minimal $A_\infty$-algebra from the Weyl-Moyal $\ast$-product algebra of polynomial functions.
For M a closed, connected, oriented manifold, we obtain the Batalin-Vilkovisky (BV) algebra of its string topology through homotopy-theoretic constructions on its based loop space. In particular, we show that the Hochschild cohomology of the chain algebra C_*ΩM carries a BV algebra structure isomorphic to that of the loop homology $\mathbb{H}_*(LM)$. Furthermore, this BV algebra structure is compatible with the usual cup product and Gerstenhaber bracket on Hochschild cohomology. To produce this isomorphism, we use a derived form of Poincaré duality with C_*ΩM-modules as local coefficient syste…
