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Hodge Decomposition For Higher Order Hochschild Homology
"... this paper is to show that the higher order Hochschild homology in charecteristic zero case have naturaral decomposition, which for d = 1 is isomorphic to the classical Hodge decomposition ([L2]). Our methods are new even for d = 1 and are based on homological properties of \Gammamodules. ..."
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this paper is to show that the higher order Hochschild homology in charecteristic zero case have naturaral decomposition, which for d = 1 is isomorphic to the classical Hodge decomposition ([L2]). Our methods are new even for d = 1 and are based on homological properties of \Gammamodules.
The deformation complex for differential graded Hopf algebras
 J. Pure Appl. Algebra
, 1996
"... Abstract. Let H be a differential graded Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the HochschildCartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H(q)structure. The d ..."
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Cited by 7 (4 self)
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Abstract. Let H be a differential graded Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the HochschildCartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H(q)structure. The direct limit of all such truncations is the appropriate setting for deforming H as a strongly homotopy associative structure. Sign complications are systematically controlled. The connection between rational perturbation theory and the deformation theory of certain free commutative differential graded algebras is clarified. 1.
Homotopy operations and rational homotopy type”, in Algebraic Topology: Categorical decomposition techniques
 Prog. in Math. 215, Birkhäuser, BostonBasel
"... In [HS] and [F1] Halperin, Stasheff, and Félix showed how an inductivelydefined sequence of elements in the cohomology of a graded commutative algebra over the rationals can be used to distinguish among the homotopy types of all possible realizations, thus providing a collection ..."
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Cited by 4 (4 self)
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In [HS] and [F1] Halperin, Stasheff, and Félix showed how an inductivelydefined sequence of elements in the cohomology of a graded commutative algebra over the rationals can be used to distinguish among the homotopy types of all possible realizations, thus providing a collection
Structure of the loop homology algebra of a closed manifold
"... The loop homology of a closed orientable manifold M of dimension d is the ordinary homology of the free loop space MS1 with degrees shifted by d, i.e. H∗(M S1) = H∗+d(M S1). Chas and Sullivan have defined a loop product on H∗(M S1) and an intersection morphism I: H∗(M S1) → H∗(ΩM). The algebra H∗(M ..."
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The loop homology of a closed orientable manifold M of dimension d is the ordinary homology of the free loop space MS1 with degrees shifted by d, i.e. H∗(M S1) = H∗+d(M S1). Chas and Sullivan have defined a loop product on H∗(M S1) and an intersection morphism I: H∗(M S1) → H∗(ΩM). The algebra H∗(M S1) is commutative and I is a morphism of algebras. In this paper we produce a model that computes the algebra H∗(M S1) and the morphism I. We show that the kernel of I is nilpotent and that the image is contained in the center of H∗(ΩM), which is in general quite small.
Homological Perturbation Theory And Computability Of Hochschild And Cyclic Homologies Of Cdgas
, 1997
"... . We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some ..."
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. We establish an algorithm computing the homology of commutative dierential graded algebras (briey, CDGAs). The main tool in this approach is given by the Homological Perturbation Theory particularized for the algebra category (see [21]). Taking into account these results, we develop and rene some methods already known about the computation of the Hochschild and cyclic homologies of CDGAs. In the last section of the paper, we analyze the plocal homology of the iterated bar construction of a CDGA (p prime). 1. Introduction. The description of eÆcient algorithms of homological computation might be considered as a very important question in Homological Algebra, in order to use those processes mainly in the resolution of problems on algebraic topology; but this subject also inuence directly on the development of non so closedareas as Cohomological Physics (in this sense, we nd useful references in [12], [24], [25]) and Secondary Calculus ([14], [27], [28]). Working in the context ...
THE VANISHING PROBLEM OF THE STRING CLASS
, 1998
"... Let be an SO.n/bundle over a simply connected manifold M with a spin structure Q! M. The string class is an obstruction to lift the structure group LSpin.n / of the loop group bundle LQ! LM to the universal central extension of LSpin.n / by the circle. We prove that the string class vanishes if an ..."
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Let be an SO.n/bundle over a simply connected manifold M with a spin structure Q! M. The string class is an obstruction to lift the structure group LSpin.n / of the loop group bundle LQ! LM to the universal central extension of LSpin.n / by the circle. We prove that the string class vanishes if and only if 1=2 the first Pontrjagin class of vanishes when M is a compact simply connected homogeneous space of rank one, a simply connected 4dimensional manifold or a finite product space of those manifolds. This result is deduced by using the EilenbergMoore spectral sequence converging to the mod p cohomology of LM whose E2term is the Hochschild homology of the mod p cohomology algebra of M. The key to the consideration is existence of a morphism of algebras, which is injective below degree 3, from an important graded commutative algebra into the Hochschild homology of a certain graded commutative algebra. 1991 Mathematics subject classification (Amer. Math. Soc.): primary 57R20; secondary 55P35, 57T35. 1.
A CHEN MODEL FOR MAPPING SPACES AND THE SURFACE PRODUCT
, 905
"... Abstract. We develop a machinery of Chen iterated integrals for higher Hochschild complexes which are complexes whose differentials are modeled by an arbitrary simplicial set much in the same way that the ordinary Hochschild differential is modeled by the circle. We use these to give algebraic model ..."
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Abstract. We develop a machinery of Chen iterated integrals for higher Hochschild complexes which are complexes whose differentials are modeled by an arbitrary simplicial set much in the same way that the ordinary Hochschild differential is modeled by the circle. We use these to give algebraic models for general mapping spaces and define and study the surface product operation on the homology of mapping spaces of surfaces of all genera into a manifold, which is an analogue of the loop product in string topology. As an application we show that this product is homotopy invariant. We prove HochschildKostantRosenberg type theorems and use them to give explicit formulae for the surface product of odd spheres and Lie groups. Contents
An algorithm for computing the first homology groups of CDGAs with linear differential
, 1999
"... We design here a primary platform for computing the basic homological information of Commutative Differential Graded Algebras (briefly, CDGAs), endowed with linear di#erential. All the algorithms have been implemented in the framework settled by Mathematica, so that we can take advantage of the use ..."
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We design here a primary platform for computing the basic homological information of Commutative Differential Graded Algebras (briefly, CDGAs), endowed with linear di#erential. All the algorithms have been implemented in the framework settled by Mathematica, so that we can take advantage of the use of symbolic computation and many other powerful tools this system provides.
The Deformation Complex for DG Hopf Algebras
, 2002
"... Abstract. Let H be a DG Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the HochschildCartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H (q)structure. The direct limit of a ..."
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Abstract. Let H be a DG Hopf algebra over a field k. This paper gives an explicit construction of a triple cochain complex that defines the HochschildCartier cohomology of H. A certain truncation of this complex is the appropriate setting for deforming H as an H (q)structure. The direct limit of all such truncations is the appropriate setting for deforming H as a strongly homotopy associative structure. Sign complications are systematically controlled. The connection between rational perturbation theory and the deformation theory of certain free commutative differential graded algebras is clarified. 1.
Obstructions to Deformations of D.G. Modules
, 1995
"... Let k be a field and n ≥ 1. There exists a differential graded kmodule (V,d) and various approximations to a differential d + td1 + t 2 d2 + · · · + t n dn on V [[t]], one of which is a nontrivial polynomial deformation, another is obstructed, and another is unobstructed at order n. The analogou ..."
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Let k be a field and n ≥ 1. There exists a differential graded kmodule (V,d) and various approximations to a differential d + td1 + t 2 d2 + · · · + t n dn on V [[t]], one of which is a nontrivial polynomial deformation, another is obstructed, and another is unobstructed at order n. The analogous problem in the category of kalgebras in characteristic zero remains a longstanding open question. 1