Hossein Abbaspour

École Polytechnique, Paliseau, Île-de-France, France

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Publications (10)0.48 Total impact

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    Hossein Abbaspour
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    ABSTRACT: We prove that the Hochschild homology (and cohomology) of a symmetric open Frobenius algebra $A$ has a natural coBV and BV structure. The underlying coalgebra and algebra structure may not be resp. counital and unital. Moreover we prove that the product and coproduct satisfy the Frobenius compatibility condition i.e. the coproduct on $HH_*(A)$ is a map of left and right $HH_*(A)$-modules. If $A$ is commutative, we also introduced a natural BV structure on the relative Hochschild homology $\widetilde{HH}_*(A)$ after a shift in degree. We anticipate that the product of this BV structure to be related to the Goresky-Hingston product on the cohomology of free loop spaces.
    09/2013;
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    Hossein Abbaspour
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    ABSTRACT: We first review various known algebraic structures on the Hochschild (co)homology of a differential graded algebras under weak Poincar\'e duality hypothesis, such as Calabi-Yau algebras, derived Poincar\'e duality algebras and closed Frobenius algebras. This includes a BV-algebra structure on $HH^*(A,A^\vee)$ or $HH^*(A,A)$, which in the latter case is an extension of the natural Gerstenhaber structure on $HH^*(A,A)$. As an example, after proving that the chain complex of the Moore loop space of a manifold $M$ is a CY-algebra and using Burghelea-Fiedorowicz-Goodwillie theorem we obtain a BV-structure on the homology of the free space. In Sections 6 we prove that these BV/coBVstructures can be indeed defined for the Hochschild homology of a symmetric open Frobenius DG-algebras. In particular we prove that the Hochschild homology and cohomology of a symmetric open Frobenius algebra is a BV and coBV-algebra. In Section 7 we exhibit a BV structure on the shifted relative Hochschild homology of a symmetric commutative Frobenius algebra. The existence of a BV-structure on the relative Hochschild homology was expected in the light of Chas-Sullivan and Goresky-Hingston results for free loop spaces. In Section 8 we present an action of Sullivan diagrams on the Hochschild (co)chain complex of a closed Frobenius DG-algebra. This recovers Tradler-Zeinalian \cite{TZ} result for closed Froebenius algebras using the isomorphism $C^*(A,A) \simeq C^*(A,A^\vee)$.
    02/2013;
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    Hossein Abbaspour, Friedrich Wagemann
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    ABSTRACT: We construct a cycle in higher Hochschild homology associated to the 2-dimensional torus which represents 2-holonomy of a non-abelian gerbe in the same way the ordinary holonomy of a principal G-bundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1-form of Baez-Schreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module mu: g -> h of the principal 2-bundle, the Lie algebra h is abelian, up to equivalence of crossed modules.
    02/2012;
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    Hossein Abbaspour, David Chataur
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    ABSTRACT: In this paper we introduce various associative products on the homology of the space of knots and singular knots in $S^n$. We prove that these products are related through a desingularization map. We also compute some of these products and prove the nontriviality of the desingularization morphism.
    02/2009;
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    Hossein Abbaspour, Thomas Tradler, Mahmoud Zeinalian
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    ABSTRACT: In this paper we construct a Lie algebra representation of the algebraic string bracket on negative cyclic cohomology of an associative algebra with appropriate duality. This is a generalized algebraic version of the main theorem of [AZ] which extends Goldman's results using string topology operations.The main result can be applied to the de Rham complex of a smooth manifold as well as the Dolbeault resolution of the endomorphisms of a holomorphic bundle on a Calabi-Yau manifold. Comment: 14 Pages
    07/2008;
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    Hossein Abbaspour, Matilde Marcolli, Thomas Tradler
    Aspects of mathematics, v.E40 (2010). 11/2006;
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    Hossein Abbaspour, Mahmoud Zeinalian
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    ABSTRACT: Let $G \to P \to M$ be a flat principal bundle over a closed and oriented manifold $M$ of dimension $m=2d$. We construct a map of Lie algebras $\Psi: \H_{2\ast} (L M) \to {\o}(\Mc)$, where $\H_{2\ast} (LM)$ is the even dimensional part of the equivariant homology of $LM$, the free loop space of $M$, and $\Mc$ is the Maurer-Cartan moduli space of the graded differential Lie algebra $\Omega^\ast (M, \adp)$, the differential forms with values in the associated adjoint bundle of $P$. For a 2-dimensional manifold $M$, our Lie algebra map reduces to that constructed by Goldman in \cite{G2}. We treat different Lie algebra structures on $\H_{2\ast}(LM)$ depending on the choice of the linear reductive Lie group $G$ in our discussion.
    03/2006;
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    Hossein Abbaspour, Ralph Cohen, Kate Gruher
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    ABSTRACT: Let G be a Poincare duality group of dimension n. For a given element g in G, let C_g denote its centralizer subgroup. Let L_G be the graded abelian group defined by (L_G)_p = oplus_{[g]}H_{p+n}(C_g) where the sum is taken over conjugacy classes of elements in G. In this paper we construct a multiplication on L_G directly in terms of intersection products on the centralizers. This multiplication makes L_G a graded, associative, commutative algebra. When G is the fundamental group of an aspherical, closed oriented n manifold M, then (L_G)_* = H_{*+n}(LM), where LM is the free loop space of M. We show that the product on L_G corresponds to the string topology loop product on H_*(LM) defined by Chas and Sullivan.
    12/2005;
  • Hossein Abbaspour
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    ABSTRACT: Let M be a connected, closed, oriented and smooth manifold of dimension d. Let LM be the space of loops in M. Chas and Sullivan introduced the loop product, an associative product of degree −d on the homology of LM. In this Note we aim at identifying 3-manifolds with “non-trivial” loop products. To cite this article: H. Abbaspour, C. R. Acad. Sci. Paris, Ser. I 338 (2004).RésuméPour M, une variété connexe, orientée et lisse de dimension d, soit LM l'espace des lacets libres de M. Chas et Sullivan ont défini un produit associatif de degré −d sur l'homologie de LM. Dans cette Note on vise à identifier les variétés de dimension 3 qui ont des produits de Chas–Sullivan « non-triviaux ». Pour citer cet article : H. Abbaspour, C. R. Acad. Sci. Paris, Ser. I 338 (2004).
    Comptes Rendus Mathematique 05/2004; 338(9):713–718. · 0.48 Impact Factor
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    Hossein Abbaspour
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    ABSTRACT: Let $M$ be a closed, oriented and smooth manifold of dimension $d$. Let $\L M$ be the space of smooth loops in $M$. Chas and Sullivan introduced loop product, a product of degree $-d$ on the homology of $LM$. In this paper we show how for three manifolds the ``nontriviality'' of the loop product relates to the ``hyperbolicity'' of the underlying manifold. This is an application of the existing powerful tool and results in three dimensional topology such as the prime decomposition, torus decomposition, Seifert theorem, torus theorem.
    11/2003;