Publications (11)2.45 Total impact
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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 GoreskyHingston product on the cohomology of free loop spaces.  [Show abstract] [Hide abstract]
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 CalabiYau algebras, derived Poincar\'e duality algebras and closed Frobenius algebras. This includes a BValgebra 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 CYalgebra and using BurgheleaFiedorowiczGoodwillie theorem we obtain a BVstructure 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 DGalgebras. In particular we prove that the Hochschild homology and cohomology of a symmetric open Frobenius algebra is a BV and coBValgebra. In Section 7 we exhibit a BV structure on the shifted relative Hochschild homology of a symmetric commutative Frobenius algebra. The existence of a BVstructure on the relative Hochschild homology was expected in the light of ChasSullivan and GoreskyHingston 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 DGalgebra. This recovers TradlerZeinalian \cite{TZ} result for closed Froebenius algebras using the isomorphism $C^*(A,A) \simeq C^*(A,A^\vee)$. 
Article: On 2Holonomy
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ABSTRACT: We construct a cycle in higher Hochschild homology associated to the 2dimensional torus which represents 2holonomy of a nonabelian gerbe in the same way the ordinary holonomy of a principal Gbundle gives rise to a cycle in ordinary Hochschild homology. This is done using the connection 1form of BaezSchreiber. A crucial ingredient in our work is the possibility to arrange that in the structure crossed module mu: g > h of the principal 2bundle, the Lie algebra h is abelian, up to equivalence of crossed modules.  [Show abstract] [Hide abstract]
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.  [Show abstract] [Hide abstract]
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 CalabiYau manifold. Comment: 14 Pages 
Article: Deformation spaces

Article: String Bracket and Flat Connections
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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 MaurerCartan 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 2dimensional 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.  [Show abstract] [Hide abstract]
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. 
Article: The loop product for 3manifolds
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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 3manifolds with “nontrivial” 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 « nontriviaux ». Pour citer cet article : H. Abbaspour, C. R. Acad. Sci. Paris, Ser. I 338 (2004).  [Show abstract] [Hide abstract]
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. 
Publication Stats
36  Citations  
2.45  Total Impact Points  
Top Journals
Institutions

2009

University of Nantes
 Jean Leray Mathematics Lab
Naoned, Pays de la Loire, France


2006

Max Planck Institute for Mathematics
Bonn, North RhineWestphalia, Germany


20032004

École Polytechnique
Paliseau, ÎledeFrance, France
