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17
Multivariable cochain operations and little ncubes
 J. Amer. Math. Soc
"... Abstract. In this paper we construct a small E ∞ chain operad S which acts naturally on the normalized cochains S ∗ X of a topological space. We also construct, for each n, a suboperad Sn which is quasiisomorphic to the normalized singular chains of the little ncubes operad. The case n = 2 leads t ..."
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Cited by 26 (1 self)
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Abstract. In this paper we construct a small E ∞ chain operad S which acts naturally on the normalized cochains S ∗ X of a topological space. We also construct, for each n, a suboperad Sn which is quasiisomorphic to the normalized singular chains of the little ncubes operad. The case n = 2 leads to a substantial simplification of our earlier proof of Deligne’s Hochschild cohomology conjecture. 1. Introduction. This paper has two goals. The first (see Theorem 2.15 and Remark 2.16(a)) is to construct a small E ∞ chain operad S which acts naturally on the normalized cochains S∗X of a topological space X. This is of interest in view of a theorem of Mandell [15, page 44] which states that if O is any E ∞ chain operad over Fp (the algebraic closure of the field with
A bordism approach to string topology
"... Abstract. Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology constructed by M. Chas and D. Sullivan, V. Godin and R. Cohen. We generalize some of these operations to spaces of maps from a sphere to a comp ..."
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Cited by 18 (1 self)
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Abstract. Using intersection theory in the context of Hilbert manifolds and geometric homology we show how to recover the main operations of string topology constructed by M. Chas and D. Sullivan, V. Godin and R. Cohen. We generalize some of these operations to spaces of maps from a sphere to a compact manifold. 1.
On Spineless Cacti, Deligne’s Conjecture and Connes–Kreimer’s Hopf Algebra
"... Abstract. We give a new direct proof of Deligne’s conjecture on the Hochschild cohomology. For this we use the cellular chain operad of normalized spineless cacti as a model for the chains of the little discs operad. Previously, we have shown that the operad of spineless cacti is homotopy equivalent ..."
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Cited by 17 (4 self)
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Abstract. We give a new direct proof of Deligne’s conjecture on the Hochschild cohomology. For this we use the cellular chain operad of normalized spineless cacti as a model for the chains of the little discs operad. Previously, we have shown that the operad of spineless cacti is homotopy equivalent to the little discs operad. Moreover, we also showed that the quasi–operad of normalized spineless cacti is homotopy equivalent to the spineless cacti operad. Now, we give a cell decomposition for the normalized spineless cacti, whose cellular chains form an operad and by our previous results a chain model for the little discs operad. The cells are indexed by bipartite black and white trees which can directly be interpreted as operations on the Hochschild cochains of an associative algebra, yielding a positive answer to Deligne’s conjecture. Furthermore, we show that the symmetric combinations of top–dimensional cells, are isomorphic to the graded pre–Lie operad. Lastly, we define the Hopf algebra of an operad which affords a direct sum. For the pre–Lie suboperad of shifted symmetric top–dimensional chains the symmetric group coinvariants of this Hopf algebra are the renormalization Hopf algebra of Connes and Kreimer.
The symmetrisation of noperads and compactification of real configuration spaces
 Adv. Math
"... It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint Sym1 given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author’s ..."
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Cited by 15 (3 self)
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It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint Sym1 given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author’s theory of higher operads, the nonsymmmetric operads are 1operads and Sym1 is the first term of the infinite series of left adjoint functors Symn, called symmetrisation functors, from noperads to symmetric operads with the property that the category of one object, one arrow,..., one (n − 1)arrow algebras of an noperad A is isomorphic to the category of algebras of Symn(A). In this paper we consider some geometrical and homotopical aspects of the symmetrisation of noperads. We follow Getzler and Jones and consider their decomposition of the FultonMacpherson operad of compactified real configuration spaces. We construct an noperadic counterpart of
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S I
"... Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCarta ..."
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Cited by 9 (4 self)
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Abstract. In this paper and its followup [MV08], we study the deformation theory of morphisms of properads and props thereby extending Quillen’s deformation theory for commutative rings to a nonlinear framework. The associated chain complex is endowed with an L∞algebra structure. Its MaurerCartan elements correspond to deformed structures, which allows us to give a geometric interpretation of these results.
On the multiplicative structure of topological Hochschild homology, Algebr
 Geom. Topol
"... Abstract We show that the topological Hochschild homology THH(R) of an Enring spectrum R is an En−1ring spectrum. The proof is based on the fact that the tensor product of the operad Ass for monoid structures and the the little ncubes operad Cn is an En+1operad, a result which is of independent ..."
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Cited by 5 (0 self)
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Abstract We show that the topological Hochschild homology THH(R) of an Enring spectrum R is an En−1ring spectrum. The proof is based on the fact that the tensor product of the operad Ass for monoid structures and the the little ncubes operad Cn is an En+1operad, a result which is of independent interest. In 1993 Deligne asked whether the Hochschild cochain complex of an associative ring has a canonical action by the singular chains of the little 2cubes operad. Affirmative answers for differential graded algebras in characteristic 0 have been found by Kontsevich and Soibelman [11], Tamarkin [15] and [16], and Voronov [18]. A more general proof, which also applies to associative ring spectra is due to McClure and Smith [14]. In [10] Kontsevich extended Deligne’s question: Does the Hochschild cochain complex of an En differential graded algebra carry a canonical En+1structure? We consider the dual problem: Given a ring R with additional structure, how much structure does the topological Hochschild homology THH(R) of R inherit from R? The close connection of THH with algebraic Ktheory and with structural questions in the category of spectra make multiplicative structures on THH desirable. In his early work on topological Hochschild homology of functors with smash product Bökstedt proved that THH of a commutative such functor is a commutative ring spectrum (unpublished). The discovery of associative, commutative and unital smash product functors of spectra simplified the definition of THH and the proof of the corresponding result for E∞ring spectra considerably (e.g. see [13]). 1 In this paper we morally prove Theorem A: For n ≥ 2, if R is an Enring spectrum then THH(R) is an En−1ring spectrum. The same result has been obtained independently by Basterra and Mandell using different techniques [2]. Why “morally”? To define THH(R) we need R to be a strictly associative spectrum. In general, Enstructures do not have a strictly associative substructure. So we have to replace R by an equivalent strictly associative ring spectrum Y, whose multiplication extends to an Enstructure. Then the statement makes sense for Y. Here is a more precise reformulation of
On a General Chain Model of the Free Loop Space And String Topology
, 2007
"... Let M be a smooth oriented manifold. The homology of M has the structure of a Frobenius algebra. This paper shows that on chain level there is a Frobeniuslike algebra structure, whose homology gives the Frobenius algebra of M. Moreover, associated to any Frobeniuslike algebra, there is a chain comp ..."
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Let M be a smooth oriented manifold. The homology of M has the structure of a Frobenius algebra. This paper shows that on chain level there is a Frobeniuslike algebra structure, whose homology gives the Frobenius algebra of M. Moreover, associated to any Frobeniuslike algebra, there is a chain complex whose homology has the structure of a Gerstenhaber algebra and a BatalinVilkovisky algebra. And if the Frobeniuslike algebra comes from M, it gives the free loop space LM and String Topology of ChasSullivan. Contents
Homotopy Gerstenhaber structures and vertex algebras
 Appl. Categ. Structures
"... Abstract. We provide a simple construction of a G∞algebra structure on an important class of vertex algebras V, which lifts the Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications to algebraic topology: the construction of a sheaf of ..."
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Cited by 2 (2 self)
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Abstract. We provide a simple construction of a G∞algebra structure on an important class of vertex algebras V, which lifts the Gerstenhaber algebra structure on BRST cohomology of V introduced by Lian and Zuckerman. We outline two applications to algebraic topology: the construction of a sheaf of G ∞ algebras on a Calabi–Yau manifold M, extending the operations of multiplication and bracket of functions and vector fields on M, and of a Lie ∞ structure related to the bracket of Courant [5].
An algebraic chain model of string topology
"... Abstract. A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and BatalinVilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant ho ..."
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Cited by 2 (1 self)
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Abstract. A chain complex model for the free loop space of a connected, closed and oriented manifold is presented, and on its homology, the Gerstenhaber and BatalinVilkovisky algebra structures are defined and identified with the string topology structures. The gravity algebra on the equivariant homology of the free loop space is also modeled. The construction includes the nonsimply connected case, and therefore gives an algebraic and chain level model of ChasSullivan’s String Topology. 1.
DEFORMATION THEORY OF REPRESENTATIONS OF PROP(ERAD)S II
"... Abstract. This paper is the followup of [MV08]. ..."