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Invariance and localization for cyclic homology of DG algebras
 J. PURE APPL. ALGEBRA
, 1998
"... We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyc ..."
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Cited by 49 (6 self)
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We show that two flat differential graded algebras whose derived categories are equivalent by a derived functor have isomorphic cyclic homology. In particular, ‘ordinary ’ algebras over a field which are derived equivalent [48] share their cyclic homology, and iterated tilting [19] [3] preserves cyclic homology. This completes results of Rickard’s [48] and Happel’s [18]. It also extends well known results on preservation of cyclic homology under Morita equivalence [10], [39], [25], [26], [41], [42]. We then show that under suitable flatness hypotheses, an exact sequence of derived categories of DG algebras yields a long exact sequence in cyclic homology. This may be viewed as an analogue of ThomasonTrobaugh’s [51] and Yao’s [58] localization theorems in Ktheory (cf. also [55]).
BatalinVilkovisky algebras and cyclic cohomology of Hopf algebras
 KTheory
"... Abstract. We show that the ConnesMoscovici cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree −2. More generally, we show that a ”cyclic operad with multiplication” is a cocyclic module whose cohomology is a BatalinVilkovisky algebra and whose cyclic cohomolo ..."
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Cited by 32 (1 self)
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Abstract. We show that the ConnesMoscovici cyclic cohomology of a Hopf algebra equipped with a character has a Lie bracket of degree −2. More generally, we show that a ”cyclic operad with multiplication” is a cocyclic module whose cohomology is a BatalinVilkovisky algebra and whose cyclic cohomology is a graded Lie algebra of degree −2. This explain why the Hochschild cohomology algebra of a symmetric algebra is a BatalinVilkovisky algebra. 1.
Homotopy Algebras and Noncommutative Geometry
, 2004
"... We study cohomology theories of strongly homotopy algebras, namely A∞, C ∞ and L∞algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of C∞algebras thus generalising previous work by Loday and GerstenhaberSchack. These results are then used to show that a C∞algebr ..."
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Cited by 25 (4 self)
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We study cohomology theories of strongly homotopy algebras, namely A∞, C ∞ and L∞algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of C∞algebras thus generalising previous work by Loday and GerstenhaberSchack. These results are then used to show that a C∞algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic C∞algebra (an ∞generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the ‘string topology’ operations (the loop product, the loop bracket and the string bracket) are homotopy invariant and can be defined on the homology or equivariant homology of an arbitrary Poincaré
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 22 (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.
The Cohomology Ring of Free Loop Spaces
"... Let X be a simply connected space and k a commutative ring. Goodwillie ..."
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Cited by 12 (2 self)
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Let X be a simply connected space and k a commutative ring. Goodwillie
ON THE NOTION OF GEOMETRIC REALIZATION
, 2003
"... Abstract. We explain why geometric realization commutes with Cartesian products and why the geometric realization of a simplicial set (resp. cyclic set) is equipped with an action of the group of orientation preserving homeomorphisms of the segment [0, 1] (resp. the circle). Key words: simplicial se ..."
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Cited by 11 (0 self)
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Abstract. We explain why geometric realization commutes with Cartesian products and why the geometric realization of a simplicial set (resp. cyclic set) is equipped with an action of the group of orientation preserving homeomorphisms of the segment [0, 1] (resp. the circle). Key words: simplicial set, cyclic set, geometric realization, cyclic homology, fiber functor Having written this article I learned that a very similar approach had been developed by A. Besser [1]. In this note there are no theorems, its only goal is to clarify the notion of geometric realization for simplicial sets [11, 10, 4, 5] and cyclic sets [3, 2, 8]. We reformulate the definitions so that the following facts become obvious: (i) geometric realization commutes with finite projective limits (e.g., with Cartesian products); (ii) the geometric realization of a simplicial set (resp. cyclic set) is equipped with an action of the group of orientation preserving homeomorphisms of the segment I: = [0,1] (resp. the circle S1). In the traditional approach [11, 10, 4, 5, 3, 2, 8] these statements are theorems, and understanding their proofs requires some efforts. Example. To a small category C there corresponds a simplicial set NC (the nerve of C) and a cyclic set NcycC (the cyclic nerve). It follows from our formula (1.1) that a point of the geometric realization NC  is a piecewise constant functor I → C. Here it is assumed that the category structure on I comes from the standard order on I, and the definition of piecewise constant functor is explained in (1.4). In §3 we give a similar description of NcycC  in which I is replaced by S 1. Convention. Unless stated otherwise, an ordered set I is always equipped with the category structure such that the number of morphisms from i ∈ I to j ∈ I equals 1 if i ≤ j and 0 otherwise.
Inertia orbifolds, configuration spaces and the ghost loop space
 Jour. of Math
"... Abstract. In this paper we define and study the ghost loop orbifold LsX of an orbifold X consisting of those loops that remain constant in the coarse moduli space of X. We construct a configuration space model for LsX using an idea of G. Segal. From this we exhibit the relation between the Hochschil ..."
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Cited by 7 (4 self)
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Abstract. In this paper we define and study the ghost loop orbifold LsX of an orbifold X consisting of those loops that remain constant in the coarse moduli space of X. We construct a configuration space model for LsX using an idea of G. Segal. From this we exhibit the relation between the Hochschild and cyclic homologies of the inertia orbifold of X (that generate the socalled twisted sectors in string theory) and the ordinary and equivariant homologies of LsX. We also show how this clarifies the relation between orbifold Ktheory, ChenRuan orbifold cohomology, Hochschild homology, and periodic cyclic homology. 1.
On the Cohomology Algebra of Free Loop Spaces
"... Let X be a simply connected space and K be any field. The normalized singular cochains N (X ; K) admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology HH N X of the space X . We prove that, endowed with this product, HH N ..."
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Cited by 7 (0 self)
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Let X be a simply connected space and K be any field. The normalized singular cochains N (X ; K) admit a natural strongly homotopy commutative algebra structure, which induces a natural product on the Hochschild homology HH N X of the space X . We prove that, endowed with this product, HH N X is isomorphic to the cohomology algebra of the free loop space of X with coefficients in K. We also show how to construct a simpler Hochschild complex which allows direct computation.