Abstract not found

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 Thomason-Trobaugh’s [51] and Yao’s [58] localization theorems in K-theory (cf. also [55]).

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.

Abstract. We show that the Connes-Moscovici 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 Batalin-Vilkovisky 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 Batalin-Vilkovisky algebra. 1.

Let X be a simply connected space and k a commutative ring. Goodwillie

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.

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 so-called twisted sectors in string theory) and the ordinary and equivariant homologies of LsX. We also show how this clarifies the relation between orbifold K-theory, Chen-Ruan orbifold cohomology, Hochschild homology, and periodic cyclic homology. 1.

Let C be a differential graded coalgebra, ΩC the Adams cobar construction and C ∨ the dual algebra. We prove that for a large class of coalgebras C there is a natural isomorphism of Gerstenhaber algebras between the Hochschild cohomologies HH ∗ (C ∨ , C ∨ ) and HH ∗ (ΩC; ΩC). This result permits to describe a Hodge decomposition of the loop space homology of a closed oriented manifold, in the sense of Chas-Sullivan, when the field of coefficients is of characteristic zero.

Abstract. In this paper we establish the existence of certain structures on the ordinary and equivariant homology of the free loop space on a manifold or, more generally, a formal Poincaré duality space. These structures; namely the loop product, the loop bracket and the string bracket, were introduced and studied by Chas and Sullivan under the general heading ‘string topology’. Our method is based on obstruction theory for C∞-algebras and rational homotopy theory. The resulting string topology operations are manifestly homotopy invariant.

Abstract. We construct a discrete model of the homotopy theory of S 1-spaces. We define a category P with objects composed of a simplicial set and a cyclic set along with suitable compatibility data. P inherits a model structure from the model structures on the categories of simplicial sets and cyclic sets. We then show that there is a Quillen equivalence between P and the model category of S 1-spaces in which weak equivalences and fibrations are maps inducing weak equivalences and fibrations on passage to all fixed point sets. 1.