Abstract not found

hep-th/yymmnnn

The interplay between geometry and topology on complex algebraic varieties is a classical theme that goes back to Lefschetz [L] and Zariski [Z] and is always present on the scene, see for instance the work by Libgober [Li]. In this paper we study complements of hypersurfaces, with a special attention to the case of hyperplane arrangements as discussed

Abstract. We prove an analogue of the Poincaré-Birkhoff-Witt theorem and of the Cartier-Milnor-Moore theorem for non-cocommutative Hopf algebras. The primitive part of a cofree Hopf algebra is a nondifferential B∞-algebra. We construct a universal enveloping functor U2 from nondifferential B∞-algebras to 2-associative algebras, i.e. algebras equipped with two associative operations. We show that any cofree Hopf algebra H is of the form U2(Prim H). We take advantage of the description of the free 2as-algebra in terms of planar trees to unravel the operad associated to nondifferential B∞-algebras.

Abstract not found

Chen's theory of iterated integrals provides a remarkable model for the di erential forms on the based loop space M of a di erentiable manifold M (Chen [10]; see also Hain-Tondeur [23] and Getzler-Jones-Petrack [21]). This article began as an attempt to nd an analogous model for 2 the complex of di erentiable forms on the double loop space M, motivated in part by the hope that this might provide an algebraic framework for understanding two-dimensional topological eld theories. Our approach is to use the formalism of operads. Operads can be de ned in any symmetric monoidal category, although we will mainly be concerned with dg-operads (di erential graded operads), that is, operads in the category of chain complexes with monoidal structure de ned by the graded tensor product. An operad is a sequence of objects a(k), k 0, carrying an action of the symmetric group Sk, with products a(k) a(j1) : : : a(jk) �! a(j1 + + jk) which are equivariant and associative | we give a precise de nition in Section 1.2. An operad such that a(k) = 0 for k 6 = 1 is a monoid: in this sense, operads are a non-linear generalization of monoids. If V is a chain complex, we may de ne an operad with EV (k) = Hom(V (k) ; V); where V (k) is the k-th tensor power of V. The symmetric group Sk acts on EV (k) through its action on V (k) , and the structure maps of EV are the obvious ones. This operad plays the same role in the theory of operads that the algebra End(V) does in the theory of associative algebras. An algebra over an operad a (or a-algebra) is a chain complex A together with a morphism of operads: a �! EA. In other words, A is equipped with structure maps k: a(k)

1.2. The axioms

Abstract: We show that the homotopy theory of differential graded algebras coincides with the homotopy theory of HZ-algebra spectra. Namely, we construct Quillen equivalences between the Quillen model categories of (unbounded) differential graded algebras and HZ-algebra spectra. We also construct Quillen equivalences between the differential graded modules and module spectra over these algebras. We use these equivalences in turn to produce algebraic models for rational stable model categories. We show that bascially any rational stable model category is Quillen equivalent to modules over a differential graded Q-algebra (with many objects). 1.

Abstract. A finite simplicial graph Γ determines a right-angled Artin group GΓ, with generators corresponding to the vertices of Γ, and with a relation vw = wv for each pair of adjacent vertices. We compute the lower central series quotients, the Chen quotients, and the (first) resonance variety of GΓ, directly from the graph Γ. 1.

The purpose of this paper is to develop some additional techniques for the weak n-categories defined by Tamsamani in [27] (which he calls n-nerves). The goal is to be able to define the internal Hom(A, B) for two n-nerves A and B, which should itself be an n-nerve. This in turn is for defining the n + 1-nerve nCAT of all n-nerves conjectured in