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Presentations "a la Coxeter" are given for all (irreducible) finite complex reflection groups. They provide presentations for the corresponding generalized braid groups (for all but six cases), which allow us to generalize some of the known properties of finite Coxeter groups and their associated braid groups, such as the computation of the center of the braid group and the construction of deformations of the finite group algebra (Hecke algebras). We introduce monodromy representations of the braid groups which factorize through the Hecke algebras, extending results of Cherednik, Opdam, Kohno and others.

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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)

this paper is the geometry and topology of the pair (M; D). The examples in which we are interested will have the features discussed in (A), (B), and (C) below. (A) Cellulations by polytopes. The divisor D cuts M into regions, called chambers,

Dedicated to the memory of Moshé Flato Abstract. The purpose of this paper is to complete Getzler-Jones ’ proof of Deligne’s Conjecture, thereby establishing an explicit relationship between the geometry of configurations of points in the plane and the Hochschild complex of an associative algebra. More concretely, it is shown that the B∞-operad, which is generated by multilinear operations known to act on the Hochschild complex, is a quotient of a certain operad associated to the compactified configuration spaces. Different notions of homotopy Gerstenhaber algebras are discussed: One of them is a B∞-algebra, another, called a homotopy G-algebra, is a particular case of a B∞-algebra, the others, a G∞-algebra, an E 1-algebra, and a weak G∞-algebra, arise from the geometry of configuration spaces. Corrections to the paper of Kimura, Zuckerman, and the author related to the use of a nonextant notion of a homotopy Gerstenhaber algebra are made. In an unpublished paper of E. Getzler and J. D. S. Jones [GJ94], the notion of a homotopy n-algebra was introduced. Unfortunately the construction that justified

. Let M be a surface, let N be a subsurface of M , and let n m be two positive integers. The inclusion of N in M gives rise to a homomorphism from the braid group B n N with n strings on N to the braid group BmM with m strings on M . We first determine necessary and sufficient conditions that this homomorphism is injective, and we characterize the commensurator, the normalizer and the centralizer of 1 N in 1 M . Then we calculate the commensurator, the normalizer, and the centralizer of B n N in BmM for large surface braid groups. 1. Introduction The classical braid groups Bm were introduced by Artin in 1926 ([Ar1], [Ar2]) and have played a remarkable role in topology, algebra, analysis, and physics. A natural generalization to braids on surfaces was introduced by Fox and Neuwirth [FoN] in 1962. The surface braid groups, for closed surfaces, were calculated in terms of generators and relations during the ensuing decade ([Bi1], [Sc], [Va], [FaV]). Since then, most progress in this...

Abstract. As a generalization of a classical result on the Alexander polynomial for fibered knots, we show in this paper that the Reidemeister torsion associated to a certain representation detects fiberedness of knots in the three sphere. 1.

this paper we develop combinatorial and representation-theoretic methods in the theory of arrangements of linear subspaces in R , which can be applied to the study of the cohomology of spaces of ordered and unordered point configurations in d-dimensional real space. In particular we provide and apply tools for determining representations of finite groups on the rational cohomology of the complement of an arrangement of linear subspaces. The connection between arrangements of linear subspaces and configuration spaces was first suggested by Bjorner [Bj4, 8.5], who was himself inspired by the work of Arnol'd [Ar1], [Ar2], [Ar3] and Vassiliev [Va]. Although the relation between hyperplane arrangements and configuration spaces had been established a long time ago by Fadell and Neuwirth [Fa-Ne], the idea of using subspace arrangements as a unifying approach in this context seems to be recent. In this paper we consider the following examples : (A) Let M be the set of all n = k + q tuples (x 1 ; : : : ; xn ) of points x i 2 R such that there is no subset E of f1; : : : ; ng of cardinality k satisfying x t = x s for all t; s 2 E. In particular M is the pure braid space (see [Fa-Ne], [Ar1]) for k = d = 2. (See [Co-La-Ma] for k = 2 and d 2: For general k the space M is first mentioned in [Co-Lu] in connection with a generalisation of the Borsuk-Ulam Theorem.) The cohomology of the spaces M was determined in [Bj-We] for d = 1; 2 (and implicitly for general d). We study the action of the symmetric group Sn on M by permuting the coordinates. For d = 2 the orbit space M ) =Sn is homeomorphic to the space of monic polynomials of degree n with no root of multiplicity k. For k = 2 this is the complement of the discriminant; it is thus the braid space...

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 1-operads and Sym1 is the first term of the infinite series of left adjoint functors Symn, called symmetrisation functors, from n-operads to symmetric operads with the property that the category of one object, one arrow,..., one (n − 1)-arrow algebras of an n-operad 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 n-operads. We follow Getzler and Jones and consider their decomposition of the Fulton-Macpherson operad of compactified real configuration spaces. We construct an n-operadic counterpart of