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Batalin–Vilkovisky algebras and twodimensional topological field theories
 265–285. AND ALGEBRAS 231
, 1994
"... Abstract: By a BatalinVilkovisky algebra, we mean a graded commutative algebra A, together with an operator A: A.+ A. such that A +1 2 = 0, and \_A,d \ — Aa is a graded derivation of A for all a e A. In this article, we show that there is a natural structure of a BatalinVilkovisky algebra on the ..."
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Cited by 122 (4 self)
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Abstract: By a BatalinVilkovisky algebra, we mean a graded commutative algebra A, together with an operator A: A.+ A. such that A +1 2 = 0, and \_A,d \ — Aa is a graded derivation of A for all a e A. In this article, we show that there is a natural structure of a BatalinVilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions. We make use of a technique from algebraic topology: the theory of operads. BatalinVilkovisky algebras are a new type of algebraic structure on graded vector spaces, which first arose in the work of Batalin and Vilkovisky on gauge fixing in quantum field theory: a BatalinVilkovisky algebra is a differential graded commutative algebra together with an operator A: A.+A such that A m+ί 2 = 0, and Δ{abc) = A(ab)c + ( V)^aA{bc) + ( l) (α ίm
The braid monodromy of plane algebraic curves and hyperplane arrangements
 COMMENT. MATH. HELVETICI
, 1997
"... To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin’s braid group Bn. Using Hansen’s polynomial covering space theory, we give a new interpretation of this construction. Next, we provide an explicit description of ..."
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Cited by 39 (9 self)
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To a plane algebraic curve of degree n, Moishezon associated a braid monodromy homomorphism from a finitely generated free group to Artin’s braid group Bn. Using Hansen’s polynomial covering space theory, we give a new interpretation of this construction. Next, we provide an explicit description of the braid monodromy of an arrangement of complex affine hyperplanes, by means of an associated “braided wiring diagram. ” The ensuing presentation of the fundamental group of the complement is shown to be TietzeI equivalent to the RandellArvola presentation. Work of Libgober then implies that the complement of a line arrangement is homotopy equivalent to the 2complex modeled on either of these presentations. Finally, we prove that the braid monodromy of a line arrangement determines the intersection lattice. Examples of Falk then show that the braid monodromy carries more information than the group of the complement, thereby answering a question of Libgober.
Homology of iterated semidirect products of free groups
 Journal of Pure and Applied Algebra
, 1998
"... Let G be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of G. This resolution is used to define representations of groups which act compatibly on G, generalizing classi ..."
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Cited by 27 (9 self)
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Let G be a group which admits the structure of an iterated semidirect product of finitely generated free groups. We construct a finite, free resolution of the integers over the group ring of G. This resolution is used to define representations of groups which act compatibly on G, generalizing classical constructions of Magnus, Burau, and Gassner. Our construction also yields algorithms for computing the homology of the Milnor fiber of a fibertype hyperplane arrangement, and more generally, the homology of the complement of such an arrangement with coefficients in an arbitrary local system.
Geometric Subgroups of Surface Braid Groups
 ANN. INST. FOURIER (GRENOBLE
, 1998
"... 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 ..."
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Cited by 20 (1 self)
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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.
Polydiagonal compactifications of configuration spaces
 J. Algebraic Geom
"... Abstract. A smooth compactification X〈n 〉 of the configuration space of n distinct labeled points in a smooth algebraic variety X is constructed by a natural sequence of blowups, with the full symmetry of the permutation group Sn manifest at each stage. The strata of the normal crossing divisor at i ..."
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Cited by 17 (0 self)
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Abstract. A smooth compactification X〈n 〉 of the configuration space of n distinct labeled points in a smooth algebraic variety X is constructed by a natural sequence of blowups, with the full symmetry of the permutation group Sn manifest at each stage. The strata of the normal crossing divisor at infinity are labeled by leveled trees and their structure is studied. This is the maximal wonderful compactification in the sense of De Concini–Procesi, and it has a stratacompatible surjection onto the Fulton–MacPherson compactification. The degenerate configurations added in the compactification are geometrically described by polyscreens similar to the screens of Fulton and MacPherson. In characteristic 0, isotropy subgroups of the action of Sn on X〈n〉 are abelian, thus X〈n 〉 may be a step toward an explicit resolution of singularities of the symmetric products X n /Sn.
Group Actions on Arrangements of Linear Subspaces and Applications to Configuration Spaces
 Trans. Amer. Math. Soc
"... this paper we develop combinatorial and representationtheoretic 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 ddimensional real space. In particular we provide and a ..."
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Cited by 16 (4 self)
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this paper we develop combinatorial and representationtheoretic 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 ddimensional 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 [FaNe], 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 [FaNe], [Ar1]) for k = d = 2. (See [CoLaMa] for k = 2 and d 2: For general k the space M is first mentioned in [CoLu] in connection with a generalisation of the BorsukUlam Theorem.) The cohomology of the spaces M was determined in [BjWe] 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...
On loop spaces of configuration spaces
 Trans. Amer. Math. Soc
"... Abstract. This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as “twistings ” between the factors. The main homological results a ..."
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Cited by 15 (0 self)
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Abstract. This article gives an analysis of topological and homological properties for loop spaces of configuration spaces. The main topological results are given by certain choices of product decompositions of these spaces, as well as “twistings ” between the factors. The main homological results are given in terms of extensions of the “infinitesimal braid relations ” or “universal YangBaxter Lie relations”. 1.
New presentations of surface braid groups
 J. of
"... In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations. 1 ..."
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Cited by 13 (2 self)
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In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations. 1