Orbits of Braid Groups on Cacti (original) (raw)

The braid monodromy of plane algebraic curves and hyperplane arrangements

Commentarii Mathematici 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 B n . 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 Tietze-I equivalent to the Randell-Arvola presentation. Work of Libgober then implies that the complement of a line arrangement is homotopy equivalent to the 2-complex 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.

Braid monodromy and topology of plane curves

Duke Mathematical Journal, 2003

In this paper we prove that braid monodromy of an affine plane curve determines the topology of a related projective plane curve. 1 2 E. ARTAL, J. CARMONA, AND J.I. COGOLLUDO

Braid monodromy of complex line arrangements

Kodai Mathematical Journal, 1999

Let V be the complex vector space C 7 , s/ an arrangement in V, i.e. a finite family of hyperplanes in V In , Moishezon associated to any algebraic plane curve <# of degree n a braid monodromy homomorphism θ F s -> B(n), where F s is a free group, B(ή) is the Artm braid group. In this paper, we will determine the braid monodromy for the case when # is an arrangement stf of complex lines in C 2 , using the notion of labyrinth of an arrangement. As a corollary we get the braid monodromy presentation for the fundamental group of the complement to the arrangement.

Some monodromy representations of generalized braid groups

Communications in Mathematical Physics, 1994

A flat connection on the trivial bundle over the complement in C" of the complexification of the system of the reflecting hyperplanes of the B n , D n Coxeter groups is built from a simple Lie algebra and its representation. The corresponding monodromy representations of the generalized braid groups XB n , XD n are computed in the simplest case.

Deformations of the braid arrangement and trees

Advances in Mathematics, 2018

We establish general counting formulas and bijections for deformations of the braid arrangement. Precisely, we consider real hyperplane arrangements such that all the hyperplanes are of the form xi − xj = s for some integer s. Classical examples include the braid, Catalan, Shi, semiorder and Linial arrangements, as well as graphical arrangements. We express the number of regions of any such arrangement as a signed count of decorated plane trees. The characteristic and coboundary polynomials of these arrangements also have simple expressions in terms of these trees. We then focus on certain "well-behaved" deformations of the braid arrangement that we call transitive. This includes the Catalan, Shi, semiorder and Linial arrangements, as well as many other arrangements appearing in the literature. For any transitive deformation of the braid arrangement we establish a simple bijection between regions of the arrangement and a set of labeled plane trees defined by local conditions. This answers a question of Gessel.

An acyclic extension of the braid group

Commentarii Mathematici Helvetici, 1991

We relate Artin's braid group B~ = lim ~ B n to a certain group F' of p/-homeomorphisms of the interval. Namely, there exists a short exact sequence 1~ B~-,A-* F'~ 1, where HkA = 0, k ~ 1.