Bronislaw Wajnryb

A simple presentation for the mapping class group of an orientable surface

LetFn.k be an orientable compact surface of genusn withk boundary components. For a suitable choice of 2n + 1 simple closed curves onFn,1 the corresponding Dehn twists generate bothMn,o andMn,1. A complete system of relations is determined for these generators and the presentations ofMn,0 andMn,1 obtained in this way are much simpler than the known presentations.

An elementary approach to the mapping class group of a surface.

We consider an oriented surface S and a cellular complex X of curves on S , dened by Hatcher and Thurston in 1980. We prove by elementary means, without Cerf theory, that the complex X is connected and simply connected. From this we derive an explicit simple presentation of the mapping class group of S , following the ideas of Hatcher{Thurston and Harer.

Orbits of Hurwitz action for coverings of a sphere with two special fibers

Abstract We consider finite, orientable, connected, branched coverings of a 2-sphere which have at most two nonsimple critical values. We prove that the equivalence class of a covering, up to a fiber preserving homeomorphism, is determined by its genus, the degree, the monodromy group and the branching data of the nonsimple critical values.

MARKOV CLASSES IN CERTAIN FINITE QUOTIENTS OF ARTIN'S BRAID GROUP

This paper studies three finite quotients of the sequence of braid groups {Bn;n = 1,2,…}. Each has the property that Markov classes in {ie160-1} = ∐Bn pass to well-defined equivalence classes in the quotient. We are able to solve the Markov problem in two of the quotients, obtaining canonical representatives for Markov classes and giving a procedure for reducing an arbitrary representative to the canonical one. The results are interpreted geometrically, and related to link invariants of the associated links and the value of the Jones polynomial on the corresponding classes.

The fundamental group of generic polynomials

IN THIS short note we shall consider complex polynomials P in one variable as maps P: Q: --* 43. In this framework a polynomial P of degree exactly (n + I) is said to be yeneric if the derivative P’ has distinct roots p,, . . . , y.. and the respective branch points w, = P(F,). . . . , I(; = P(y,,), are also all distinct. Generic polynomials of degree (n + 1) form an open set V” in an atline space of dimension (n + 2). and one can write down (cf. Q I) an equation for the complement of rl,. The main object of this note is to establish the following.

A braidlike presentation of Sp(n, p)

Forn even andp an odd prime a symplectic group Sp(n, p) is a quotient of the Artin braid groupBn+1. Ifs1, …,sn are standard generators ofBn+1 then the kernel of the corresponding epimorphism is the normal closure of just four elements:s1p,(s1s2)6,s1(p+1)/2s24s1(p−1)/2s2−2s1−1s22 and (s1s2s3)4A−1s1−2A, whereA=s2s3−1s2(p−1)/2s4s32s4, all of them lying in the subgroupB5. Sp(n, p) acts on a vector space and the image of the subgroupBn ofBn+1 in Sp(n, p), denoted Sp(n−1,p), is a stabilizer of one vector. A sequence of inclusions …Bk+1·Bk … induces a sequence of inclusions …Sp(k,p)·Sp(k−1,p)…, which can be used to study some finite-valued invariants of knots and links in the 3-sphere via the Markov theorem.

Moduli spaces and braid monodromy types of bidouble covers of the quadric

Bidouble covers

Multivariate Polynomials: A Spanning Question

\pi : S \mapsto Q

Special parity of perfect matchings in bipartite graphs

of the quadric Q are parametrized by connected families depending on four positive integers a,b,c,d. In the special case where b=d we call them abc-surfaces. Such a Galois covering

Divisor class group descent for affine krull domains

\pi