On surface braid groups
by
Luis Paris
Université de Bourgogne

Throughout the lecture M will denote a compact surface, non necessarily oriented and possibly with boundary. Choose a finite collection P={p₁, ..., p_m} of m points (called punctures) in the interior of M. Define a braid with m strings on M based at P to be a collection b={b₁, ..., b_m} of m paths, b_i:[0, 1] --> M such that:

(1) b_i(0)=p_i and b_i(1) in P for all i in {1, ..., m};

(2) b_i(t) =/= b_j(t) for i, j in {1, ..., m}, i =/= j, and t in [0, 1].

There is a natural notion of homotopy of braids. The braid group on m strings based at P is the group B(M, P) of homotopy classes of braids based at P. The group operation is concatenation of braids, generalizing the construction of the fundamental group.

After explaining the link between braid groups and configuration spaces, we will talk about torsion, centers, presentations, and other combinatorial aspects of these groups.

http://math.u-bourgogne.fr/topolog/paris

Date received: May 4, 1999