Contents

1 Introduction

2 Basic definitions of surface braid groups
2.1 Surface braids as a collection of strings
2.2 Surface braids as trajectories of non-colliding particles
2.3 Surface braid groups as the fundamental group of configuration spaces
2.4 Relationship between braid and mapping class groups

3 Some properties of surface braid groups
3.1 Exact sequences of braid groups
3.2 Presentations of surface braid groups
3.3 The centre of surface braid groups
3.4 Embeddings of surface braid groups
3.5 Braid combing and the splitting problem
3.6 Homotopy type of the configuration spaces of S² and RP² and periodicity
3.7 Orderability
3.8 Linearity

4 Braid groups of the sphere and the projective plane
4.1 Basic properties
4.2 Finite subgroups of the braid groups of S² and RP²
4.3 Virtually cyclic subgroups of the braid groups of S² and RP²

5 K-theory of surface braid groups
5.1 Generalities
5.2 The K-theoretic Farrell-Jones Conjecture for braid groups of aspherical surfaces
5.3 The Farrell-Jones Conjecture for the braid groups of S² and RP²
5.4 General remarks for computations
5.5 Computational results
5.6 Remarks