Braid Groups

This chapter introduces the reader to Artin's classical braid groups Bₙ. The group Bₙ is isomorphic to the mapping class group of a disk with n marked points. Since disks are planar, the braid groups lend themselves to special pictorial representations. This gives the theory of braid groups its own special flavor within the theory of mapping class groups. The chapter begins with a discussion of three equivalent ways of thinking about the braid group, focusing on Artin's classical definition, fundamental groups of configuration spaces, and the mapping class group of a punctured disk. It then presents some classical facts about the algebraic structure of the braid group, after which a new proof of the Birman–Hilden theorem is given to relate the braid groups to the mapping class groups of closed surfaces.