Configuration Spaces and Braid Groups

Abstract

The main thrust of these notes is 3-fold: (1) An analysis of certain $K(π,1)$'s that arise from the connections between configuration spaces, braid groups, and mapping class groups, (2) a function space interpretation of these results, and (3) a homological analysis of the cohomology of some of these groups for genus zero, one, and two surfaces possibly with marked points, as well as the cohomology of certain associated function spaces. An example of the type of results given here is an analysis of the space k particles moving on a punctured torus up to equivalence by the natural $SL(2,\mathbb{Z})$ action.