Suppose you are handed a finite sheeted (possibly branched) covering space between ccompact 2-manifolds. A natural question to ask is what is the relationship between the mapping class group of the covering surface and the mapping class group of the base surface?
In this talk, we will take a journey through this question for surfaces with boundary. Along the way, we will encounter a appearances from the fundamental groupoid, a classical theorem of Birman and Hilden, the Burau representation for braid groups, and new embeddings of the braid group in mapping class groups.
This is joint work with Alan McLeay.