Superelliptic Covers and the Lifting Mapping Class Group

Wednesday 6 April 2016 from 12:00–13:00 in Carslaw 535A

Please join us for lunch after the talk!

Abstract: Given a finite sheeted (possibly branched) covering space over a surface, one can ask the following question:  Which homeomorphisms of the base space lift to homeomorphisms of the total space?  If we take the quotient of this question by isotopy, it becomes a much more interesting one:  What can we say about the subgroup of the mapping class group of the base space that consists of isotopy classes of homeomorphisms that lift to the total space?  This subgroup is the lifting mapping class group.

This question was completely answered by Birman and Hilden when the deck group is the two element group generated by a fixed hyperelliptic involution.  In this case, everything lifts.  Interestingly, this does not happen in general.

In this talk, I will give a brief introduction to the mapping class group of a surface and a history of this lifting problem.  I will then focus on the on the lifting mapping class group in the case of the superelliptic covers, which are \(k\)-sheeted generalisations of the 2-sheeted covering spaces studied by Birman and Hilden.  Time permitting, I will outline some interesting questions that arise from this work.

This is joint work with Rebecca Winarski.