**SMS scnews item created by Stephan Tillmann at Tue 5 Apr 2016 1048**

Type: Seminar

Distribution: World

Expiry: 5 Jul 2016

**Calendar1: 6 Apr 2016 1200-1300**

**CalLoc1: Carslaw 535A**

CalTitle1: Superelliptic Covers and the Lifting Mapping Class Group

Auth: tillmann@p710.pc (assumed)

### Geometry & Topology

# Superelliptic Covers and the Lifting Mapping Class Group

### Tyrone Ghaswala (Waterloo)

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.