Tyrone Ghaswala, Department of Pure Mathematics, University of Waterloo
“The Superelliptic Covers and the Lifting Mapping Class Group”
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.
This talk will focus on the superelliptic covers, which are n-sheeted generalisations of the 2-sheeted covering spaces studied by Birman and Hilden. In particular, a technique for computing presentations and the index of the lifting mapping class group will be developed. These have a large potential to be applied to many more families of covering spaces.
No familiarity with the mapping class group of a surface will be assumed. This is joint work with Rebecca Winarski.
MC 5403