Lifting Homeomorphisms of Cyclic Branched Covers of Spheres

Geometry Topology Seminar
Monday, August 22, 2016 - 2:05pm
1 hour (actually 50 minutes)
Skiles 006
University of Wisconsin at Milwaukee
Birman and Hilden ask: given finite branched cover X over the 2-sphere, does every homeomorphism of the sphere lift to a homeomorphism of X?  For covers of degree 2, the answer is yes, but the answer is sometimes yes and sometimes no for higher degree covers.  In joint work with Ghaswala, we completely answer the question for cyclic branched covers.  When the answer is yes, there is an embedding of the mapping class group of the sphere into a finite quotient of the mapping class group of X.  In a family where the answer is no, we find a presentation for the group of isotopy classes of homeomorphisms of the sphere that do lift, which is a finite index subgroup of the mapping class group of the sphere.  Our family introduces new examples of orbifold Picard groups of subloci of Teichmuller space that are finitely generated but not cyclic.