Presentation by Dr. Elif Medetoğulları

TED University

Faculty of Education

"On geometry and topology of low dimensional manifolds”

By

Dr. Elif Medetoğulları 

Date: Friday, June 24, 2021

Time: 10:00-11:00

Place: Zoom

Abstract: Low dimensional manifolds are objects that are locally similar to Euclidean spaces with dimensions less than or equal to four. Even though they locally look like Euclidean space, their global properties may be far away from it. There are different research areas and methods such as contact structures, mapping class groups, branched covers, open book decompositions to understand the geometry and topology of low dimensional manifolds. All of these areas which are from algebra, geometry, and topology are related to each other. In this presentation, a brief overview of these areas will be given. Besides this, I will mention one of our latest works on the Birman-Hilden properties of branched covers where branched cover is a map from a surface, which is a two-dimensional manifold, called covering surface, to another surface, called base surface, satisfying certain properties. The Birman-Hilden theory is interested in the relation between the homeomorphisms of the base surface and the homeomorphisms of the covering surface. Homeomorphisms of base surface that somehow lift to homeomorphisms of covering surface are called liftable homeomorphisms. For some branched covers in some sense, all homeomorphisms of covering surfaces are coming from liftable homeomorphism then these are called coverings having Birman-Hilden property. I will mention our result about this property and answer the question for which branched covers of a surface, all homeomorphism of this surface lifts to a homeomorphism of its covering surface for the cases when the base is a sphere or real projective plane.