2012 Redbud Geometry/Topology Conference

Oklahoma State University, Stillwater OK

Investigators

Abstract

This conference will feature a number of talks by Tao Li and by other experts on Heegaard splittings and ranks of fundamental groups. A Heegaard splitting is a decomposition of a 3-dimensional manifold into two simple pieces called handlebodies along a surface. A Heegaard splitting defines a presentation for the fundamental group of the 3-manifold in which the number of generators is equal to the genus of the surface. Thus the rank of the 3-manifold (the minimal number of generators) is less than or equal to the genus of the minimal Heegaard surface. In most cases, rank and genus are equal, but a small number of examples have been found in which the rank is strictly less than the genus. Tao Li recently discovered the first examples of hyperbolic 3-manifolds with rank strictly less than genus. This raises the question of how common these examples are and how big the difference between rank and genus can be. When we look at the surface of the earth from ground level, it appears to be a flat plane that goes on forever in all directions. However, from world navigation and viewing the earth from space, we know that it is, in fact, a completely different shape - a sphere, or the surface of a ball. Similarly, the universe that we live in appears to be a flat 3-dimensional space that goes on forever in all directions. Unfortunately we have no way of checking what the shape actually is, but we can study the possible shapes that it could be. This is the focus of 3-dimensional topology. This conference will bring together experts from around the world to discuss recent developments at the interface between algebraic methods and geometric methods for studying 3-dimensional spaces. http://comp.uark.edu/~matthewd/redbud/ http://www.math.okstate.edu/~jjohnson/redbud/