The structure of high distance Heegaard splittings
(Oklahoma State University)
Abstract: A Heegaard splitting is a decomposition of a three-dimensional manifold into two simple pieces called handlebodies, glued along an embedded surface. The notion of distance for a Heegaard splitting of a three-dimensional manifold $M$, introduced by John Hempel, has proved to be a very powerful tool for understanding the geometry and topology of $M$. I will describe how distance, and a slight generalization known as subsurface projection distance, can be used to explore the connection between geometry and topology at the center of the modern theory hyperbolic three-manifolds. In particular, Schalremann-Tomova showed that if a Heegaard splitting for $M$ has high distance then it will be the only irreducible Heegaard splitting of $M$ with genus less than a certain bound. I will explain this result in terms of both a geometric proof and a topological proof. Then, using the notion of subsurface distance, I will describe a construction of a manifold with multiple distinct low-distance Heegaard splittings of the same (small) genus, and a manifold with both a high distance, low-genus Heegaard splitting and a distinct, irreducible high-genus, low-distance Heegaard splitting.