Faculty Mentor: Corrin Clarkson
Heegaard Floer homology is a topological invariant i.e. it is a way of measuring topological complexity. Given such a tool, it is natural to ask whether or not it is precise enough to distinguish between distinct topological spaces. In general, Heegaard Floer homology is good a distinguishing between spaces, but it is also known to have blind spots. In fact, there are infinitely many examples of distinct manifolds having the same Heegaard Floer homology. Determining whether this invariant distinguishes between a particular pair of manifolds can be challenging due to the difficult nature of computing Heegaard Floer homology. This is why we will focus on torus bundles.
Torus bundles are a particularly nice class of three-manifolds. There is a correspondence between these manifolds and elements of SL(2,Z) which allows us to use linear algebra to describe them. Moreover, there are relatively efficient algorithms for computing the Heegaard Floer homology of these manifolds. The question we will be exploring is the following. Is Heegaard Floer homology a complete invariant of torus bundles i.e. does it distinguish between any two distinct torus bundles?