One of the outstanding challenges in 3-manifold theory is to relate the modern Heegaard Floer invariants to the fundamental group. Recently, a conjectural picture has emerged from the work of Boyer-Gordon-Watson: a closed, irreducible rational homology sphere M is an L-space (i.e. it has the simplest possible Heegaard Floer homology) if and only if its fundamental group is not left-orderable. Whereas there has been encouraging evidence supporting the truth of the conjecture, the problem remains poorly understood for key classes of 3-manifolds.

 

In this talk, we focus on negative-definite graph manifolds (one of these poorly understood classes): for these, Nemethi constructs a lattice cohomology, an invariant inspired in ideas from singularity theory and conjecturally isomorphic to Heegaard Floer homology. Using the combinatorial tractability of lattice cohomology, we produce several comprehensive families of manifolds against which to test the Boyer-Gordon-Watson conjecture. Then, using either horizontal foliation arguments or direct manipulation of the fundamental group, we prove that they do indeed satisfy the conjecture.