On lattice cohomology and left-orderability
It has been recently conjectured by Boyer-Gordon-Watson that a closed, orientable, irreducible $3$-manifold $M$ is a Heegaard Floer $L$-space if and only if $π_1(M)$ is not left-orderable. In this article, we study this conjecture from the point of view of lattice cohomology, an invariant introduced by Némethi which is conjecturally isomorphic to the $HF^+$ version of Heegaard Floer homology. Using the invariant's combinatorial tractability as a stepping stone, we produce some interesting quite general families of negative-definite graph manifolds against which to test the Boyer-Gordon-Watson conjecture. Then, using horizontal foliation arguments and direct manipulation of the fundamental group, we prove that these families do indeed satisfy the conjecture.