Bordered Heegaard Floer Homology and Graph Manifolds

We use the techniques of bordered Heegaard Floer homology to investigate the Heegaard Floer homology of graph manifolds. Bordered Heegaard Floer homology allows us to split a graph manifold into pieces and perform computations for each piece separately. The resulting invariants can then be combined by a simple algebraic procedure to recover 𝐻𝐹. Graph manifolds by definition decompose into pieces which are S¹-bundles over surfaces. This decomposition makes them particularly well suited to the divide-and-conquer techniques of bordered Heegaard Floer homology. In fact, the problem reduces to computing bordered Heegaard Floer invariants of just two pieces. The first invariant is the type D trimodule associated to the trivial S¹-bundle over the pair of pants 𝑃. The second is a bimodule that is necessary for self-gluing, when two torus boundary components of a bordered manifold are glued to each other. We explicitly compute both of these multimodules. We then describe an algorithm for computing 𝐻𝐹 of any graph manifold using these results. The algorithm has been implemented in Python, and we give some example computations. We also use this algorithm to inductively prove that the bordered invariants of graph manifolds with torus boundary have a particularly simple form when the plumbing graphs are trees with no bad vertices.