We consider a class of manifolds with torus boundary admitting bordered Heegaard Floer homology of a particularly simple form; namely, the type D structure may be described graphically by a disjoint union of loops. We develop a calculus for studying bordered invariants of this form and, in particular, provide a complete description of slopes giving rise to L–space Dehn fillings as well as necessary and sufficient conditions for L–spaces resulting from identifying two such manifolds along their boundaries. As an application, we show that Seifert-fibred spaces with torus boundary fall into this class, leading to a proof that, among graph manifolds containing a single JSJ torus, the property of being an L–space is equivalent to non-left-orderability of the fundamental group and to the nonexistence of a coorientable taut foliation.
All Science Journal Classification (ASJC) codes
- Geometry and Topology