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A calculus for bordered Floer homology

Jonathan Hanselman and Liam Watson

Geometry & Topology 27 (2023) 823–924

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.

Heegaard Floer, bordered Floer homology, 3–manifolds, L–space conjecture
Mathematical Subject Classification 2010
Primary: 57M27
Received: 24 September 2015
Revised: 10 July 2020
Accepted: 20 August 2021
Published: 8 June 2023
Proposed: Peter Ozsváth
Seconded: Ciprian Manolescu, Cameron Gordon
Jonathan Hanselman
Department of Mathematics
University of Texas at Austin
Austin, TX
United States
Department of Mathematics
Princeton University
Princeton, NJ
United States
Liam Watson
School of Mathematics and Statistics
University of Glasgow
United Kingdom
Department of Mathematics
University of British Columbia
Vancouver BC

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