Volume 16, issue 6 (2016)

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Strong Heegaard diagrams and strong L–spaces

Joshua Evan Greene and Adam Simon Levine

Algebraic & Geometric Topology 16 (2016) 3167–3208

We study a class of 3–manifolds called strong L–spaces, which by definition admit a certain type of Heegaard diagram that is particularly simple from the perspective of Heegaard Floer homology. We provide evidence for the possibility that every strong L–space is the branched double cover of an alternating link in the three-sphere. For example, we establish this fact for a strong L–space admitting a strong Heegaard diagram of genus 2 via an explicit classification. We also show that there exist finitely many strong L–spaces with bounded order of first homology; for instance, through order eight, they are connected sums of lens spaces. The methods are topological and graph-theoretic. We discuss many related results and questions.

$3$–manifolds, Heegaard diagrams, Heegaard Floer homology, L–spaces
Mathematical Subject Classification 2010
Primary: 57M27, 57R58
Received: 9 December 2014
Accepted: 2 June 2016
Published: 15 December 2016
Joshua Evan Greene
Department of Mathematics
Boston College
Maloney Hall, Fifth Floor
Chestnut Hill, MA 02467
United States
Adam Simon Levine
Department of Mathematics
Princeton University
Fine Hall
Washington Road
Princeton, NJ 08544
United States