#### 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
##### Abstract

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.

##### Keywords
$3$–manifolds, Heegaard diagrams, Heegaard Floer homology, L–spaces
##### Mathematical Subject Classification 2010
Primary: 57M27, 57R58
##### Publication
Received: 9 December 2014
Accepted: 2 June 2016
Published: 15 December 2016
##### Authors
 Joshua Evan Greene Department of Mathematics Boston College Maloney Hall, Fifth Floor Chestnut Hill, MA 02467 United States http://www2.bc.edu/joshua-e-greene Adam Simon Levine Department of Mathematics Princeton University Fine Hall Washington Road Princeton, NJ 08544 United States http://www.math.princeton.edu/~asl2