On lattice cohomology and left-orderability

2013/08/08 by Mauro Mauricio, Mauricio, Mauro
Mathematics · #Advanced Operator Algebra Research #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.1308.1890

openalex publication_date 2013/08/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

It has been recently conjectured by Boyer-Gordon-Watson that a closed, orientable, irreducible 3-manifold M is a Heegaard Floer L-space if and only if π₁(M) is not left-orderable. In this article, we study this conjecture from the point of view of lattice cohomology, an invariant introduced by Némethi which is conjecturally isomorphic to the HF⁺ version of Heegaard Floer homology. Using the invariant's combinatorial tractability as a stepping stone, we produce some interesting quite general families of negative-definite graph manifolds against which to test the Boyer-Gordon-Watson conjecture. Then, using horizontal foliation arguments and direct manipulation of the fundamental group, we prove that these families do indeed satisfy the conjecture.

Citations

• Holomorphic disks and topological invariants for closed three-manifolds
• On L-spaces and left-orderable fundamental groups

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• The equivalence of lattice and Heegaard Floer homology
• Foliations, orders, representations, L-spaces and graph manifolds
• On L-spaces and non left-orderable 3-manifold groups
• On the involutive Heegaard Floer homology of negative semi-definite plumbed 3-manifolds with b₁=1
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