[Paper Review] On lattice cohomology and left-orderability

Mauro Mauricio|arXiv (Cornell University)|Aug 8, 2013

Geometric and Algebraic Topology11 references3 citations

TL;DR

This paper investigates the Boyer-Gordon-Watson conjecture linking Heegaard Floer L-spaces to non-left-orderability of the fundamental group, using lattice cohomology to analyze negative-definite graph manifolds. It proves that insulated plumbing graphs without very bad vertices are L-spaces and have non-left-orderable fundamental groups, while graphs with very bad vertices or E₈ subgraphs are not L-spaces and have left-orderable fundamental groups, supporting the conjecture in broad families of 3-manifolds.

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 $π_1(M)$ 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.

Motivation & Objective

• To provide evidence for the Boyer-Gordon-Watson conjecture that a 3-manifold is a Heegaard Floer L-space if and only if its fundamental group is not left-orderable.
• To use lattice cohomology as a combinatorial tool to analyze the L-space property in negative-definite plumbing graphs.
• To establish families of graph manifolds where the conjecture holds by proving left-orderability or non-left-orderability of the fundamental group.
• To connect taut foliations and left-orderability through the structure of the fundamental group in plumbing graphs.
• To extend known results on Seifert-fibered spaces and graph-manifold integral homology spheres to broader classes of negative-definite graphs.

Proposed method

• Utilizes lattice cohomology, a combinatorial invariant conjecturally isomorphic to HF⁺, to determine whether a plumbing graph Γ is a lattice cohomology L-space.
• Applies Laufer’s algorithm to test the vanishing of the reduced lattice cohomology, which corresponds to the L-space condition.
• Introduces the concept of 'insulated' graphs—negative-definite trees with no very bad vertices and specific deficiency and neighbor constraints.
• Employs group-theoretic arguments based on vertex deficiency and neighbor relations to analyze left-orderability of π₁(Γ).
• Uses inductive reasoning on tree height, applying lemmas on group generators to show that left-orderability leads to contradiction unless all vertices satisfy condition A or A*.
• Applies the fact that a root cannot be maximal in a left-ordering if the tree is re-rooted, leading to contradiction if left-ordering exists.

Experimental results

Research questions

• RQ1 Does the Boyer-Gordon-Watson conjecture hold for negative-definite plumbing graphs without very bad vertices?
• RQ2 Can lattice cohomology be used to systematically identify L-spaces among graph manifolds?
• RQ3 What conditions on the plumbing graph ensure that π₁(Γ) is not left-orderable?
• RQ4 How do taut foliations and left-orderability relate in the context of lattice cohomology and plumbing graphs?
• RQ5 Under what conditions does a plumbing graph with a very bad vertex or E₈ subgraph fail to be an L-space?

Key findings

• If a plumbing graph Γ has a very bad vertex, then π₁(Γ) is left-orderable and Γ is not a Heegaard Floer L-space.
• If a plumbing graph Γ contains a proper negative-definite E₈ subgraph, then π₁(Γ) is left-orderable and Γ is not a Heegaard Floer L-space.
• For insulated graphs (no very bad vertices, with specific deficiency and neighbor constraints), π₁(Γ) is not left-orderable, and Γ is a Heegaard Floer L-space.
• The proof relies on showing that a left-ordering leads to a contradiction via re-rooting the tree, implying that no consistent total order can exist.
• The fundamental group of a leaf in a minimal plumbing graph is nontrivial, which is essential for the inductive argument on vertex conditions.
• The result holds under the assumption of negative-definiteness; the conjecture fails without this condition, as shown by counterexamples like the Poincaré homology sphere.