Correction terms, $\mathbb Z_2$--Thurston norm, and triangulations

TL;DR

This paper links Heegaard Floer homology correction terms to lower bounds on the genus of one-sided Heegaard splittings and the $Z_2$--Thurston norm, providing new tools to estimate 3-manifold complexity.

Contribution

It establishes a novel connection between correction terms in Heegaard Floer homology and the $Z_2$--Thurston norm, leading to bounds on 3-manifold complexity.

Findings

Lower bounds on genus of one-sided Heegaard splittings

Lower bounds on $Z_2$--Thurston norm

Computed $Z_2$--Thurston norm for double branched covers

Abstract

We show that the correction terms in Heegaard Floer homology give a lower bound to the the genus of one-sided Heegaard splittings and the $\mathbb Z_2$--Thurston norm. Using a result of Jaco--Rubinstein--Tillmann, this gives a lower bound to the complexity of certain closed $3$--manifolds. As an application, we compute the $\mathbb Z_2$--Thurston norm of the double branched cover of some closed 3--braids, and give upper and lower bounds for the complexity of these manifolds.