A Parametrization of Integral Homology 3-Spheres by the Fourth Johnson subgroup

TL;DR

This paper characterizes integral homology 3-spheres via the fourth Johnson subgroup and provides an intrinsic description of the related equivalence relation, including a new Lagrangian trace map and computations of subgroup images.

Contribution

It offers an intrinsic description of the equivalence relation on the fourth Johnson subgroup using handlebody subgroups and introduces a novel antisymmetric Lagrangian trace map.

Findings

Intrinsic description of the equivalence relation on the fourth Johnson subgroup.

Introduction of an antisymmetric Lagrangian trace map.

Computed the image of third Johnson handlebody subgroups by the third Johnson homomorphism.

Abstract

By results of Morita, Pitsch and, more recently, Faes, it is known that any integral homology 3-sphere can be constructed as a Heegaard splitting with a gluing map an element of the fourth Johnson subgroup. In this work we prove that the equivalence relation on the fourth Johnson subgroup induced by this construction admits an intrinsic description in terms of the fourth Johnson handlebody subgroups. In addition, we give an ``antisymmetic'' Lagrangian trace map inspired in the Lagrangian trace map introduced by Faes and compute the image of the third Johnson handlebody subgroups by the third Johnson homomorphism.