Cubiquitous Lattices and Branched Covers bounding rational balls

TL;DR

This paper investigates cubiquitous lattices as obstructions to rational homology 3-spheres bounding rational homology 4-balls, introducing the Wu obstruction and classifying cubiquitous sublattices.

Contribution

It develops a geometric Wu obstruction to classify cubiquitous lattices with orthogonal bases and applies it to branched covers of torus links.

Findings

Classified cubiquitous sublattices with orthogonal bases.

Introduced the Wu obstruction as a tool for cubiquity analysis.

Applied the obstruction to infinite lattice families.

Abstract

Greene and Owens explore cubiquitous lattices as an obstruction to rational homology 3-spheres bounding rational homology 4-balls. The purpose of this article is to better understand which sublattices of $\mathbb{Z}^n$ are cubiquitous with the aim of effectively using their cubiquity obstruction. We develop a geometric obstruction (called the Wu obstruction) to cubiquity and use it as tool to completely classify which sublattices with orthogonal bases are cubiquitous. We then apply this result the double branched covers of alternating connected sums of torus links. Finally, we explore how the Wu obstruction can be used in conjunction with contractions to obstruct the cubiquity of infinite families of lattices.