Using rational homology circles to construct rational homology balls

TL;DR

This paper develops methods to construct rational homology 4-balls from plumbed 3-manifolds, leading to new examples of rational homology spheres that do not bound integer homology 4-balls, expanding understanding of 4-manifold topology.

Contribution

It introduces a systematic approach to identify and construct rational homology 3-spheres bounding rational homology 4-balls, including infinite families and new examples.

Findings

Infinite families of rational homology 3-spheres bounding rational homology 4-balls.

New examples of integer homology 3-spheres bounding rational but not integer homology 4-balls.

A simple method for constructing plumbed 3-manifolds with these properties.

Abstract

Motivated by Akbulut-Larson's construction of Brieskorn spheres bounding rational homology 4-balls, we explore plumbed 3-manifolds that bound rational homology circles and use them to construct infinite families of rational homology 3-spheres that bound rational homology 4-balls. Some of these rational homology 3-spheres are new examples of integer homology 3-spheres that bound rational homology 4-balls, but do not bound integer homology 4-balls. In particular, we find infinite families of torus bundles over the circle that bound rational homology circles, provide a simple method for constructing more general plumbed 3-manifolds that bound rational homology circles, and show that, for example, $-1$-surgery along any unknotting number one knot $K$ with a positive crossing that can be switched to unknot $K$ bounds a rational homology 4-ball.