# Horizontal decompositions, II

## TL;DR

This paper classifies certain 4-manifolds with low Euler characteristic and genus-one handlebody decompositions, and constructs new rational homology balls that embed into the complex projective plane, linking almost-complex and symplectic embeddings.

## Contribution

It completes the classification of specific 4-manifolds and introduces a large family of rational homology balls with embedding properties into P^2.

## Key findings

- Classified all 4-manifolds with Euler characteristic < 4 and genus-one decompositions.
- Constructed a large family of rational homology balls embedding into P^2.
- Established the equivalence of almost-complex and symplectic embeddings for these balls.

## Abstract

We complete the classification of the smooth, closed, oriented 4-manifolds having Euler characteristic less than four and a horizontal handlebody decomposition of genus one. We use the classification result to find a large family of rational homology ball smoothings of cyclic quotient singularities which can be smoothly embedded into the complex projective plane. Our family contains all such rational balls previously known to embed into $\mathbb{CP}^2$ and infinitely many more. We also show that a rational ball of our family admits an almost-complex embedding in $\mathbb{CP}^2$ if and only if it admits a symplectic embedding.