The Farey tree and embeddings of lens spaces and rational balls in $\mathbb{CP}^2$

TL;DR

This paper explores embeddings of rational homology balls into complex projective planes using Farey tree structures, extending previous constructions and providing new explicit examples related to a conjecture of Kollár.

Contribution

It introduces a recursive Kirby calculus method to embed multiple rational homology balls in P^2, expanding on prior work and offering explicit constructions for homotopy P^2s.

Findings

Embedded triples of rational homology balls with lens space boundaries are constructed.

A recursive Kirby calculus approach is developed for these embeddings.

New explicit examples of embeddings into homotopy P^2s are provided.

Abstract

Motivated by a conjecture of Koll\'ar, we study embeddings of multiple rational homology balls in $\mathbb{CP}^2$. To each node of the Farey tree, we associate such an embedding of three rational homology balls with lens space boundary, extending earlier work of the second author and of Lisca and Parma, using a recursive Kirby calculus argument. We also give further explicit constructions of embeddings of triples of rational homology balls into homotopy $\mathbb{CP}^2$s.