Smooth Embeddings of Rational Homology Balls

TL;DR

This paper demonstrates that many smooth 4-manifolds contain embeddings of rational homology balls $B_n$ for odd $n \,\geq\, 3$, providing explicit examples using Kirby calculus.

Contribution

It establishes the presence of all $B_n$ for odd $n \geq 3$ in a broad class of smooth 4-manifolds and constructs explicit embeddings.

Findings

All $B_n$ for odd $n \geq 3$ embed in certain smooth 4-manifolds.

Explicit Kirby calculus methods for embeddings.

Examples of families with embedded rational homology balls.

Abstract

The rational homology balls $B_n$ appeared in Fintushel and Stern's rational blow-down construction [FS] and were subsequently used (e.g. Fintushel-Stern[FS4], Park[Pa2]) to construct exotic smooth manifolds with small Euler numbers. We show that a large class of smooth 4-manifolds have all of the $B_n$'s for odd $n \geq 3$ embedded in them. In particular, we give explicit examples, using Kirby calculus, of several families of smooth embeddings of the rational homology balls $B_n$.