Exceptional cosmetic surgeries on homology spheres

TL;DR

This paper investigates exceptional cosmetic surgeries on homology spheres, establishing bounds on the number of such surgeries and characterizing their properties, contributing to the understanding of the cosmetic surgery conjecture in 3-manifold topology.

Contribution

It proves that under certain conditions, there is at most one pair of exceptional truly cosmetic slopes, and that toroidal truly cosmetic surgeries on integer homology spheres are themselves integer homology spheres.

Findings

At most one pair of exceptional truly cosmetic slopes exists under specified conditions.

Toroidal truly cosmetic surgeries on integer homology spheres are integer homology spheres.

The results restrict the types of surgeries that can produce cosmetic effects on homology spheres.

Abstract

The cosmetic surgery conjecture is a longstanding conjecture in 3-manifold theory. We present a theorem about exceptional cosmetic surgery for homology spheres. Along the way we prove that if the surgery is not a small seifert $\mathbb{Z}/2\mathbb{Z}$-homology sphere or a toroidal irreducible non-Seifert surgery then there is at most one pair of exceptional truly cosmetic slope. We also prove that toroidal truly cosmetic surgeries on integer homology spheres must be integer homology spheres.