Rational homology disk smoothings of surface singularities; the exceptional cases

TL;DR

This paper completes the classification of certain surface singularities with rational homology disk smoothings by explicitly determining the fundamental groups for three exceptional families, revealing new non-abelian cases.

Contribution

It provides explicit constructions and fundamental group calculations for the previously unresolved exceptional families of surface singularities with rational homology disk smoothings.

Findings

Fundamental group of $ extbf{B}_2^3(p)$ is non-abelian.

Fundamental groups of $ extbf{C}_2^3(p)$ and $ extbf{C}_3^3(p)$ are abelian.

Complete classification of these singularities' Milnor fibers.

Abstract

It is known (Stipsicz-Szab\'o-Wahl) that there are exactly three triply-infinite and seven singly-infinite families of weighted homogeneous normal surface singularities admitting a rational homology disk ($\mathbb{Q}$HD) smoothing, i.e., having a Milnor fibre with Milnor number zero. Some examples are found by an explicit "quotient construction", while others require the "Pinkham method". The fundamental group of the Milnor fibre has been known for all except the three exceptional families $\mathcal B_2^3(p), \mathcal C^3_2(p),$ and $\mathcal C^3_3(p)$. In this paper, we settle these cases. We present a new explicit construction for the $\mathcal B_2^3(p)$ family, showing the fundamental group is non-abelian (as occurred previously only for the $\mathcal A^4(p), \mathcal B^4(p)$ and $\mathcal C^4(p)$ cases). We show that the fundamental groups for $ \mathcal C^3_2(p)$ and $\mathcal C^3_3(p)$ are abelian, hence easily computed; using the Pinkham method here requires precise calculations for the fundamental group of the complement of a plane curve.