Rational homology disk smoothings of surface singularities; the exceptional cases
Abstract
It is known (StipsiczSzabóWahl) that there are exactly three triplyinfinite and seven singlyinfinite 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 nonabelian (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.
 Publication:

arXiv eprints
 Pub Date:
 August 2021
 arXiv:
 arXiv:2108.09209
 Bibcode:
 2021arXiv210809209A
 Keywords:

 Mathematics  Algebraic Geometry;
 Mathematics  Geometric Topology;
 14H20;
 32S50;
 57M05
 EPrint:
 24 pages