Fundamental group of rational homology disk smoothings of surface singularities

Enrique Artal Bartolo and Jonathan Wahl

Journal of Singularities
volume 24 (2022), 126-144

Received: 22 February 2022. Received in revised form: 11 June 2022.

DOI: 10.5427/jsing.2022.24e

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Abstract:

It is known that there are exactly three triply-infinite and seven singly-infinite families of weighted homogeneous normal surface singularities admitting a rational homology disk 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 three exceptional families. In this paper, we settle these cases. We present a new explicit construction for one of the exceptional families, showing the fundamental group is non-abelian (as occurred previously only for three families). We show that the fundamental groups for the remaining two exceptional families are abelian, hence easily computed; using the Pinkham method here requires precise calculations for the fundamental group of the complement of a plane curve.

2020 Mathematical Subject Classification:

14H20, 32S50, 57M05

Key words and phrases:

surface singularity, rational homology sphere, Milnor fibre

Author(s) information:

Enrique Artal Bartolo
Departamento de Matemáticas-IUMA
Facultad de Ciencias
Universidad de Zaragoza
c/ Pedro Cerbuna 12
E-50009 Zaragoza SPAIN
email: artal@unizar.es

Jonathan Wahl
Department of Mathematics
The University of North Carolina
Chapel Hill, NC 27599-3250
email: jmwahl@email.unc.edu