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On rational homology disk smoothings of valency $4$ surface singularities

Jonathan Wahl

Geometry & Topology 15 (2011) 1125–1156

Thanks to recent work of Stipsicz, Szabó and the author and of Bhupal and Stipsicz, one has a complete list of resolution graphs of weighted homogeneous complex surface singularities admitting a rational homology disk (“ HD”) smoothing, that is, one with Milnor number 0. They fall into several classes, the most interesting of which are the 3 classes whose resolution dual graph has central vertex with valency 4. We give a uniform “quotient construction” of the HD smoothings for those classes; it is an explicit –Gorenstein smoothing, yielding a precise description of the Milnor fibre and its non-abelian fundamental group. This had already been done for two of these classes; what is new here is the construction of the third class, which is far more difficult. In addition, we explain the existence of two different HD smoothings for the first class.

We also prove a general formula for the dimension of a HD smoothing component for a rational surface singularity. A corollary is that for the valency 4 cases, such a component has dimension 1 and is smooth. Another corollary is that “most” H–shaped resolution graphs cannot be the graph of a singularity with a HD smoothing. This result, plus recent work of Bhupal and Stipsicz, is evidence for a general conjecture:

Conjecture The only complex surface singularities admitting a HD smoothing are the (known) weighted homogeneous examples.

surface singularity, rational homology disk fillings, smoothing surface singularities, Milnor fibre
Mathematical Subject Classification 2010
Primary: 14B07, 14J17, 32S30
Received: 9 February 2011
Revised: 9 February 2011
Accepted: 11 April 2011
Published: 30 June 2011
Proposed: Richard Thomas
Seconded: Ronald Fintushel, Yasha Eliashberg
Jonathan Wahl
Department of Mathematics
The University of North Carolina
Chapel Hill NC 27599-3250