#### Volume 15, issue 2 (2011)

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

### Jonathan Wahl

Geometry & Topology 15 (2011) 1125–1156
##### Abstract

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 (“$ℚHD$”) 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 $ℚHD$ smoothings for those classes; it is an explicit $ℚ$–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 $ℚHD$ smoothings for the first class.

We also prove a general formula for the dimension of a $ℚHD$ 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 $ℚHD$ smoothing. This result, plus recent work of Bhupal and Stipsicz, is evidence for a general conjecture:

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

##### Keywords
surface singularity, rational homology disk fillings, smoothing surface singularities, Milnor fibre
##### Mathematical Subject Classification 2010
Primary: 14B07, 14J17, 32S30