#### Volume 9, issue 4 (2009)

 Download this article For screen For printing
 Recent Issues
 The Journal About the Journal Editorial Board Editorial Interests Subscriptions Submission Guidelines Submission Page Policies for Authors Ethics Statement ISSN (electronic): 1472-2739 ISSN (print): 1472-2747 Author Index To Appear Other MSP Journals
Symplectic surgeries and normal surface singularities

### David T Gay and András I Stipsicz

Algebraic & Geometric Topology 9 (2009) 2203–2223
##### Abstract

We show that every negative definite configuration of symplectic surfaces in a symplectic $4$–manifold has a strongly symplectically convex neighborhood. We use this to show that if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration and if the complex singularity with resolution diffeomorphic to a neighborhood of the configuration has a smoothing, then the configuration can be symplectically replaced by the smoothing of the singularity. This generalizes the symplectic rational blowdown procedure used in recent constructions of small exotic $4$–manifolds.

##### Keywords
symplectic rational blow-down, surface singularity, symplectic neighborhood
##### Mathematical Subject Classification 2000
Primary: 57R17
Secondary: 14E15, 14J17
##### Publication
Received: 9 December 2008
Revised: 25 August 2009
Accepted: 9 September 2009
Published: 27 October 2009
##### Authors
 David T Gay Department of Mathematics and Applied Mathematics University of Cape Town Private Bag X3 Rondebosch 7701 South Africa András I Stipsicz Hungarian Academy of Sciences Renyi Institute of Mathematics Reáltanoda utca 13–15 Budapest 1053 Hungary Mathematics Department, Columbia University 2990 Broadway, New York, NY 10027 USA http://www.renyi.hu/~stipsicz