Double node neighborhoods and families of simply connected -manifolds with

Authors:
Ronald Fintushel and Ronald J. Stern

Journal:
J. Amer. Math. Soc. **19** (2006), 171-180

MSC (2000):
Primary 14J26, 53D05, 57R55, 57R57

Published electronically:
August 15, 2005

MathSciNet review:
2169045

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We introduce a new technique that is used to show that the complex projective plane blown up at 6, 7, or 8 points has infinitely many distinct smooth structures. None of these smooth structures admits smoothly embedded spheres with self-intersection , i.e., they are minimal. In addition, none of these smooth structures admits an underlying symplectic structure. Shortly after the appearance of a preliminary version of this article, Park, Stipsicz, and Szabo used the techniques described herein to show that the complex projective plane blown up at 5 points has infinitely many distinct smooth structures. In the final section of this paper we give a construction of such a family of examples.

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Additional Information

**Ronald Fintushel**

Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

Email:
ronfint@math.msu.edu

**Ronald J. Stern**

Affiliation:
Department of Mathematics, University of California, Irvine, California 92697

Email:
rstern@math.uci.edu

DOI:
http://dx.doi.org/10.1090/S0894-0347-05-00500-X

Keywords:
$4$-manifold,
Seiberg-Witten invariant,
knot surgery,
rational blowdown

Received by editor(s):
January 13, 2005

Published electronically:
August 15, 2005

Additional Notes:
The first author was partially supported by NSF Grant DMS0305818 and the second author by NSF Grant DMS0204041

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.