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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Double node neighborhoods and families of simply connected $4$-manifolds with $b^+=1$


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
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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 $-1$, 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
PII: S 0894-0347(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.