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

Fintushel, Ronald; Stern, Ronald

doi:10.1090/S0894-0347-05-00500-X

Double node neighborhoods and families of simply connected $4$-manifolds with $b^+=1$
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by Ronald Fintushel and Ronald J. Stern

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

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

Published electronically: August 15, 2005

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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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