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

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- by Ronald Fintushel and Ronald J. Stern PDF
- J. Amer. Math. Soc.
**19**(2006), 171-180 Request permission

## 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.## References

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*On Park’s exotic smooth four-manifolds*, to appear in “Proceedings of the 2004 McMaster Conference on Geometry and Topology of Manifolds”.

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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
- 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
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**19**(2006), 171-180 - MSC (2000): Primary 14J26, 53D05, 57R55, 57R57
- DOI: https://doi.org/10.1090/S0894-0347-05-00500-X
- MathSciNet review: 2169045