Double node neighborhoods and families of simply connected 4-manifolds with b^+ = 1
By R. Fintushel, Ronald J. Stern. 2004
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JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 19, Number 1, Pages 171–180 S 0894-0347(05)00500-X Article electronically published on August 15, 2005 DOUBLE NODE NEIGHBORHOODS AND FAMILIES OF SIMPLY CONNECTED 4-MANIFOLDS WITH b + = 1 RONALD FINTUSHEL AND RONALD J. STERN 1. Introduction A basic question of 4-manifold topology is whether the complex projective plane, CP 2 , admits exotic smooth structures. Thus one is interested in knowing the smallest m for which CP 2 #m CP admits an exotic smooth structure. In the late 1980s, Dieter Kotschick [K] proved that the Barlow surface, which was known to be homeomorphic to CP 2 # 8CP , is not diffeomorphic to it. In the following years the subject of simply connected smooth 4-manifolds with b + = 1 languished because of a lack of suitable examples. However, largely due to a beautiful paper of Jongil Park [P], who found the first examples of exotic simply connected 4-manifolds with b + = 1 and b − = 7, the past year has found renewed interest in this subject. Peter Ozsvath and Zoltan Szabo [OS] proved that Park’s manifold is minimal, and Andras Stipsicz and Szabo [SS] used a technique similar to Park’s to construct an exotic manifold with b + = 1 and b − = 6. The rational elliptic surface E(1) ∼ CP 2 #9 CP admits infinitely many distinct smooth structures. Up to now it has not been known whether CP 2 #m CP can have an infinite family of smooth structures when m < 9. It is the purpose of this paper to introduce a new technique which we use to show that for m = 6, 7, 8, CP 2 #m CP does have an infinite family of smooth structures. The construction of these examples to some extent follows the ideas of Park [P] after performing suitable knot surgeries [FS3] on E(1). The essential new idea is that after certain knot surgeries on E(1), one is able to find an immersed sphere of self-intersection −1 representing the ‘pseudo-section’. This is accomplished by studying ‘double node neighborhoods’ as described below. Shortly after the appearance of a preliminary version of this article, Park, Stip- sicz, and Szabo [PSS] used the techniques described herein to give even better examples: an infinite family of pairwise nondiffeomorphic smooth 4-manifolds all homeomorphic to CP 2 #5 CP . In the final section of this paper we give a quick construction of such a family of examples. Received by the editors January 13, 2005. 2000 Mathematics Subject Classification. Primary 14J26, 53D05, 57R55, 57R57. Key words and phrases. 4-manifold, Seiberg-Witten invariant, knot surgery, rational blowdown. The first author was partially supported by NSF Grant DMS0305818 and the second author by NSF Grant DMS0204041. c 2005 American Mathematical Society Reverts to public domain 28 years from publication
Subfields: Geometry and Topology
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Citations in the corpus
- An exotic smooth structure on CP^2+6CP^2-bar (2004, unknown): “1 After the submission of this paper the results of Theorem 1.1 and Corollary 1.2 have been improved by finding infinitely many exotic smooth structures on CP 2 #nCP 2 for n ≥ 5, see [ 6 , 18].”
- Manifolds of Positive Scalar Curvature: A Progress Report (2006, result): “Finally, there are some other results that imply something interesting about the space R+(M).”
- Exotic smooth structures on small 4-manifolds with odd signatures (2007, background): “An infinite family of non-symplectic irreducible 4-manifolds homeomorphic to CP(2)#6CP(2) was first constructed in [12].”
- Constructing infinitely many smooth structures on small 4‐manifolds (2007, methodology): “Infinitely many exotic examples were later constructed by Fintushel and Stern in [ 11 ].”