Surgery along star-shaped plumbings and exotic smooth structures on 4–manifolds

Abstract

We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective spaces blown up eight, seven, and six times. We also show how this operation can be used in conjunction with knot surgery to construct an infinite family of minimal exotic smooth structures on the complex projective space blown-up seven times. 53Dxx; 57R57

Deformation of singular fibers of genus two fibrations and small exotic symplectic 4-manifolds

“…Similar results starting from the ellptic surface E(n) for n ≥ 3 were obtained in [1]. See also a related more recent work in [37,28], which again uses elliptic fibrations on E(1). One of the key ingredients in the above mentioned articles was the use of Kodaira's classification of singular fibers in elliptic fibrations.…”

“…There has been lots of activity in the discovery of exotic smooth structures on simply connected 4-manifolds with small Euler characteristic in the period of last 15 years. The following references are among many dealing with this subject [44,49,16,46,50,45,3,2,6,5,4,7,37,56,8,28]. In this article, we will be primarily interested in the construction of exotic smooth structures on simply connected 4-manifolds with small Euler characteristic using the rational-blowdown surgery [15,43].…”

Unchaining surgery and topology of symplectic 4-manifolds

“…Constructions of small simply-connected symplectic 4-manifolds with b + 2 ≤ 3 via rational blowdowns has a fairly long and rich history, pioneered by the works of Fintushel-Stern and Jongil Park [24,42,23,43]. Similar constructions via monodromy substitutions in positive factorizations of Lefschetz pencils was first given by Endo and Gurtas [18], who observed that lantern substitutions (see Lemma 2.1) amount to a rational blowdown of a symplectic (−4)-sphere -which since then, has been extended to many other substitutions corresponding to blowdowns of more general configurations of spheres [19,26,33]. The hardship of the latter approach is to have explicit positive factorizations of pencils that contain an enough number of lantern configurations for rational blowdowns, and was so far successfully applied to genus-2 Lefschetz pencils in [18] and [4].…”

Small exotic 4-manifolds from lines and quadrics in $\mathbb{CP}^{2}$

“…Lemma 5 essentially stated as [11,Lemma 5.4] presents a standard symplectic structure on CP 2 #kCP 2 and calculates the sign of the required cup product to be negative. Lemma 6 shows that this product has to be negative for any symplectic structure on CP 2 #8CP 2 or CP 2 #9CP 2 , and this is a rather special result for CP 2 #mCP 2 given 2 ≤ m ≤ 9.…”