Surgery along star-shaped plumbings and exotic smooth structures on 4-manifolds (Q2630676)

The original \textit{rational blow-down} operation [\textit{R. Fintushel} and \textit{R. J. Stern}, J. Differ. Geom. 46, No. 2, 181--235 (1997; Zbl 0896.57022)] -- and its generalizations -- [\textit{J. Park}, Bull. Aust. Math. Soc. 56, No. 3, 363--384 (1997; Zbl 0892.57020)], [\textit{A. I. Stipsicz} et al., J. Topol. 1, No. 2, 477--517 (2008; Zbl 1143.32018)], [\textit{M. Bhupal} and \textit{A. I. Stipsicz}, Am. J. Math. 133, No. 5, 1259--1297 (2011; Zbl 1227.32033)] amount to removing the neighborhood of a union of spheres which intersect according to a particular plumbing tree and regluing a rational ball which has the same boundary as the neighborhood. In the present paper the authors define a new cut and paste operation, called \textit{star surgery} which is a strong generalization of the rational blow-down. The star surgery operation is similarly defined. First identify (symplectic) spheres which intersect according to a star-shaped graph with a negativity condition on the central vertex. The star surgery operation cuts out a neighborhood of these spheres and replaces it by an alternate symplectic filling of the induced contact boundary. These alternate fillings always have smaller Euler characteristic than the neighborhood of spheres and are negative-definite. Using the star surgery operation the authors deduce multiple constructions of symplectic smoothly exotic complex projective spaces blown up eight, seven, and six times. The authors also show 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. The main results are included in these three theorems. Theorem 1.1. There is a minimal symplectic 4-manifold \(X\) which is homeomorphic but not diffeomorphic to \(\mathbb{C}P^2\#8 \overline{\mathbb{C}P^2}\) and which is obtained by a star surgery. The symplectic Kodaira dimension of \(X\) is 2. Theorem 1.2. There are constructions of symplectic exotic copies of \(\mathbb{C}P^2\# 7 \overline{\mathbb{C}P^2}\) and \(\mathbb{C}P^2\# 6 \overline{\mathbb{C}P^2}\) obtained by performing star surgery operations on blow-ups of \(E(1)\). Theorem 1.3. For every \(n\geq 2\) there exist smooth minimal mutually nondiffeomorphic 4-manifolds \(Y_n\) which are all homeomorphic to \(\mathbb{C}P^2\# 7 \overline{\mathbb{C}P^2}\). These manifolds are obtained by a star surgery.