Small exotic 4–manifolds

Akhmedov, Anar¹

¹Mathematics Department, Columbia University, New York, NY 10027, USA

Algebraic and Geometric Topology, Tome 8 (2008) no. 3, pp. 1781-1794

Résumé

In this article, we construct the first example of a simply-connected minimal symplectic 4–manifold that is homeomorphic but not diffeomorphic to 3ℂℙ2#7ℂℙ¯2. We also construct the first exotic minimal symplectic ℂℙ2#5ℂℙ¯2.

Détail

DOI : 10.2140/agt.2008.8.1781

Keywords: 4–manifolds, exotic smooth structures, symplectic 4–manifolds, fiber sum, Seiberg–Witten invariants

Bibliographie

[1] A Akhmedov, Construction of symplectic cohomology $S^{2}\times S^{2}$, Gökova Geometry-Topology Proc 14 (2007) 36

[2] G Burde, H Zieschang, Knots, de Gruyter Studies in Mathematics 5, Walter de Gruyter Co. (1985)

[3] R Fintushel, R J Stern, Rational blowdowns of smooth $4$-manifolds, J. Differential Geom. 46 (1997) 181

[4] R Fintushel, R J Stern, Families of simply connected 4–manifolds with the same Seiberg–Witten invariants, Topology 43 (2004) 1449

[5] R Fintushel, R J Stern, Double node neighborhoods and families of simply connected 4–manifolds with $b^+=1$, J. Amer. Math. Soc. 19 (2006) 171

[6] M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357

[7] R E Gompf, A new construction of symplectic manifolds, Ann. of Math. $(2)$ 142 (1995) 527

[8] M J D Hamilton, D Kotschick, Minimality and irreducibility of symplectic four-manifolds, Int. Math. Res. Not. (2006) 13

[9] D Kotschick, The Seiberg–Witten invariants of symplectic four-manifolds (after C H Taubes), Astérisque (1997) 4, 195

[10] T J Li, S T Yau, Embedded surfaces and Kodaira Dimension, preprint

[11] Y Matsumoto, Lefschetz fibrations of genus two—a topological approach, from: "Topology and Teichmüller spaces (Katinkulta, 1995)", World Sci. Publ., River Edge, NJ (1996) 123

[12] J D Mccarthy, J G Wolfson, Symplectic normal connect sum, Topology 33 (1994) 729

[13] B Ozbagci, A I Stipsicz, Noncomplex smooth $4$–manifolds with genus–$2$ Lefschetz fibrations, Proc. Amer. Math. Soc. 128 (2000) 3125

[14] J Park, Seiberg–Witten invariants of generalised rational blow-downs, Bull. Austral. Math. Soc. 56 (1997) 363

[15] J Park, Simply connected symplectic 4–manifolds with $b^+_2=1$ and $c^2_1=2$, Invent. Math. 159 (2005) 657

[16] J Park, Exotic smooth structures on $3\mathbb C\mathrm{P}^2\# 8\overline{\mathbb{C}\mathrm{P}}{}^2$, Bull. Lond. Math. Soc. 39 (2007) 95

[17] J Park, A I Stipsicz, Z Szabó, Exotic smooth structures on $\mathbb{CP}^2\#5\overline{\mathbb{CP}^2}$, Math. Res. Lett. 12 (2005) 701

[18] A I Stipsicz, Z Szabó, An exotic smooth structure on $\mathbb C\mathbb P^2\#6\overline\mathbb{C\mathbb P^2}$, Geom. Topol. 9 (2005) 813

[19] A I Stipsicz, Z Szabó, Small exotic 4–manifolds with $b^+_2=3$, Bull. London Math. Soc. 38 (2006) 501

[20] C H Taubes, The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994) 809

[21] M Usher, Minimality and symplectic sums, Int. Math. Res. Not. (2006) 17

[22] M Usher, Kodaira dimension and Symplectic Sums, to appear in Commentarii Mathematici Helvetici

[23] E Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994) 769