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

Karakurt, Çağri¹ ; Starkston, Laura²

¹Department of Mathematics, Bogazici University, 34342 Istanbul, Turkey
²Mathematics Department, Stanford University, 450 Serra Mall, Building 380, Stanford, CA 78712-1202, United States

Algebraic and Geometric Topology, Tome 16 (2016) no. 3, pp. 1585-1635

Résumé

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.

Détail

DOI : 10.2140/agt.2016.16.1585

Classification : 53Dxx, 57R57
Keywords: symplectic, exotic, rational blow-down, Seiberg–Witten

Bibliographie

[1] A Akhmedov, Small exotic 4–manifolds, Algebr. Geom. Topol. 8 (2008) 1781

[2] A Akhmedov, R İ Baykur, B D Park, Constructing infinitely many smooth structures on small 4–manifolds, J. Topol. 1 (2008) 409

[3] A Akhmedov, B D Park, Exotic smooth structures on small 4–manifolds, Invent. Math. 173 (2008) 209

[4] A Akhmedov, B D Park, Exotic smooth structures on small 4–manifolds with odd signatures, Invent. Math. 181 (2010) 577

[5] S Baldridge, P Kirk, A symplectic manifold homeomorphic but not diffeomorphic to CP2#3CP2, Geom. Topol. 12 (2008) 919

[6] M Bhupal, B Özbaǧci, Symplectic fillings of lens spaces as Lefschetz fibrations (2013)

[7] M Bhupal, A I Stipsicz, Weighted homogeneous singularities and rational homology disk smoothings, Amer. J. Math. 133 (2011) 1259

[8] V Colin, P Ghiggini, K Honda, Equivalence of Heegaard Floer homology and embedded contact homology via open book decompositions, Proc. Natl. Acad. Sci. USA 108 (2011) 8100

[9] S K Donaldson, Irrationality and the h–cobordism conjecture, J. Differential Geom. 26 (1987) 141

[10] H Endo, Y Z Gurtas, Lantern relations and rational blowdowns, Proc. Amer. Math. Soc. 138 (2010) 1131

[11] H Endo, T E Mark, J Van Horn-Morris, Monodromy substitutions and rational blowdowns, J. Topol. 4 (2011) 227

[12] J B Etnyre, B Özbaǧci, Invariants of contact structures from open books, Trans. Amer. Math. Soc. 360 (2008) 3133

[13] R Fintushel, B D Park, R J Stern, Reverse engineering small 4–manifolds, Algebr. Geom. Topol. 7 (2007) 2103

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

[15] R Fintushel, R J Stern, Knots, links, and 4–manifolds, Invent. Math. 134 (1998) 363

[16] 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

[17] R Fintushel, R J Stern, Six lectures on four 4–manifolds, from: "Low dimensional topology" (editors T S Mrowka, P S Ozsváth), IAS/Park City Math. Ser. 15, Amer. Math. Soc. (2009) 265

[18] R Fintushel, R J Stern, Pinwheels and nullhomologous surgery on 4–manifolds with b+ = 1, Algebr. Geom. Topol. 11 (2011) 1649

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

[20] R Friedman, J W Morgan, On the diffeomorphism types of certain algebraic surfaces. I, J. Differential Geom. 27 (1988) 297

[21] D Gay, T E Mark, Convex plumbings and Lefschetz fibrations, J. Symplectic Geom. 11 (2013) 363

[22] D T Gay, A I Stipsicz, Symplectic rational blow-down along Seifert fibered 3–manifolds, Int. Math. Res. Not. 2007 (2007) 20

[23] D T Gay, A I Stipsicz, Symplectic surgeries and normal surface singularities, Algebr. Geom. Topol. 9 (2009) 2203

[24] R E Gompf, Handlebody construction of Stein surfaces, Ann. of Math. 148 (1998) 619

[25] D Kotschick, On manifolds homeomorphic to CP2#8CP2, Invent. Math. 95 (1989) 591

[26] Ç Kutluhan, Y J Lee, C H Taubes, HF = HM I : Heegaard Floer homology and Seiberg–Witten Floer homology (2010)

[27] T J Li, The Kodaira dimension of symplectic 4–manifolds, from: "Floer homology, gauge theory, and low-dimensional topology" (editors D A Ellwood, P S Ozsváth, A I Stipsicz, Z Szabó), Clay Math. Proc. 5, Amer. Math. Soc. (2006) 249

[28] T J Li, A Liu, Symplectic structure on ruled surfaces and a generalized adjunction formula, Math. Res. Lett. 2 (1995) 453

[29] D Mcduff, D Salamon, Introduction to symplectic topology, , The Clarendon Press Oxford University Press (1995)

[30] M Michalogiorgaki, Rational blow-down along Wahl type plumbing trees of spheres, Algebr. Geom. Topol. 7 (2007) 1327

[31] J W Morgan, The Seiberg–Witten equations and applications to the topology of smooth four-manifolds, 44, Princeton Univ. Press (1996)

[32] P Ozsváth, Z Szabó, The symplectic Thom conjecture, Ann. of Math. 151 (2000) 93

[33] P Ozsváth, Z Szabó, Holomorphic triangle invariants and the topology of symplectic four-manifolds, Duke Math. J. 121 (2004) 1

[34] H Park, A I Stipsicz, Smoothings of singularities and symplectic surgery (2012)

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

[36] J Park, Simply connected symplectic 4–manifolds with b2+ = 1 and c12 = 2, Invent. Math. 159 (2005) 657

[37] J Park, A I Stipsicz, Z Szabó, Exotic smooth structures on CP2#5CP2, Math. Res. Lett. 12 (2005) 701

[38] U Persson, Configurations of Kodaira fibers on rational elliptic surfaces, Math. Z. 205 (1990) 1

[39] L P Roberts, Rational blow-downs in Heegaard–Floer homology, Commun. Contemp. Math. 10 (2008) 491

[40] L Starkston, Comparing star surgery to rational blow-down (2014)

[41] L Starkston, Symplectic fillings of Seifert fibered spaces, Trans. Amer. Math. Soc. 367 (2015) 5971

[42] A I Stipsicz, Z Szabó, An exotic smooth structure on CP2#6CP2, Geom. Topol. 9 (2005) 813

[43] A I Stipsicz, Z Szabó, J Wahl, Rational blowdowns and smoothings of surface singularities, J. Topol. 1 (2008) 477

[44] M Symington, Symplectic rational blowdowns, J. Differential Geom. 50 (1998) 505

[45] M Symington, Generalized symplectic rational blowdowns, Algebr. Geom. Topol. 1 (2001) 503

[46] Z Szabó, Exotic 4–manifolds with b2+ = 1, Math. Res. Lett. 3 (1996) 731

[47] C Wendl, Strongly fillable contact manifolds and J–holomorphic foliations, Duke Math. J. 151 (2010) 337