Inventiones mathematicae

, Volume 181, Issue 3, pp 577–603

Exotic smooth structures on small 4-manifolds with odd signatures

Article

Abstract

Let M be \((2n-1)\mathbb{CP}^{2}\#2n\overline{\mathbb{CP}}{}^{2}\) for any integer n≥1. We construct an irreducible symplectic 4-manifold homeomorphic to M and also an infinite family of pairwise non-diffeomorphic irreducible non-symplectic 4-manifolds homeomorphic to M. We also construct such exotic smooth structures when M is \(\mathbb{CP}{}^{2}\#4\overline {\mathbb{CP}}{}^{2}\) or \(3\mathbb{CP}{}^{2}\#k\overline{\mathbb{CP}}{}^{2}\) for k=6,8,10.

Mathematics Subject Classification (2000)

57R55 57R17 

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References

  1. 1.
    Akhmedov, A.: Small exotic 4-manifolds. Algebraic Geom. Topol. 8, 1781–1794 (2008) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Akhmedov, A., Baldridge, S., Baykur, R.İ., Kirk, P., Park, B.D.: Simply connected minimal symplectic 4-manifolds with signature less than −1. J. Eur. Math. Soc. 12, 133–161 (2010) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Akhmedov, A., Baykur, R.İ., Park, B.D.: Constructing infinitely many smooth structures on small 4-manifolds. J. Topol. 1, 409–428 (2008) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Akhmedov, A., Park, B.D.: Exotic smooth structures on small 4-manifolds. Invent. Math. 173, 209–223 (2008) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Auroux, D., Donaldson, S.K., Katzarkov, L.: Luttinger surgery along Lagrangian tori and non-isotopy for singular symplectic plane curves. Math. Ann. 326, 185–203 (2003) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Baldridge, S., Kirk, P.: A symplectic manifold homeomorphic but not diffeomorphic to \({\mathbb{CP}}^{2}\#3\overline{\mathbb{CP}}^{2}\). Geom. Topol. 12, 919–940 (2008) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Baldridge, S., Kirk, P.: Constructions of small symplectic 4-manifolds using Luttinger surgery. J. Differ. Geom. 82, 317–361 (2009) MATHMathSciNetGoogle Scholar
  8. 8.
    Birman, J.S.: Braids, Links, and Mapping Class Groups. Annals of Mathematics Studies, vol. 82. Princeton University Press, Princeton (1974) Google Scholar
  9. 9.
    Fadell, E., Van Buskirk, J.: The braid groups of E 2 and S 2. Duke Math. J. 29, 243–257 (1962) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Fintushel, R., Park, B.D., Stern, R.J.: Reverse engineering small 4-manifolds. Algebraic Geom. Topol. 7, 2103–2116 (2007) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Fintushel, R., Stern, R.J.: Symplectic surfaces in a fixed homology class. J. Differ. Geom. 52, 203–222 (1999) MATHMathSciNetGoogle Scholar
  12. 12.
    Fintushel, R., Stern, R.J.: Double node neighborhoods and families of simply connected 4-manifolds with b +=1. J. Am. Math. Soc. 19, 171–180 (2006) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Freedman, M.H.: The topology of four-dimensional manifolds. J. Differ. Geom. 17, 357–453 (1982) MATHGoogle Scholar
  14. 14.
    Gompf, R.E.: A new construction of symplectic manifolds. Ann. Math. 142, 527–595 (1995) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Gompf, R.E., Stipsicz, A.I.: 4-Manifolds and Kirby Calculus. Graduate Studies in Mathematics, vol. 20. Am. Math. Soc., Providence (1999) MATHGoogle Scholar
  16. 16.
    Hambleton, I., Kreck, M.: Cancellation, elliptic surfaces and the topology of certain four-manifolds. J. Reine Angew. Math. 444, 79–100 (1993) MATHMathSciNetGoogle Scholar
  17. 17.
    Hamilton, M.J.D., Kotschick, D.: Minimality and irreducibility of symplectic four-manifolds. Int. Math. Res. Not. 2006, Art. ID 35032 (2006) 13 pp. CrossRefMathSciNetGoogle Scholar
  18. 18.
    Kulikov, V.S.: The fundamental group of the complement of a hypersurface in ℂn. Math. USSR-Izv. 38, 399–418 (1992) CrossRefMathSciNetGoogle Scholar
  19. 19.
    Li, T.-J.: Smoothly embedded spheres in symplectic 4-manifolds. Proc. Am. Math. Soc. 127, 609–613 (1999) MATHCrossRefGoogle Scholar
  20. 20.
    Libgober, A.: Homotopy groups of the complements to singular hypersurfaces, II. Ann. Math. 139, 117–144 (1994) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Luttinger, K.M.: Lagrangian tori in ℝ4. J. Differ. Geom. 42, 220–228 (1995) MATHMathSciNetGoogle Scholar
  22. 22.
    Morgan, J.W., Mrowka, T.S., Szabó, Z.: Product formulas along T 3 for Seiberg-Witten invariants. Math. Res. Lett. 4, 915–929 (1997) MATHMathSciNetGoogle Scholar
  23. 23.
    Stern, R.J.: Lecture at the Topology of 4-Manifolds Conference in honor of R. Fintushel’s 60th birthday, Tulane University, November 10–12, 2006 Google Scholar
  24. 24.
    Stipsicz, A.I., Szabó, Z.: An exotic smooth structure on \({\mathbb{CP}}^{2}\#6\overline {\mathbb{CP}}^{2}\). Geom. Topol. 9, 813–832 (2005) MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Park, B.D.: Constructing infinitely many smooth structures on \(3{\mathbb{CP}}^{2}\#n\overline{\mathbb{CP}}^{2}\). Math. Ann. 322, 267–278 (2002); Erratum. Math. Ann. 340, 731–732 (2008) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Park, J.: Exotic smooth structures on \(3{\mathbb{CP}}^{2}\#8\overline {\mathbb{CP}}^{2}\). Bull. Lond. Math. Soc. 39, 95–102 (2007) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Taubes, C.H.: The Seiberg-Witten invariants and symplectic forms. Math. Res. Lett. 1, 809–822 (1994) MATHMathSciNetGoogle Scholar
  28. 28.
    Usher, M.: Minimality and symplectic sums. Int. Math. Res. Not. 2006, Art. ID 49857 (2006) 17 pp. CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of Pure MathematicsUniversity of WaterlooWaterlooCanada

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