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EXOTIC SMOOTH STRUCTURE ON ℂℙ2#13ℂℙ2

  • Cho, Yong-Seung ;
  • Hong, Yoon-Hi
  • Published : 2006.07.01

Abstract

In this paper, we construct a new exotic smooth 4-manifold X which is homeomorphic, but not diffeomorphic, to ${\mathbb{C}}\mathbb{P}^2{\sharp}13\overline{\mathbb{C}\mathbb{P}}^2$. Moreover the manifold X has vanishing Seiberg-Witten invariants for all $Spin^c$-structures of X and has no symplectic structure.

Keywords

Seiberg-Witten invariant;symplectic 4-manifold;antisymplectic involution;double branched cover

References

  1. R. Barlow, A simply connected surface of general type with $p_g$ = 0, Invent. Math. 79 (1985), no. 2, 293-301 https://doi.org/10.1007/BF01388974
  2. W. Barth, C. Peters, and A. Van de Ven, Compact Complex Surfaces, Springer, Heidelberg, 1984
  3. G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972
  4. Y. S. Cho, Cyclic group actions on gauge theory, Differential Geom. Appl. 6 (1996), no. 1, 87-99 https://doi.org/10.1016/0926-2245(96)00009-5
  5. Y. S. Cho and D. Joe, Anti-symplectic involutions with Lagrangian fixed loci and their quotients, Proc. Amer. Math. Soc. 130 (2002), no. 9, 2797-2801
  6. Y. S. Cho and Y. H. Hong, Cyclic group actions on 4-manifold, Acta Math. Hungar. 94 (2002), no. 4, 333-350 https://doi.org/10.1023/A:1015647713638
  7. Y. S. Cho and Y. H. Hong, Anti-symplectic involutions on non-Kahler symplectic 4-manifolds, Preprint
  8. S. Donaldson, La topologie differentielle des surfaces complexes, C. R. Acad. Sci. Paris Ser. I Math. 301 (1985), no. 6, 317-320
  9. R. Fintushel and R. J. Stern, Double node neighborhoods and families of simply connected 4-manifolds with $b^+$ = 1, J. Amer. Math. Soc. 19 (2006), no. 1, 171-180 https://doi.org/10.1090/S0894-0347-05-00500-X
  10. R. Friedman and J. W. Morgan, On the diffeomorphism types of certain algebraic surfaces. I, J. Differential Geom. 27 (1988), no. 2, 297-369 https://doi.org/10.4310/jdg/1214441784
  11. R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math.(2) 142 (1995), no. 3, 527-595 https://doi.org/10.2307/2118554
  12. R. E. Gompf and A. I. Stipsciz, 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, Vol. 20, AMS Providence, Rhode Island, 1999
  13. D. Kotschick, On manifolds homeomorphic to $CP^2#8\overline{CP2}$, Invent. Math. 95 (1989), no. 3, 591-600 https://doi.org/10.1007/BF01393892
  14. J. Park, Simply connected symplectic 4-manifolds with $b_2^+\;=\;1$ and $c^2_1\;=\;2$, Invent. Math. 159 (2005), no. 3, 657-667 https://doi.org/10.1007/s00222-004-0404-1
  15. J. Park, A. I. Stipsicz, and Z. Szabo, Exotic smooth structures on $CP^2 #5\overline{CP^2}$ , Math. Res. Lett. 12 (2005), no. 5-6, 701-712 https://doi.org/10.4310/MRL.2005.v12.n5.a7
  16. R. Silhol, Real algebraic surfaces, Lecture Notes in Math. vol. 1392, Springer- Verlag, 1989
  17. A. Stipsicz and Z. Szabo, An exotic smooth structure on $CP^2#6\overline{CP^2}$, Geom. Topol. 9 (2005), 813-832 https://doi.org/10.2140/gt.2005.9.813
  18. Z. Szabo, Exotic 4-manifolds with $b_2^+$ = 1, Math. Res. Lett. 3 (1996), no. 6, 731-741 https://doi.org/10.4310/MRL.1996.v3.n6.a2
  19. S. Wang, Gaugy theory and involutions, Oxford University Thesis, 1990
  20. M. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357-453 https://doi.org/10.4310/jdg/1214437136
  21. R. E. Gompf and T. S. Mrowka, Irreducible 4-manifolds need not be complex, Ann. of Math.(2) 138 (1993), no. 1, 61-111 https://doi.org/10.2307/2946635
  22. S. Akbulut, On quotients of complex surfaces under complex conjugation, J. Reine Angew. Math. 447 (1994), 83-90
  23. Y. S. Cho and Y. H. Hong, Seiberg-Witten invariants and (anti-)symplectic involutions, Glasg. Math. J. 45 (2003), no. 3, 401-413 https://doi.org/10.1017/S0017089503001344
  24. R. Kirby, Problems in low-dimensional topology, AMS/IP Stud. Adv. Math., 2.2, Geometric topology (Athens, GA, 1993), 35-473
  25. I. Dolgachev, Algebraic surfaces with $p_g\;=\;q\;=\;0$, in Algebraic surfaces, CIME 1977, Liguori Napoli, 1981, 97-215