EXOTIC SMOOTH STRUCTURE ON ℂℙ2#13ℂℙ2

Title & Authors
EXOTIC SMOOTH STRUCTURE ON ℂℙ2#13ℂℙ2
Cho, Yong-Seung; Hong, Yoon-Hi;

Abstract
In this paper, we construct a new exotic smooth 4-manifold X which is homeomorphic, but not diffeomorphic, to $\small{{\mathbb{C}}\mathbb{P}^2{\sharp}13\overline{\mathbb{C}\mathbb{P}}^2}$. Moreover the manifold X has vanishing Seiberg-Witten invariants for all $\small{Spin^c}$-structures of X and has no symplectic structure.
Keywords
Seiberg-Witten invariant;symplectic 4-manifold;antisymplectic involution;double branched cover;
Language
English
Cited by
References
1.
S. Akbulut, On quotients of complex surfaces under complex conjugation, J. Reine Angew. Math. 447 (1994), 83-90

2.
R. Barlow, A simply connected surface of general type with $p_g$ = 0, Invent. Math. 79 (1985), no. 2, 293-301

3.
W. Barth, C. Peters, and A. Van de Ven, Compact Complex Surfaces, Springer, Heidelberg, 1984

4.
G. E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972

5.
Y. S. Cho, Cyclic group actions on gauge theory, Differential Geom. Appl. 6 (1996), no. 1, 87-99

6.
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

7.
Y. S. Cho and Y. H. Hong, Cyclic group actions on 4-manifold, Acta Math. Hungar. 94 (2002), no. 4, 333-350

8.
Y. S. Cho and Y. H. Hong, Seiberg-Witten invariants and (anti-)symplectic involutions, Glasg. Math. J. 45 (2003), no. 3, 401-413

9.
Y. S. Cho and Y. H. Hong, Anti-symplectic involutions on non-Kahler symplectic 4-manifolds, Preprint

10.
I. Dolgachev, Algebraic surfaces with $p_g\;=\;q\;=\;0$, in Algebraic surfaces, CIME 1977, Liguori Napoli, 1981, 97-215

11.
S. Donaldson, La topologie differentielle des surfaces complexes, C. R. Acad. Sci. Paris Ser. I Math. 301 (1985), no. 6, 317-320

12.
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

13.
M. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), no. 3, 357-453

14.
R. Friedman and J. W. Morgan, On the diffeomorphism types of certain algebraic surfaces. I, J. Differential Geom. 27 (1988), no. 2, 297-369

15.
R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math.(2) 142 (1995), no. 3, 527-595

16.
R. E. Gompf and T. S. Mrowka, Irreducible 4-manifolds need not be complex, Ann. of Math.(2) 138 (1993), no. 1, 61-111

17.
R. E. Gompf and A. I. Stipsciz, 4-Manifolds and Kirby Calculus, Graduate Studies in Mathematics, Vol. 20, AMS Providence, Rhode Island, 1999

18.
R. Kirby, Problems in low-dimensional topology, AMS/IP Stud. Adv. Math., 2.2, Geometric topology (Athens, GA, 1993), 35-473

19.
D. Kotschick, On manifolds homeomorphic to $CP^2#8\overline{CP2}$, Invent. Math. 95 (1989), no. 3, 591-600

20.
J. Park, Simply connected symplectic 4-manifolds with $b_2^+\;=\;1$ and $c^2_1\;=\;2$, Invent. Math. 159 (2005), no. 3, 657-667

21.
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

22.
R. Silhol, Real algebraic surfaces, Lecture Notes in Math. vol. 1392, Springer- Verlag, 1989

23.
A. Stipsicz and Z. Szabo, An exotic smooth structure on $CP^2#6\overline{CP^2}$, Geom. Topol. 9 (2005), 813-832

24.
Z. Szabo, Exotic 4-manifolds with $b_2^+$ = 1, Math. Res. Lett. 3 (1996), no. 6, 731-741

25.
S. Wang, Gaugy theory and involutions, Oxford University Thesis, 1990