# A NOTE ON INDECOMPOSABLE 4-MANIFOLDS

• Published : 2005.11.01

#### Abstract

In this note we show that there is an anti-symplectic involution $\sigma\;:\;X\;\to\;X$ on a simply-connected, closed, non-Kahler and symplectic 4-manifold X with a disjoint union of Riemann surfaces ${\amalg}^n_{i=1}{\Sigma}_i,\;n\;{\ge}\;2$ as a fixed point set. Also we show that its quotient X/$\sigma$ is homeomorphic to $\mathbb{CP}^2{\sharp}r\mathbb{CP}^2$ but not diffeomorphic to $\mathbb{CP}^2{\sharp}r\mathbb{CP}^2,\;r\;=\;b_2^-(X/{\sigma})$.

#### References

1. S. Akbulut, On quotients of complex surfaces under complex conjugation, J. Reine. Angew. Math. 447 (1994), 83-90
2. G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York and London, 1972
3. Y. S. Cho, Cyclic group actions on gauge theory, Diffential. Geom. Appl. 6 (1996), 87-99 https://doi.org/10.1016/0926-2245(96)00009-5
4. 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
5. Y. S. Cho and Y. H. Hong, Seiberg-Witten theory and anti-symplectic involutions, Glasg. Math. J. 45 (2003), 401-413 https://doi.org/10.1017/S0017089503001344
6. Y. S. Cho and Y. H. Hong, Anti-symplectic involutions on non-Kahler symplectic 4-manifolds, Preprint
7. M. Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982), 357-454 https://doi.org/10.4310/jdg/1214437136
8. R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math. 142 (1995), 527-595 https://doi.org/10.2307/2118554
9. R. E. Gompf and T. S. Mrowka, Irreducible 4-manifolds need not be complex, Ann. of Math. 138 (1993), 61-111 https://doi.org/10.2307/2946635
10. R. E. Gompf and A. I. Stipsciz, 4-Manifolds and Kirby Calculus, Grad. Stud. Math.
11. R. Kirby, Problems in low-dimensional topology, Berkeley, 1995
12. P. B. Kronheimer and T. S. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), 797-808 https://doi.org/10.4310/MRL.1994.v1.n6.a14
13. J. W. Morgan, T. S. Mrowka, and Z. Szabo, Product formulas along $T^3$ for Seiberg-Witten invariants, Math. Res. Lett. 4 (1997), 915-929 https://doi.org/10.4310/MRL.1997.v4.n6.a11
14. J. W. Morgan, Z. Szabo, and C. Taubes, A product formula for the Seiberg- Witten invariants and the generalized Thom Conjecture, J. Differential Geom. 44 (1996), 706-788
15. B. Ozbagci and A. I. Stipsciz, Non complex smooth 4-manifolds with genus 2- Lefschetz fibration
16. A. I. Stipsciz, Manifolds not containing Gomph nuclei, Acta Math. 83 (1998), 107-113
17. W. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), 467-468
18. C. H. Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), 809-822 https://doi.org/10.4310/MRL.1994.v1.n6.a15
19. C. H. Taubes, The Seiberg-Witten invariants and the Gromov invariants, Math. Res. Lett. 2 (1995), 221-238 https://doi.org/10.4310/MRL.1995.v2.n2.a10
20. S. Wang, Gauge theory and involutions, Oxford University Thesis, 1990
21. S. Wang, A Vanishing theorem for Seiberg-Witten invariants, Math. Res. Lett. 2 (1995), 305-310 https://doi.org/10.4310/MRL.1995.v2.n3.a7