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New symplectic 4–manifolds with b+=1

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In 1995 Dusa McDuff and Dietmar Salamon conjectured the existence of symplectic 4–manifolds (X,ω) which satisfy b+=1, K2=0, K·ω>0, and which fail to be of Lefschetz type. This is equivalent to finding a symplectic, homology T2×S2 manifold with nontorsion canonical class and a cohomology ring which is not isomorphic to the cohomology ring of T2×S2. They needed such examples to complete a list of possible symplectic 4–manifolds with b+=1. In that same year Tian-Jun Li and Ai-ko Liu, working from a different point of view, questioned whether there existed symplectic 4–manifolds with b+=1 with Seiberg- Witten invariants that did not depend on the chamber structure of the moduli space. The purpose of this paper is to construct an infinite number of examples which satisfy both requirements.

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References

  1. Baldridge, S.: Seiberg–Witten invariants of 4-manifolds with free circle actions. Commun. Contemp. Math. 3, 341–353 (2001)

    Article  Google Scholar 

  2. Baldridge, S.: Seiberg–Witten invariants, orbifolds, and circle actions. Transactions of the American Mathematical Society 355(4), 1669–1697 (2002)

    Article  Google Scholar 

  3. Baldridge, S., Li, T.J.: Geography of symplectic 4manifolds with Kodaira dimension one. Algebraic and geometric Topology, 5, 355–368 (2005)

    Google Scholar 

  4. Fintushel, R., Stern, R.J.: Double node neighborhoods and families of simply connected 4-manifolds with b+=1 December 2004, E-print GT0412126

  5. Geiges, H.: Symplectic Structures on T2-bundles over T2. Duke Math. J. 67(3), 539–555 (1992)

    Article  Google Scholar 

  6. Gompf, R., Stipsicz, A.: `4–Manifolds and Kirby Calculus', Graduate Studies in Mathematics 20, American Mathematical Society, Providence, Rhode Island, 1999.

  7. Fernández, M., Gray, A., Morgan, J.: Compact symplectic manifolds with free circle actions, and Massey products, Michigan Math. J. 38, 271–283 (1991)

    Google Scholar 

  8. Kotschick, D.: On manifolds homeomorphic to Invent. Math. 95, 591–600 (1989)

    Google Scholar 

  9. Li, T.J.: Symplectic 4–manifolds with Kodaira dimension zero, preprint.

  10. Li, T.J., Liu, A.: General wall crossing formula. Math. Res. Lett. 2(6), 797–810 (1995)

    Google Scholar 

  11. Liu, A.: Some new applications of the general wall crossing formula. Math. Res. Lett. 3, 569–585 (1996)

    Google Scholar 

  12. Meng, G., Taubes, C.: = Milnor Torsion. Math. Res. Lett. 3, 661–674 (1996)

    Google Scholar 

  13. McDuff, D., Salamon, D.: A survey of symplectic 4-manifolds with b+2=1. Turkish Jour. Math. 20, 47–61 (1996)

    Google Scholar 

  14. Ozsváth, P., Sabó, Z.: On Park's exotic smooth four-manifolds, November 2004, E-print GT/0411218.

  15. Park, J.: Non–complex symplectic 4–manifolds with b+2 =1. Bull. London Math. Soc. 36, 231–240 (2004)

    Article  Google Scholar 

  16. Park, J.: Simply connected symplectic 4-manifolds with b+=1 and c12=2. Invent. Math. (to appear), E-print GT/0311395.

  17. Park, J., Stipsicz, A., Sabó, Z.: Exotic smooth structures on December 2004, E-print GT/0412216.

  18. Stipsicz, A., Sabó, Z.: An exotic smooth structure on November 2004, E-print GT/0411258.

  19. Taubes, C.: The Seiberg–Witten invariants and symplectic forms. Math. Res. Lett. 1(6), 809–822 (1994)

    Google Scholar 

  20. Thurston, W.: Some simple examples of symplectic manifolds. Proc. Amer. Math. Soc. 55, 467–468 (1976)

    Google Scholar 

Download references

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Correspondence to Scott Baldridge.

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The author was partially supported by NSF grant DMS-0406021.

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Baldridge, S. New symplectic 4–manifolds with b+=1. Math. Ann. 333, 633–643 (2005). https://doi.org/10.1007/s00208-005-0684-9

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  • DOI: https://doi.org/10.1007/s00208-005-0684-9

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