The geography problem for irreducible spin four-manifolds

Park, B. Doug; Szabó, Zoltán

doi:10.1090/S0002-9947-00-02467-3

The geography problem for irreducible spin four-manifolds
HTML articles powered by AMS MathViewer

by B. Doug Park and Zoltán Szabó PDF

Trans. Amer. Math. Soc. 352 (2000), 3639-3650 Request permission

Abstract:

We study the geography problem for smooth irreducible simply-connected spin four-manifolds. For a large class of homotopy types, we exhibit both symplectic and non-symplectic representatives. We also compute the Seiberg-Witten invariants of all the four-manifolds we construct.

References

S. K. Donaldson, An application of gauge theory to four-dimensional topology, J. Differential Geom. 18 (1983), no. 2, 279–315. MR 710056
Ronald Fintushel and Ronald J. Stern, Surgery in cusp neighborhoods and the geography of irreducible $4$-manifolds, Invent. Math. 117 (1994), no. 3, 455–523. MR 1283727, DOI 10.1007/BF01232253
R. Fintushel and R. Stern: Knots, links and 4-manifolds, Invent. Math. 134 (1998), 363–400.
Ronald Fintushel and Ronald J. Stern, Rational blowdowns of smooth $4$-manifolds, J. Differential Geom. 46 (1997), no. 2, 181–235. MR 1484044
Robert E. Gompf, Nuclei of elliptic surfaces, Topology 30 (1991), no. 3, 479–511. MR 1113691, DOI 10.1016/0040-9383(91)90027-2
Robert E. Gompf, A new construction of symplectic manifolds, Ann. of Math. (2) 142 (1995), no. 3, 527–595. MR 1356781, DOI 10.2307/2118554
Robert E. Gompf and Tomasz S. Mrowka, Irreducible $4$-manifolds need not be complex, Ann. of Math. (2) 138 (1993), no. 1, 61–111. MR 1230927, DOI 10.2307/2946635
Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
R. Kirby: Problems in low-dimensional topology, Geometric Topology, AMS/IP Stud. Adv. Math. 2.2, 35-473, Amer. Math. Soc., 1997.
P. B. Kronheimer and T. S. Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994), no. 6, 797–808. MR 1306022, DOI 10.4310/MRL.1994.v1.n6.a14
John W. Morgan, The Seiberg-Witten equations and applications to the topology of smooth four-manifolds, Mathematical Notes, vol. 44, Princeton University Press, Princeton, NJ, 1996. MR 1367507
John W. Morgan and Zoltán Szabó, Embedded genus-$2$ surfaces in four-manifolds, Duke Math. J. 89 (1997), no. 3, 577–602. MR 1470343, DOI 10.1215/S0012-7094-97-08924-9
John W. Morgan, Zoltán Szabó, and Clifford Henry Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996), no. 4, 706–788. MR 1438191
J. Park: Ph.D. Thesis, Michigan State University, 1995.
András Stipsicz, Elementary computations of Donaldson series, Turkish J. Math. 19 (1995), no. 2, 159–166. MR 1349568
Zoltán Szabó, Simply-connected irreducible $4$-manifolds with no symplectic structures, Invent. Math. 132 (1998), no. 3, 457–466. MR 1625732, DOI 10.1007/s002220050230
Zoltán Szabó, Exotic $4$-manifolds with $b^+_2=1$, Math. Res. Lett. 3 (1996), no. 6, 731–741. MR 1426531, DOI 10.4310/MRL.1996.v3.n6.a2
Z. Szabó: Exotic smooth structures on $(2n+1) (S^2\times S^2)$, in preparation.
Zoltán Szabó, Irreducible four-manifolds with small Euler characteristics, Topology 35 (1996), no. 2, 411–426. MR 1380507, DOI 10.1016/0040-9383(95)00014-3
Clifford Henry Taubes, The Seiberg-Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994), no. 6, 809–822. MR 1306023, DOI 10.4310/MRL.1994.v1.n6.a15
Clifford Henry Taubes, More constraints on symplectic forms from Seiberg-Witten invariants, Math. Res. Lett. 2 (1995), no. 1, 9–13. MR 1312973, DOI 10.4310/MRL.1995.v2.n1.a2
W. P. Thurston, Some simple examples of symplectic manifolds, Proc. Amer. Math. Soc. 55 (1976), no. 2, 467–468. MR 402764, DOI 10.1090/S0002-9939-1976-0402764-6
Edward Witten, Monopoles and four-manifolds, Math. Res. Lett. 1 (1994), no. 6, 769–796. MR 1306021, DOI 10.4310/MRL.1994.v1.n6.a13