Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The geography of irreducible 4-manifolds


Author: Jongil Park
Journal: Proc. Amer. Math. Soc. 126 (1998), 2493-2503
MSC (1991): Primary 57N13, 57R55
DOI: https://doi.org/10.1090/S0002-9939-98-04762-5
MathSciNet review: 1487335
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we investigate the existence and the uniqueness problems for simply connected irreducible $4$-manifolds. By taking fiber sums along an embedded surface of square $0$ and by a rational blow-down procedure, we construct many new irreducible $4$-manifolds which have infinitely many distinct smooth structures. Furthermore, we prove that all but at most finitely many lattice points lying in the non-positive signature region with $2e+3 sign \geq 0$ are covered by these irreducible $4$-manifolds.


References [Enhancements On Off] (What's this?)

  • [C] Z. Chen, On the geography of surfaces: Simply connected minimal surfaces with positive index, Math. Ann. 277, 141-164 (1987). MR 88c:14057
  • [FS1] R. Fintushel and R. Stern, Surgery in cusp neighborhoods and the geography of irreducible 4-manifolds, Inventiones Math. 117, 455-523 (1994). MR 95f:57040
  • [FS2] R. Fintushel and R. Stern, Rational blowdowns of smooth 4-manifolds, J. Diff. Geo., to appear.
  • [G] R. Gompf, A new construction of symplectic manifolds, Annals of Math. 142, 527-595 (1995). MR 96j:57025
  • [K] D. Kotschick, The Seiberg-Witten invariants of symplectic four-manifolds [after C.H. Taubes], Seminaire Bourbaki. 48eme annee, no 812 (1995-96), Astérisque, no. 241, Soc. Math. France, Paris, 1997, pp. 195-220. CMP 98:01
  • [L] W. Lorek, private communication.
  • [MMS] J. Morgan, T. Mrowka and Z. Szabó, Product formulas along $T^{3}$ for Seiberg-Witten invariants, in preparation.
  • [MST] J. Morgan, Z. Szabó and C. Taubes, A product formula for the Seiberg-Witten invariants and the generalized Thom conjecture, J. Diff. Geo. 44, 706-788 (1996). MR 97m:57052
  • [P] U. Persson, An introduction to the geography of surfaces of general type, Proc. Symp. Pure Math. vol 46, 195-217 (1987). MR 89a:14057
  • [S] A. Stipsicz, A note on the geography of symplectic manifolds, Turkish J. Math. 20 no 1, 135-139 (1996). MR 97m:57035
  • [T] C. Taubes, The Seiberg-Witten and the Gromov invariants, Mathematical Research Letters 2, 221-238 (1995). MR 96a:57076

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57N13, 57R55

Retrieve articles in all journals with MSC (1991): 57N13, 57R55


Additional Information

Jongil Park
Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92697-3875
Address at time of publication: Department of Mathematics, Kon-Kuk University, Kwangjin-gu Mojin-dong 93-1, Seoul 143-701, Korea
Email: jpark@math.uci.edu, jipark@kkucc.konkuk.ac.kr

DOI: https://doi.org/10.1090/S0002-9939-98-04762-5
Keywords: Fiber sum, irreducible manifold, rational blow-down
Received by editor(s): January 23, 1997
Communicated by: Leslie Saper
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society