Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Higher type adjunction inequalities for Donaldson invariants


Author: Vicente Muñoz
Journal: Trans. Amer. Math. Soc. 353 (2001), 2635-2654
MSC (2000): Primary 57R57; Secondary 57R58
DOI: https://doi.org/10.1090/S0002-9947-01-02793-3
Published electronically: March 15, 2001
MathSciNet review: 1828464
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract:

We prove new adjunction inequalities for embedded surfaces in four-manifolds with non-negative self-intersection number using the Donaldson invariants. These formulas are completely analogous to the ones obtained by Ozsváth and Szabó using the Seiberg-Witten invariants. To prove these relations, we give a fairly explicit description of the structure of the Fukaya-Floer homology of a surface times a circle. As an aside, we also relate the Floer homology of a surface times a circle with the cohomology of some symmetric products of the surface.


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

  • 1. A. Bertram and M. Thaddeus, On the quantum cohomology of a symmetric product of an algebraic curve, math.AG/9803026
  • 2. S. Dostoglou and D. Salamon, Self-dual instantons and holomorphic curves, Annals of Math., 139 1994, 581-640. MR 95g:58050
  • 3. R. Fintushel and R. J. Stern, The blow-up formula for Donaldson invariants, Annals of Math. 143 1996, 529-546. MR 97i:57036
  • 4. K. Fukaya, Instanton homology for oriented $3$-manifolds, Adv. Studies in Pure Mathematics, Ed. Y. Matsumoto and S. Morita.
  • 5. P. B. Kronheimer and T. S. Mrowka, Embedded surfaces and the structure of Donaldson's polynomial invariants, J. Diff. Geom. 41 1995, 573-734. MR 96e:57019
  • 6. I. G. MacDonald, Symmetric products of an algebraic curve, Topology, 1 1962, 319-343. MR 27:1445
  • 7. V. Muñoz, Gluing formulae for Donaldson invariants for connected sums along surfaces, Asian J. Math. 1 1997, 785-800. MR 99m:57027
  • 8. V. Muñoz, Ring structure of the Floer cohomology of $\Sigma \times {{\mathbb S}}^1$, Topology, 38 1999, 517-528. MR 99m:57028
  • 9. V. Muñoz, Fukaya-Floer homology of $\Sigma \times {{\mathbb S}}^1$ and applications, to appear in J. Diff. Geom.
  • 10. V. Muñoz, Basic classes for four-manifolds not of simple type, Comm. Anal. Geom. 8 2000, 653-670. CMP 2000:16
  • 11. V. Muñoz and B-L. Wang, Seiberg-Witten-Floer homology of a surface times a circle, math.DG/9905050.
  • 12. P. Osváth and Z. Szabó, Higher type adjunction inequalities in Seiberg-Witten theory, math.DG/0005268.
  • 13. E. Witten, Monopoles and four-manifolds, Math. Research Letters, 1 1994, 769-796. MR 96d:57035

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57R57, 57R58

Retrieve articles in all journals with MSC (2000): 57R57, 57R58


Additional Information

Vicente Muñoz
Affiliation: Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain
Address at time of publication: Departamento de Matemáticas, Facultad de Ciencias, Universidad Autónoma de Madrid, 28049 Madrid, Spain
Email: vicente.munoz@uam.es

DOI: https://doi.org/10.1090/S0002-9947-01-02793-3
Keywords: 4-manifolds, adjunction inequalities, Donaldson invariants, Fukaya-Floer homology
Received by editor(s): February 23, 1999
Published electronically: March 15, 2001
Additional Notes: Partially supported by DGES through Spanish Research Project PB97-1095
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society