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

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Abstract | References | Similar Articles | Additional Information

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.

**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 -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 ,*Topology,***38**1999, 517-528. MR**99m:57028****9.**V. Muñoz, Fukaya-Floer homology of 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**

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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