Publication:
Fukaya-Floer homology of Σ×S1 and applications.

Loading...
Thumbnail Image
Full text at PDC
Publication Date
1999
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
International Press
Citations
Google Scholar
Research Projects
Organizational Units
Journal Issue
Abstract
We determine the Fukaya-Floer (co)homology groups of the three-manifold y = S x S 1 , where S is a Riemann surface of genus g > 1. These are of two kinds. For the 1-cycle S1 C Y, we compute the Fukaya-Floer cohomology HFF*(Y, S1) and its ring structure, which is a sort of deformation of the Floer cohomology HF*(Y). On the other hand, for 1-cycles ö C 'S CY, we determine the Fukaya-Floer homology HFF*(Y,S) and its i?-F*(Y)-module structure. We give the following applications: We show that every four-manifold with 6+ > 1 is of finite type. Four-manifolds which arise as connected sums along surfaces of fourmanifolds with 6i = 0 are of simple type and we give constraints on their basic classes. We find the invariants of the product of two Riemann surfaces both of genus greater than or equal to one.
Description
UCM subjects
Unesco subjects
Keywords
Citation
P. J. Braam, Floer homology groups for homology three-spheres, Adv. in Math. 88 (1991) 131-144. P. Braam & S. K. Donaldson, Fukaya-Floer homology and gluing formulae for polynomial invariants, The Floer memorial volume, Progr. Math. 133 (1994)257-281. S. K. Donaldson, On the work of Andreas Floer, Jahresber. Deutsch. Math.Verein, 95 (1993) 103-120. Floer homology and algebraic geometry, Vector bundles in algebraic geometry,London Math. Soc. Lecture Notes Ser., Vol. 208, Cambridge Univ. Press,Cambridge, 1995, 119-138. S. K. Donaldson, M. Furuta & D. Kotschick, Floer homology groups in Yang-Mills theory, Unpublished book. S. K. Donaldson & P. B. Kronheimer, The geometry of 4-manifolds, Oxford Univ.Press, 1990. S. Dostoglou & D. Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. 139 (1994) 581-640. [8] R. Fintushel & R. J. Stern, Donaldson invariants of ^-manifolds with simple type,J. Differential Geom. 42 (1995) 577-633. The blow-up formula for Donaldson invariants, Ann. of Math. 143 (1996)529-546. A. Floer, Instanton homology and Dehn surgery, The Floer memorial volume.Progr. Math. 133 (1994) 77-97. W. Fulton & J. Harris, Representation Theory, a first course, Graduate Texts in Math. Vol. 129, Springer, Berlin. K. Fukaya, Instanton homology for oriented ^-manifolds, Adv. Stud. Pure Math.,(éd. Y. Matsumoto and S. Morita). K. A. Froyshov, Some equivariant aspects of Yang-Mills-Floer theory, math.DG/9903083. A. D. King & P. E. Newstead, On the cohomology ring of the moduli space of rank 2 vector bundles on a curve, Topology 37 (1998) 407-418. D. Kotschick, SO(3) invariants for 4-manifolds with b^ = 1, Proc. Lond. Math.Soc. 63 (1991) 426-448. P. B. Kronheimer & T. S. Mrowka, Embedded surfaces and the structure of Donaldson's polynomial invariants, J.Differential Geom. 41 (1995) 573-734. The structure of Donaldson's invariants for four-manifolds not of simple type,http://www.math.harvard.edu/HTML/Individuals/Peter_Kronheimer.html. J. W. Morgan & Z. Szabó, Embedded tori in four-manifolds,Topology, 38 (1999)479-496. V. Munoz, Gauge theory and complex manifolds, Oxford D. Phil. Thesis, 1996. Donaldson invariants for connected sums along surfaces of genus 2,Topology Appi. 107 (2000) 215-232. Gluing formulae for Donaldson invariants for connected sums along surfaces,Asian J. of Math., 1 (1997) 785-800. Wall crossing formulae for algebraic surfaces with positive irregularity,J. London Math. Soc. (2) 61 (2000) 259-274. Ring structure of the Floer cohomology of S x S1, Topology,38 (1999)517-528. Quantum cohomology of the moduli space of stable bundles over a Riemann surface, Duke Math. J. 98 (1999) 525-540. Basic classes for four-manifolds not of simple type, To appear in Comm.Anal. Geom. D. Salamon, Lagrangian intersections, 3-manifolds with boundary and the Atiyah-Floer conjecture, Proc. Internat. Congress of Math., 1 Birkhäuser, Boston, 1994,526-536. B. Siebert & G. Tian, Recursive relations for the cohomology ring of moduli spaces of stable bundles, Turkish J. Math. 19 (1995) 131-144. On quantum cohomology rings of Fano manifolds and a formula of Vafa and Intriligator, Asian J. Math. 1 (1997) 679-695. A. Stipsicz, Computations of Donaldson invariants via cut and paste techniques,Ph. D. Thesis, Rutgers Univ., NJ, 1994. W. Wieczorek, Immersed spheres and finite type of Donaldson invariants, math.DG/9811116.
Collections