Journal of the American Mathematical Society

Author(s): Mikhail Khovanov; Paul Seidel
Journal: J. Amer. Math. Soc. 15 (2002), 203-271.
MSC (2000): Primary 18G10, 53D40, 20F36
Posted: September 24, 2001
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Abstract: We consider the derived categories of modules over a certain family

( $m \geq 1$ ) of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type

We show that each of these two rather different objects encodes the topology of curves on an

-punctured disc. We prove that the braid group $B_{m+1}$ acts faithfully on the derived category of

-modules, and that it injects into the symplectic mapping class group of the Milnor fibers. The philosophy behind our results is as follows. Using Floer cohomology, one should be able to associate to the Milnor fibre a triangulated category (its construction has not been carried out in detail yet). This triangulated category should contain a full subcategory which is equivalent, up to a slight difference in the grading, to the derived category of

-modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them.

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Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 18G10, 53D40, 20F36

Mikhail Khovanov
Affiliation: Department of Mathematics, University of California at Davis, Davis, California 95616-8633
Email: mikhail@math.ucdavis.edu

Paul Seidel
Affiliation: Department of Mathematics, Ecole Polytechnique, F-91128 Palaiseau, France -- and -- School of Mathematics, Institute for Advanced Study, Einstein Drive, Princeton, New Jersey 08540
Email: seidel@math.polytechnique.fr, seidel@math.ias.edu

DOI: 10.1090/S0894-0347-01-00374-5
PII: S 0894-0347(01)00374-5
Received by editor(s): July 6, 2000
Received by editor(s) in revised form: April 3, 2001
Posted: September 24, 2001
Additional Notes: The first author was supported by NSF grants DMS 96-27351 and DMS 97-29992 and, later on, by the University of California at Davis. The second author was supported by NSF grant DMS-9304580 and by the Institut Universitaire de France.
Copyright of article: Copyright 2001, American Mathematical Society

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ISSN: 1088-6834(e) ISSN: 0894-0347(p)

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