2000/06/30 by Mikhail Khovanov, Paul Seidel · 1 citation
Mathematics · #math.QA #math.SG #msc:53D40 #msc:16G20 #msc:18E30
published as J. Amer. Math. Soc. 15 (2002), no. 1, 203--271 (electronic) · 72 pages, 28 eps figures, latex. Several misprints were corrected
arxiv created 2001/06/13 · arxiv updated 2009/11/30
We consider the derived categories of modules over a certain family Am of graded rings, and Floer cohomology of Lagrangian intersections in the symplectic manifolds which are the Milnor fibres of simple singularities of type Am. We show that each of these two rather different objects encodes the topology of curves on an (m+1)-punctured disc. We prove that the braid group Bm+1 acts faithfully on the derived category of Am-modules and that it injects into the symplectic mapping class group of 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). This triangulated category should contain a full subcategory which is equivalent, up to a slight difference in the grading, to the derived category of Am-modules. The full embedding would connect the two occurrences of the braid group, thus explaining the similarity between them.