Zusammenfassung für den Vortrag am 16.04.2009 (14:45 Uhr)Oberseminar GEOMETRIE
R.C.A.M. van der Vorst (Vrije Universiteit Amsterdam, Netherlands)
Braid classes and their Floer homology
Area-preserving diffeomorphisms of a 2-disc can be regarded as time-1 maps of (non-autonomous) Hamiltonian systems on . In this 3-dimensional setting we can think of flow-lines of the Hamilton equations as closed braids in the solid torus . In the spirit of positive braid classes and flat-knot types we define braid classes and use Floer’s variational approach on these spaces to define a chain-complex and the associated Floer homology invariants. This yields a Morse type theory for braided solutions of the Hamilton equations and in particular for periodic points of area-preserving diffeomorphisms of the 2-disc. The ideas presented here carry over to general two dimensional compact symplectic manifolds (M, ) (with or without boundary). In the case of the 2-disc we prove a reduction result with respect to Gar- side’s normal form for braids and conjugacy classes of braids.