Summer Lectures in Geometry 

18/06/2001, 11:00 — 12:00 — Amphitheatre Pa2, Mathematics Building
François Lalonde, Université de Montréal

Symplectic fibrations and quantum homology (I)

By the very definition of a symplectic form, every symplectic manifold is locally fibered. Donaldson's theorem on Lefschetz pencils shows that a somehow similar statement (with singularities) is actually true at a global level. This means that symplectic fibrations play a fundamental role in symplectic topology and geometry. The goal of these lectures is to introduce to the theory of non- singular symplectic fibrations in arbitrary dimensions. Here, by “symplectic fibration”, we mean a symplectic manifold which is fibered by symplectic submanifolds. I will show that this is essentially equivalent to a topological fibration \((M,\omega) \to P \to B\) with structure group equal to the group of Hamiltonian diffeomorphisms of \(M\) (at least when \(B\) is a simply connected symplectic manifold). I will discuss the topology of such fibrations, showing in particular that their rational cohomology splits in a many interesting cases (this is a strong generalisation of works of Kirwan and Atiyah-Bott). This gives hard obstructions to the construction of new symplectic manifolds by twisted products of two symplectic manifolds. I will sketch the proof, based on a geometric interpretation of the Seidel map in quantum homology. If time permits, I will mention the consequences of this on the discovery of new rigidity phenomenon in Hamiltonian dynamics. Most of this work is the result of a collaboration with McDuff and Polterovich.