Lectures on four-dimensional Dehn twists

Let M be a closed symplectic manifold, with symplectic form ω. A symplectic automorphism is a diffeomorphism φ : M → M such that φω = ω. We equip the group Aut(M) = Aut(M,ω) of all such maps with the C-topology. Like the whole diffeomorphism group, this is an infinite-dimensional Lie group in a very loose sense: it has a well-defined Lie algebra, which consists of closed oneforms on M , but the exponential map is not locally onto. We will be looking at the homotopy type of Aut(M), and in particular the symplectic mapping class group π0(Aut(M)).