The Symplectomorphism Group of a Blow Up

We study the relation between the symplectomorphism group SympM of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp f M and Diff f M of its one point blow up f M . There are three main arguments. The first shows that for any oriented M the natural map from π1(M) to π0(Diff M) is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group π1(Diff M) that persist into the space of self homotopy equivalences of f M . Since it uses purely homological arguments, it applies to c-symplectic manifolds (M,a), that is, to manifolds of dimension 2n that support a class a ∈ H(M ;R) such that a 6= 0. The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp f M using characteristic classes defined by parametric Gromov–Witten invariants. Some results about many point blow ups are also obtained. For example we show that if M is the 4-torus with k-fold blow up f Mk (where k > 0) then π1(Diff Mk) is not generated by the groups π1 `