The symplectomorphism group of a blow up

Abstract

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp $\widetilde{M}$ and Diff $\widetilde{M}$ of its one point blow up $\widetilde{M}$ . There are three main arguments. The first shows that for any oriented M the natural map from $\pi_1(M)$ to $\pi_0({\rm Diff}\widetilde{M})$ is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group $\pi_1({\rm Diff}\widetilde{M})$ that persist into the space of self-homotopy equivalences of $\widetilde{M}$ . 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 $a\in H^2(M;{\mathbb{R}})$ such that $a^n\ne 0$ . The third argument uses the symplectic structure on M and detects nontrivial elements in the (higher) homology of BSymp, 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 four-torus with k-fold blow up $\widetilde{M}_k$ (where k >  0) then $\pi_1({\rm Diff} \widetilde{M}_k)$ is not generated by the groups $\pi_1\left({\rm Symp}\, (\widetilde{M}_k, \widetilde{\omega})\right)$ as $\widetilde{\omega}$ ranges over the set of all symplectic forms on $\widetilde{M}_k$ .