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 \(\widetilde{M}_k\) .