The symplectomorphism group of a blow up

Details The symplectomorphism group of a blow up The symplectomorphism group of a blow up

2006-10-04

arxiv.org/abs/math/0610142v3

Mathematics

Symplectic Geometry

29 pages; v3 with minor revisions, to be published in Geom. Dedicata

Scientific paper

We study the relation between the symplectomorphism group Symp M of a closed connected symplectic manifold M and the symplectomorphism and diffeomorphism groups Symp \TM and Diff \TM of its one point blow up \TM. There are three main arguments. The first shows that for any oriented M the natural map from pi_1(M) to pi_0(Diff \TM) is often injective. The second argument applies when M is simply connected and detects nontrivial elements in the homotopy group pi_1(Diff \TM) that persist into the space of self homotopy equivalences of \TM. 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;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 \TM 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 \TM_k (where k>0) then pi_1(Diff \TM_k) is not generated by the groups pi_1\Symp (\TM_k, \Tom) as \Tom ranges over the set of all symplectic forms on \TM_k.

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