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Symplectic mapping class groups of rational 4-manifolds

Symplectic mapping class groups of rational 4-manifolds

The symplectic isotopy problem is a question about automorphisms of a compact symplectic manifold. It asks whether the relation of symplectic isotopy between such automorphism is finer than the relation of diffeotopy (smooth isotopy). After Arnold and Seidel, the Lagrangian Dehn twists are proved to be exotic symplectomorphisms. A question of Donaldson asks whether Lagrangian Dehn twists generate the full symplectic mapping class groups. In this talk, we explain how to compute the symplectic mapping class group for rational 4-manifolds, in particular, those supporting log Calabi-Yau structures. This leads to an affirmative answer to Donaldson's question and proves that all toric surfaces have trivial symplectic Torelli groups. As a by-product, we show the existence of a Hamiltonian Z/2-action on a symplectic four-manifold which does not extend to an S^1-action. This is joint work with Tian-Jun Li and Weiwei Wu.

## Department of Mathematics and Statistics