In his PhD thesis, Paul Seidel found examples of symplectomorphisms which are smoothly isotopic to the identity, but not isotopic to the identity within the symplectomorphism group.
These symplectomorphisms are known as Dehn twists, and they exist for those symplectic 4-manifolds which admit an embedded Lagrangian 2-sphere. Later Seidel proved that Dehn twists are of infinite order in the symplectic mapping class groups of certain symplectic K3 surfaces. In spite of this deep result of Seidel, and other results that followed it, we still have no general way to construct non-isotopic symplectomorphisms for 4-manifolds. For instance, there are 4-manifolds which do not contain Lagrangian spheres, yet it is believed that they have a non-trivial symplectic mapping class group.
In this talk we will introduce and study a new type of symplectomorphisms for symplectic 4-manifolds and discuss some examples of symplectic 4-manifolds for which these new twists are infinite order elements in the symplectic mapping class groups.
This is a joint work with Vsevolod Shevchishin.