269 research outputs found
Unlinking and unknottedness of monotone Lagrangian submanifolds
Under certain topological assumptions, we show that two monotone Lagrangian
submanifolds embedded in the standard symplectic vector space with the same
monotonicity constant cannot link one another and that, individually, their
smooth knot type is determined entirely by the homotopy theoretic data which
classifies the underlying Lagrangian immersion. The topological assumptions are
satisfied by a large class of manifolds which are realised as monotone
Lagrangians, including tori. After some additional homotopy theoretic
calculations, we deduce that all monotone Lagrangian tori in the symplectic
vector space of odd complex dimension at least five are smoothly isotopic.Comment: 31 page
Exotic spheres and the topology of symplectomorphism groups
We show that, for certain families of diffeomorphisms of
high-dimensional spheres, the commutator of the Dehn twist along the
zero-section of with the family of pullbacks
gives a noncontractible family of compactly-supported symplectomorphisms. In
particular, we find examples: where the Dehn twist along a parametrised
Lagrangian sphere depends up to Hamiltonian isotopy on its parametrisation;
where the symplectomorphism group is not simply-connected, and where the
symplectomorphism group does not have the homotopy-type of a finite CW-complex.
We show that these phenomena persist for Dehn twists along the standard
matching spheres of the -Milnor fibre. The nontriviality is detected by
considering the action of symplectomorphisms on the space of parametrised
Lagrangian submanifolds. We find related examples of symplectic mapping classes
for and of an exotic symplectic structure on
standard at infinity.Comment: 17 pages, 3 figures; v2 streamlined version. Accepted for publication
by Journal of Topolog
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