Spherical Lagrangians via ball packings and symplectic cutting

Abstract

In this paper we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, S^2 or RP^2, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction this is a natural extension of McDuff's connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of the symplectic Torelli group, classifying Lagrangian isotopy classes in the presence of knotting, and detecting Floer-theoretically essential Lagrangian tori in the del Pezzo surfaces.Comment: 25 pages, 2 figures; v2: minor corrections and clarifications, added discussion after Corollary 1.2. To appear in Selecta Mathematic

Monotone Lagrangians in of minimal Maslov number n + 1

“…By the latter two items, we conclude that HF 2 (L, L; Z/2) is a Z/2, generated by CO 0 (h) * 1 L = CO 0 (h). Now, applying the diagram (3), relating i * to CO 0 , we obtain the commuting diagram (8) H 2 (CP n ; Z/2) H 2 (L; Z/2) QH 2 (CP n ; Z/2) HF 2 (L, L; Z/2) i *…”

The affine invariant of proper semitoric integrable systems

“…This invariant encodes the singular affine structure induced by the (singular) Lagrangian fibration F : M → R 2 on the base F (M ). Its construction and properties appear in Theorems B, C, D. This affine structure also plays a role in parts of symplectic topology, mirror symmetry, and algebraic geometry, see for instance Auroux [Au09], Borman-Li-Wu [BLW13], Kontsevich--Soibelman [KS06]. Integrable systems exhibiting semitoric features appear in the theory of symplectic quasi-states, see Eliashberg-Polterovich [EP10].…”

Exact Lagrangians in $$A_n$$ A n -surface singularities

“…In this section we introduce a technique of producing symplectomorphism alluded in [20,3], which we call the ball-swapping, and try to address its relation with the Dehn twists.…”