Lagrangian Klein bottles in S2xS2Bonn symplectic geometry seminar

by Nikolas Adaloglou (Leiden)

Tuesday Dec 3, 2024, 12:15 PM → 1:15 PM Europe/Berlin

Endenicher Allee 60, Nebengebäude/N0-003 - Seminarraum (Mathezentrum)

Using (almost) toric fibrations and their visible Lagrangians we can
construct many novel and interesting examples of Lagrangian submanifolds of
symplectic 4 manifolds.  Naturally, one can ask whether visible Lagrangians are all
the Lagrangians that exist, or, in other words, how faithfull the pictures coming
from almost toric fibrations are. I will answer this question for Klein bottles in
(S2xS2,ω_λ), i.e.  the product of two spheres where the first factor has area 1
and the other factor has area λ. In particular, I will first construct a visible
Lagrangian Klein bottle when λ<2.  Then I will show  that no  Lagrangian Klein
bottles exists otherwise. The key input for obstructing the existence of the Klein
bottles is Luttinger surgery along with techniques of  (compact) pseudoholomorpic
curves and Seiberg-Witten theory.

This is joint work with J. Evans.