#### Volume 14, issue 2 (2014)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1472-2739 ISSN (print): 1472-2747
Displacing Lagrangian toric fibers by extended probes

### Miguel Abreu, Matthew Strom Borman and Dusa McDuff

Algebraic & Geometric Topology 14 (2014) 687–752
##### Abstract

In this paper we introduce a new way of displacing Lagrangian fibers in toric symplectic manifolds, a generalization of McDuff’s original method of probes. Extended probes are formed by deflecting one probe by another auxiliary probe. Using them, we are able to displace all fibers in Hirzebruch surfaces except those already known to be nondisplaceable, and can also displace an open dense set of fibers in the weighted projective space $ℙ\left(1,3,5\right)$ after resolving the singularities. We also investigate the displaceability question in sectors and their resolutions. There are still many cases in which there is an open set of fibers whose displaceability status is unknown.

##### Keywords
symplectic manifolds, Lagrangian tori, nondisplaceable, toric manifolds
##### Mathematical Subject Classification 2010
Primary: 53D12
Secondary: 14M25, 53D40