Four dimensions from two in symplectic topology

By Margaret Symington. 2002

186 citations recorded by the source corpus.

This paper introduces two-dimensional diagrams that are slight general- izations of moment map images for toric four-manifolds and catalogs techniques for reading topological and symplectic properties of a symplectic four-manifold from these diagrams. The paper offers a purely topological approach to toric manifolds as well as methods for visualizing other manifolds, including the K3 surface, and for visualizing certain surgeries. Most of the techniques extend to higher dimensions.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/da9c72c33a213ffa25b24de17fbb6e9592ec91d0

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Constructing integrable systems of semitoric type (2009, background): “These structures have been studied in the context of integrable systems (in particularby Zung [23]), but also became a central concept in the works by Symington [18], Symington-Leung [15] in the context of symplectic geometry and topology, and by Gross-Siebert [10], [11], [12] and [13], among others”
Dirichlet Branes and Mirror Symmetry (2009, background): “In fact, in this case the compactification can be carried out in the symplectic category (see [435, 90]) but we will not give details here.”
Infinitely many exotic monotone Lagrangian tori in ℂℙ2 (2014, methodology): “See section 9 of [14], for understanding how to see the respective lens spaces as contact manifolds.”
Symplectic theory of completely integrable Hamiltonian systems (2011, unknown): “That is what Symington does in [99] to show that neighborhoods of focus-focus fibers are always symplectomorphic (after forgetting the foliation, of course).”
Almost toric symplectic four-manifolds (2003, methodology): “Almost toric manifolds, introduced by the second author in [39], are symplectic manifolds equipped with a fibration structure that generalizes toric manifolds while retaining some of their geometric features and rigidity.”
Four-dimensional symplectic cobordisms containing three-handles (2006, unknown): “dgay@maths.uct.ac.za Proposed: Yasha Eliashberg Received: 22 June 2006 Seconded: Peter Ozsváth, Tomasz Mrowka Accepted: 13 October 2006 Geometry & Topology, Volume 10 (2006)”
Infinitely many monotone Lagrangian tori in del Pezzo surfaces (2016, methodology): “The idea is to construct different almost toric fibrations (denoted from here by ATF) [23,31] of CP2.”
The wall-crossing formula and Lagrangian mutations (2017, methodology): “Almost toric fibrations on symplectic 4-manifolds were introduced by Symington [33], and have recently been used by Vianna [35, 36, 37] to construct infinitely many monotone Lagrangian tori in del Pezzo surfaces.”