Toward a topological characterization of symplectic manifolds

By Robert E. Gompf. 2002

53 citations recorded by the source corpus.

A topological condition is given, characterizing which closed manifolds in dimensions < 8 (and conjecturally in general) admit symplectic structures. The condition is the existence of a certain fibration-like structure called a hyperpencil. A deformation class of hyperpencils on a manifold X of any even dimension is shown to determine an isotopy class of symplectic structures on X. This provides an inverse (at least in dimensions < 8) to Donaldson's program for constructing linear systems on symplectic manifolds. It follows that (at least in dimensions < 8) the set of deformation classes of hyperpencils canonically maps onto the set of isotopy classes of rational symplectic forms up to positive scale, topologically determining a dense subset of all symplectic forms up to an equivalence relation on hyperpencils. Other applications of the main techniques are presented, including the construction of symplectic structures on domains of locally holomorphic maps, and on high-dimensional Lefschetz pencils and other linear systems.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1c20eba4d959b430a73a61178c71f7d386653555

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In Computational Theory and Mathematics

1. Tian-Jun Li
2. Weiyi Zhang
3. Simon Donaldson
4. L. Katzarkov

In Geometry and Topology

1. L. Katzarkov
2. D. Auroux
3. Tian-Jun Li
4. Weiyi Zhang
5. Simon Donaldson
6. P. Seidel
7. Yankı Lekili
8. G'abor Sz'ekelyhidi

In Mathematical Physics

1. L. Katzarkov
2. D. Auroux
3. Weiyi Zhang
4. Simon Donaldson
5. Tian-Jun Li
6. G'abor Sz'ekelyhidi
7. I. Smith
8. P. Seidel

Citations in the corpus

• Singular Lefschetz pencils (2004, background): “In any case, the point we wish to bring out, in formulating things the way we have, is that the near-symplectic condition has a meaning independent of Riemannian geometry.”
• Planar open books, monodromy factorizations and symplectic fillings (2009, unknown): “Because a compatible symplectic structure on a Lefschetz fibration is unique up to symplectic deformation [10], it follows that all Stein structures on X are symplectic deformation equivalent; by Theorem 1.”
• Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds (2007, background): “In such a situation, on the one hand, there is the construction of tamed symplectic forms in [16], and on the other hand, there are new subtle obstructions being Kähler in [35].”