Taubes's Gromov invariant

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In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold.

Taubes proved that is equivalent to the Seiberg-Witten equations, in a series of four long papers. Much of the analytical complexity connected to this invariant comes from properly counting multiply-covered pseudoholomorphic curves.

Embedded contact homology is an attempt to generalize these results to noncompact four-manifolds that are a compact contact three-manifold cross the real numbers; conjecturally, a certain count of embedded holomorphic curves gives the differential for a homology theory isomorphic to Seiberg-Witten-Floer homology.

[edit] Reference

• Taubes, Clifford (2000). Seiberg-Witten and Gromov Invariants in Symplectic 4-manifolds. Boston: International Press. ISBN 1-57146-061-6