Taubes's Gromov Invariant
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High Quality Content by WIKIPEDIA articles! In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold. (Multiple covers of 2-tori with self-intersection 1 are also counted.) Taubes proved the information contained in this invariant is equivalent to invariants derived from 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. The crux is a topologically defined index...
High Quality Content by WIKIPEDIA articles! In mathematics, the Gromov invariant of Clifford Taubes counts embedded (possibly disconnected) pseudoholomorphic curves in a symplectic 4-manifold. (Multiple covers of 2-tori with self-intersection 1 are also counted.) Taubes proved the information contained in this invariant is equivalent to invariants derived from 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. The crux is a topologically defined index for pseudoholomorphic curves which controls embeddedness and bounds the Fredholm index. Embedded contact homology is a generalization due to Michael Hutchings of these results to noncompact four-manifolds that are a compact contact three-manifold cross the real numbers; by a theorem of Taubes a certain count of embedded holomorphic curves (and multiply covered trivial cylinders) defines a symplectic field theory-like invariant isomorphic to Seiberg Witten Floer homology. It relies upon an analogous "ECH index" for symplectizations.
