Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. 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.

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