Taubes's Gromov invariant

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.

References

• Taubes, Clifford (2000). Seiberg–Witten and Gromov Invariants in Symplectic 4-manifolds. Boston: International Press. ISBN 1571460616. 
• Taubes, Clifford (2008). Embedded contact homology and Seiberg–Witten Floer homology I. arXiv:0811.3985.