In mathematics, the Gromov invariant of

Clifford Taubes Clifford Henry Taubes (born February 21, 1954) is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology. His brother is the journalist Gary Taube ...

counts embedded (possibly disconnected)

pseudoholomorphic curve In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or ''J''-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equations, Cauchy–Riemann equa ...

s in a symplectic

4-manifold In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a ...

, where the curves are holomorphic with respect to an auxiliary compatible

almost complex structure In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex ...

. (Multiple covers of 2-tori with self-intersection 0 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 so that the result is invariant of the choice of almost complex structure. The crux is a topologically defined index for pseudoholomorphic curves which controls embeddedness and bounds the Fredholm index.

Embedded contact homology In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer i ...

is an extension due to Michael Hutchings of this work to noncompact four-manifolds of the form

Y \times \R

, where ''Y'' is a compact

contact Contact may refer to: Interaction Physical interaction * Contact (geology), a common geological feature * Contact lens or contact, a lens placed on the eye * Contact sport, a sport in which players make contact with other players or objects * ...

3-manifold In mathematics, a 3-manifold is a space that locally looks like Euclidean 3-dimensional space. A 3-manifold can be thought of as a possible shape of the universe. Just as a sphere looks like a plane to a small enough observer, all 3-manifolds lo ...

. ECH is a

symplectic field theory The term "symplectic" is a calque of "complex" introduced by Hermann Weyl in 1939. In mathematics it may refer to: * Symplectic Clifford algebra, see Weyl algebra * Symplectic geometry * Symplectic group * Symplectic integrator * Symplectic mani ...

-like invariant; namely, it is the homology of a chain complex generated by certain combinations of Reeb orbits of a contact form on ''Y'', and whose differential counts certain embedded pseudoholomorphic curves and multiply covered pseudoholomorphic cylinders with "ECH index" 1 in

Y \times \R

. The ECH index is a version of Taubes's index for the cylindrical case, and again, the curves are pseudoholomorphic with respect to a suitable almost complex structure. The result is a topological invariant of ''Y'', which Taubes proved is isomorphic to monopole

Floer homology In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analogue of finite-dimensional Morse homology. Andreas Floer in ...

, a version of Seiberg–Witten homology for ''Y''.

References

* * Symplectic topology 4-manifolds {{geometry-stub