Taubes's Gromov invariant
Encyclopedia
In mathematics
, the Gromov invariant of Clifford Taubes
counts embedded (possibly disconnected) pseudoholomorphic curve
s 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 cylinder
s) defines a symplectic field theory-like invariant isomorphic to Seiberg–Witten–Floer homology. It relies upon an analogous "ECH index" for symplectizations.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Gromov invariant of Clifford Taubes
Clifford Taubes
Clifford Henry Taubes is the William Petschek Professor of Mathematics at Harvard University and works in gauge field theory, differential geometry, and low-dimensional topology.-Early career:Taubes received his Ph.D...
counts embedded (possibly disconnected) pseudoholomorphic curve
Pseudoholomorphic curve
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study of...
s in a symplectic 4-manifold
4-manifold
In mathematics, 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different...
. (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 cylinder
Trivial cylinder
In geometry and topology, trivial cylinders are certain pseudoholomorphic curves appearing in certain cylindrical manifolds.In Floer homology and its variants, chain complexes or differential graded algebras are generated by certain combinations of closed orbits of vector fields...
s) defines a symplectic field theory-like invariant isomorphic to Seiberg–Witten–Floer homology. It relies upon an analogous "ECH index" for symplectizations.