In
mathematics, the Gromov invariant of
Clifford Taubes counts embedded (possibly disconnected)
pseudoholomorphic curves 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 dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. ...
, where the curves are holomorphic with respect to an auxiliary compatible
almost complex structure. (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
In mathematics, Fredholm operators are certain operators that arise in the Fredholm theory of integral equations. They are named in honour of Erik Ivar Fredholm. By definition, a Fredholm operator is a bounded linear operator ''T'' : ''X ...
.
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 int ...
is an extension due to
Michael Hutchings of this work to noncompact four-manifolds of the form
, 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 ...
. ECH is a
symplectic field theory-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
. 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 i ...
, a version of Seiberg–Witten homology for ''Y''.
References
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*
Symplectic topology
4-manifolds
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