In mathematics

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...

, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifold

Symplectic manifold

In mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form, ω, called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology...

s, which glues two given manifolds into a single new one. It is a symplectic version of connected sum

Connected sum

In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each...

mation along a submanifold, often called a fiber sum.

The symplectic sum is the inverse of the symplectic cut

Symplectic cut

In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that glues two manifolds together into one. The symplectic cut...

, which decomposes a given manifold into two pieces. Together the symplectic sum and cut may be viewed as a deformation of symplectic manifolds, analogous for example to deformation to the normal cone

Blowing up

In mathematics, blowing up or blowup is a type of geometric transformation which replaces a subspace of a given space with all the directions pointing out of that subspace. For example, the blowup of a point in a plane replaces the point with the projectivized tangent space at that point...

 in algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

.

The symplectic sum has been used to construct previously unknown families of symplectic manifolds, and to derive relationships among the Gromov-Witten invariant

Gromov-Witten invariant

In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold...

s of symplectic manifolds.

Definition

Let and be two symplectic -manifolds and a symplectic -manifold, embedded as a submanifold into both and via

such that the Euler class

Euler class

In mathematics, specifically in algebraic topology, the Euler class, named after Leonhard Euler, is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is...

es of the normal bundle

Normal bundle

In differential geometry, a field of mathematics, a normal bundle is a particular kind of vector bundle, complementary to the tangent bundle, and coming from an embedding .-Riemannian manifold:...

s are opposite:

In the 1995 paper that defined the symplectic sum, Robert Gompf proved that for any orientation

Orientation (mathematics)

In mathematics, orientation is a notion that in two dimensions allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra, the notion of orientation makes sense in arbitrary dimensions...

-reversing isomorphism

there is a canonical isotopy class of symplectic structures on the connected sum

meeting several conditions of compatibility with the summands . In other words, the theorem defines a symplectic sum operation whose result is a symplectic manifold, unique up to isotopy.

To produce a well-defined symplectic structure, the connected sum must be performed with special attention paid to the choices of various identifications. Loosely speaking, the isomorphism is composed with an orientation-reversing symplectic involution of the normal bundles of (or rather their corresponding punctured unit disk bundles); then this composition is used to glue

Quotient space

In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

  to along the two copies of .

Generalizations

In greater generality, the symplectic sum can be performed on a single symplectic manifold containing two disjoint copies of , gluing the manifold to itself along the two copies. The preceding description of the sum of two manifolds then corresponds to the special case where consists of two connected components, each containing a copy of .

Additionally, the sum can be performed simultaneously on submanifolds of equal dimension and meeting transversally.

Other generalizations also exist. However, it is not possible to remove the requirement that be of codimension two in the , as the following argument shows.

A symplectic sum along a submanifold of codimension requires a symplectic involution of a -dimensional annulus. If this involution exists, it can be used to patch two -dimensional balls together to form a symplectic -dimensional sphere

Sphere

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

. Because the sphere is a compact

Compact space

In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...

 manifold, a symplectic form on it induces a nonzero cohomology

Cohomology

In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...

 class

But this second cohomology group is zero unless . So the symplectic sum is possible only along a submanifold of codimension two.

Identity element

Given with codimension-two symplectic submanifold , one may projectively complete the normal bundle of in to the -bundle

This contains two canonical copies of : the zero-section , which has normal bundle equal to that of in , and the infinity-section , which has opposite normal bundle. Therefore one may symplectically sum with ; the result is again , with now playing the role of :

So for any particular pair there exists an identity element

Identity (mathematics)

In mathematics, the term identity has several different important meanings:*An identity is a relation which is tautologically true. This means that whatever the number or value may be, the answer stays the same. For example, algebraically, this occurs if an equation is satisfied for all values of...

  for the symplectic sum. Such identity elements have been used both in establishing theory and in computations; see below.

Symplectic sum and cut as deformation

It is sometimes profitable to view the symplectic sum as a family of manifolds. In this framework, the given data , , , , , determine a unique smooth -dimensional symplectic manifold and a fibration

Fibration

In topology, a branch of mathematics, a fibration is a generalization of the notion of a fiber bundle. A fiber bundle makes precise the idea of one topological space being "parameterized" by another topological space . A fibration is like a fiber bundle, except that the fibers need not be the same...

in which the central fiber is the singular space

obtained by joining the summands along , and the generic fiber is a symplectic sum of the . (That is, the generic fibers are all members of the unique isotopy class of the symplectic sum.)

Loosely speaking, one constructs this family as follows. Choose a nonvanishing holomorphic section of the trivial complex line bundle

Then, in the direct sum

with representing a normal vector to in , consider the locus of the quadratic equation

for a chosen small . One can glue both (the summands with deleted) onto this locus; the result is the symplectic sum .

As varies, the sums naturally form the family described above. The central fiber is the symplectic cut of the generic fiber. So the symplectic sum and cut can be viewed together as a quadratic deformation of symplectic manifolds.

An important example occurs when one of the summands is an identity element . For then the generic fiber is a symplectic manifold and the central fiber is with the normal bundle of "pinched off at infinity" to form the -bundle . This is analogous to deformation to the normal cone along a smooth divisor

Divisor (algebraic geometry)

In algebraic geometry, divisors are a generalization of codimension one subvarieties of algebraic varieties; two different generalizations are in common use, Cartier divisors and Weil divisors...

  in algebraic geometry. In fact, symplectic treatments of Gromov-Witten theory often use the symplectic sum/cut for "rescaling the target" arguments, while algebro-geometric treatments use deformation to the normal cone for these same arguments.

However, the symplectic sum is not a complex operation in general. The sum of two Kähler manifolds need not be Kähler.

History and applications

The symplectic sum was first clearly defined in 1995 by Robert Gompf. He used it to demonstrate that any finitely presented group appears as the fundamental group

Fundamental group

In mathematics, more specifically algebraic topology, the fundamental group is a group associated to any given pointed topological space that provides a way of determining when two paths, starting and ending at a fixed base point, can be continuously deformed into each other...

 of a symplectic four-manifold. Thus the category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 of symplectic manifolds was shown to be much larger than the category of Kähler manifolds.

Around the same time, Eugene Lerman proposed the symplectic cut as a generalization of symplectic blow up and used it to study the symplectic quotient and other operations on symplectic manifolds.

A number of researchers have subsequently investigated the behavior of 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 under symplectic sums, proving various versions of a symplectic sum formula for Gromov-Witten invariants. Such a formula aids computation by allowing one to decompose a given manifold into simpler pieces, whose Gromov-Witten invariants should be easier to compute. Another approach is to use an identity element to write the manifold as a symplectic sum

A formula for the Gromov-Witten invariants of a symplectic sum then yields a recursive formula for the Gromov-Witten invariants of .