Symplectic sum
In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a submanifold, often called a fiber sum. The symplectic sum is the inverse of 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 in algebraic geometry. The symplectic sum has been used to construct previously unknown families  of symplectic manifolds, and to derive relationships among the Gromov–Witten invariants of symplectic manifolds. Definition. Let formula_1 and formula_2 be two symplectic formula_3-manifolds and formula_4 a symplectic formula_5-manifold, embedded as a submanifold into both formula_1 and formula_2 via such that the Euler classes of the normal bundles are opposite: In the 1995 paper that defined the symplectic sum, Robert Gompf proved that for any orientation-reversing isomorphism there is a canonical isotopy class of symplectic structures on the connected sum meeting several conditions of compatibility with the summands formula_12. 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 formula_13 is composed with an orientation-reversing symplectic involution of the normal bundles of formula_4 (or rather their corresponding punctured unit disk bundles); then this composition is used to glue formula_1 to formula_2 along the two copies of formula_4. Generalizations. In greater generality, the symplectic sum can be performed on a single symplectic manifold formula_18 containing two disjoint copies of formula_4 , 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 formula_20 consists of two connected components, each containing a copy of formula_4. Additionally, the sum can be performed simultaneously on submanifolds formula_22 of equal dimension and meeting formula_4 transversally. Other generalizations also exist. However, it is not possible to remove the requirement that formula_4 be of codimension two in the formula_12, as the following argument shows. A symplectic sum along a submanifold of codimension formula_26 requires a symplectic involution of a formula_26-dimensional annulus. If  this involution exists, it can be used to patch two formula_26-dimensional balls together to form a symplectic formula_26-dimensional sphere. Because the sphere is a compact manifold, a symplectic form formula_30 on it induces a nonzero cohomology class But this second cohomology group is zero unless formula_32. So the symplectic sum is possible only along a submanifold of codimension two. Identity element. Given formula_18 with codimension-two symplectic submanifold formula_4, one may projectively complete the normal bundle of formula_4 in formula_18 to the formula_37-bundle This formula_39 contains two canonical copies of formula_4: the zero-section formula_41, which has  normal bundle equal to that of formula_4 in formula_18, and the infinity-section formula_44, which has opposite normal bundle. Therefore, one may symplectically sum formula_45 with formula_46; the result is again formula_18, with formula_41 now playing the role of formula_4: So for any particular pair formula_45 there exists an identity element formula_39 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 formula_1, formula_2, formula_4 , formula_56, formula_57, formula_13 determine a unique smooth formula_59-dimensional symplectic manifold formula_60 and a fibration in which the central fiber is the singular space obtained by joining the summands formula_12 along formula_4, and the generic fiber formula_65 is a symplectic sum of the formula_12. (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 formula_67 of the trivial complex line bundle Then, in the direct sum with formula_70 representing a normal vector to formula_4 in formula_12, consider the locus of  the quadratic equation for a chosen small formula_74. One can glue both formula_75 (the summands with formula_4 deleted) onto this locus; the result is the symplectic sum formula_65. As formula_78 varies, the sums formula_65 naturally form the family formula_80 described above. The central fiber formula_81 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 formula_39. For then the generic fiber is a symplectic manifold formula_18 and the central fiber is formula_18  with the normal bundle of formula_4 "pinched off at infinity" to form the formula_37-bundle formula_39. This is analogous to deformation to the normal cone along a smooth divisor formula_4 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 of a symplectic four-manifold. Thus the category 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 curves 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 formula_39 to write the manifold formula_18 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 formula_18.