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 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 . References Robert Gompf: A new construction of symplectic manifolds, Annals of Mathematics 142 (1995), 527-595 Dusa McDuff and Dietmar Salamon: Introduction to Symplectic Topology (1998) Oxford Mathematical Monographs, Dusa McDuff and Dietmar Salamon: J-Holomorphic Curves and Symplectic Topology (2004) American Mathematical Society Colloquium Publications, Symplectic topology