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Every finitely presented group is the fundamental group of a closed symplectic four-manifold. The construction starts from products of surfaces, realizes relator words by embedded loops after stabilization, and takes symplectic sums with copies of the rational elliptic surface along square-zero tori. Since the complement of a regular elliptic fiber in the rational elliptic surface is simply connected, van Kampen's theorem imposes the desired generators and relators.

The symplectic neighborhood theorem says that a symplectomorphism between compact symplectic submanifolds, together with a symplectic vector bundle isomorphism of their symplectic normal bundles, extends to a symplectomorphism of neighborhoods. To outline the proof, use tubular neighborhoods to extend the bundle identification smoothly. The two pulled-back symplectic forms agree as bilinear forms along the zero section. The relative Poincaré lemma writes their difference as $d\alpha$, with $\alpha$ vanishing there. Their linear interpolation ${\omega }_t$ remains nondegenerate on a sufficiently small neighborhood. Solvingand integrating $V_t$ gives the relative Moser theorem; the flow fixes the submanifold and carries one form to the other.

For the symplectic fiber sum along a square-zero surface, the self-intersection is the Euler number of the oriented rank-two normal bundle. Thus each normal bundle is trivial. Rescale one ambient symplectic form if necessary to equalize the areas of the two copies of $C$. The Moser theorem for the surface then supplies a base identification preserving their area forms. The symplectic neighborhood theorem identifies each neighborhood with a product carryingRemove small disk neighborhoods and glue collars with the base identification andfor a suitable positive constant $a$. Both signs reverse, so the normal two-forms match; the base forms already match. They define a closed nondegenerate two-form across the neck, agreeing with the ambient forms elsewhere. This constructs the symplectic sum. Its construction uses the area normalization and the chosen normal-bundle gluing.

For a smooth counterexample, take $X=Y={\mathbb{CP}}^2$, and in a small four-ball in each choose an unknotted sphere bounding a three-ball. These spheres are null-homologous and have self-intersection zero. Their standard smooth fiber sum isIndeed, the complement of the standard $S^2\times D^2$ in $S^4$ is $S^1\times D^3$; doubling these complements gives $S^1\times S^3$, and the two ambient projective-plane summands remain as connected sums. This is the smooth fiber sum along unknotted null-homologous spheres.

In the induced orientation, both summands in $Z={\mathbb{CP}}^2\#({\mathbb{CP}}^2\#(S^1\times S^3))$ have positive positive index of the intersection form. The symplectic connected-sum obstruction therefore rules out a symplectic form: connected-sum vanishing of the Seiberg–Witten invariant of a four-manifold contradicts Taubes nonvanishing theorem. In the opposite orientation the intersection form is negative definite, also ruling out a symplectic form, whose class has positive square. Thus the smooth fiber sum need not be symplectic.

Let $Q\subset {\mathbb{CP}}^2$ be a smooth complex conic. Its homology class is $2H$, so its self-intersection number isRescale the Fubini-Study form so that $Q$ and $C$ have the same symplectic area. Their normal bundles have opposite Euler numbers, $+4$ and $-4$, so the symplectic sum can be formed along $Q$ and $C$.

Concretely, remove tubular neighborhoods $\nu (C)$ and $\nu (Q)$ and glue the boundaries by a fiber-reversing bundle map. The symplectic neighborhood theorem supplies the standard models needed for the gluing, and the symplectic-sum construction supplies a symplectic form onThe second piece is a rational homology ball. Thus this operation replaces the neighborhood of the $-4$ sphere by that rational ball and is the symplectic rational blowdown of a minus-four sphere.

If symplectic four-manifolds contain symplectic copies of the same closed surface with self-intersection zero, their trivial symplectic normal bundles may be identified with reversed circle fibers. Removing the two tubular neighborhoods and gluing their boundaries gives a symplectic sum carrying a natural symplectic form.

A symplectic sphere of self-intersection $-4$ can be rationally blown down by taking the symplectic sum with ${\mathbb{CP}}^2$ along the sphere and a smooth conic of self-intersection $+4$. Equivalently, its disk-bundle neighborhood is replaced by the rational ball ${\mathbb{CP}}^2\setminus \nu (Q)$ whose boundary is the lens space $L(4,1)$.