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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.

The symplectic neighborhood theorem says that a symplectomorphism between closed symplectic submanifolds which lifts to an isomorphism of their symplectic normal bundles extends to a symplectomorphism of neighborhoods.

Let $C_X\subset X$ and $C_Y\subset Y$ be the given copies of $C$. Their self-intersection numbers are the Euler classes of their oriented normal bundles, so square zero makes both normal bundles trivial. The neighborhood theorem identifies neighborhoods with $C\times D^2$. Remove their interiors and identify the boundary circle bundles by a map covering the chosen identification $C_X\cong C_Y$ and reversing the normal-circle orientation. On collars, the two forms have the modeland the radial coordinate can be reversed while the circle coordinate is reversed so that the forms glue. A collar application of Moser's trick removes any discrepancy. This proves that the symplectic fiber sum along a square-zero surface $X{\#}_{C}Y$ has a natural symplectic form.

The displayed relation is an ordinary product relation, sowhere the relation eliminates $b=ac{a}^{-1}$. The Gompf realization theorem constructs a closed symplectic four-manifold $M_{\Gamma }$ with this fundamental group. Concretely, its construction starts from a product of a sufficiently high-genus surface and a torus, represents the two surviving generators and the relations by loops, and crosses the relevant loops with circle factors to obtain square-zero tori. Symplectic sums with copies of the rational elliptic surface kill the unwanted generators and impose the relations: the complement of a regular elliptic fiber is simply connected, so the Seifert-van Kampen theorem gives exactly ${\pi }_1(M_{\Gamma })=\Gamma \cong F_2$.

Finally, ${\mathbb{CP}}^{48}$ is simply connected and has real dimension $96$. With the product symplectic form,is a closed symplectic manifold of real dimension $100$ and has fundamental group $\Gamma$.

The symplectic neighborhood theorem says that a neighborhood of a compact symplectic submanifold is determined, up to symplectomorphism, by the restricted symplectic form and its symplectic normal bundle. More precisely, if $f:S_0\to S_1$ is a symplectomorphism and an isomorphism of symplectic normal bundles covers $f$, then that bundle isomorphism extends to a symplectomorphism between neighborhoods of $S_0$ and $S_1$.

If a compatible almost complex structure is prescribed along a closed symplectic submanifold $N\subset M$ and preserves $TN$, it extends to one on $M$. Choose a compatible structure separately on $TN$ and its symplectic normal bundle, use the associated metric along $N$, extend that metric to $M$, and apply the metric construction of a compatible almost complex structure.

If a symplectomorphism between compact symplectic submanifolds lifts to an isomorphism of their symplectic normal bundles, then it extends to a symplectomorphism between neighborhoods. Thus the germ of a symplectic neighborhood is determined by the restricted form and the symplectic normal bundle.

A symplectic vector bundle has a smooth fiberwise symplectic form, making each fiber a symplectic vector space. Its structure group reduces to the symplectic group. A symplectic normal bundle is an important example.