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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 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 $\Gamma$.