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A closed connected three-manifold has a handle decomposition with one zero-handle and one three-handle. To arrange this, cancel the zero-handles along a spanning tree of connecting one-handles; perform the corresponding construction in the dual handle decomposition to consolidate the three-handles. This does not require the three-manifold to be orientable.

If there are $r$ one-handles and $s$ two-handles, the Euler characteristic isMod-two Poincare duality pairs the Betti numbers of a closed odd-dimensional manifold, giving $\chi (Y)=0$. Thus $r=s$. The one-handles give $r$ generators of a group of the fundamental group, the attaching circles of the two-handles give $r$ relators, and the three-handle does not change the fundamental group. Therefore ${\pi }_1(Y)$ admits a balanced presentation.

Now letbe any finite group presentation. Start with a four-dimensional zero-handle and $r$ one-handles. Its fundamental group is the free group on the $x_i$. Represent the finitely many relators by disjoint embedded circles in its three-dimensional boundary; a small perturbation makes the circles disjoint. Choose framings of an embedded sphere and attach $s$ two-handles. The resulting compact connected oriented four-manifold $W$ has ${\pi }_1(W)\cong G$ by the Seifert-van Kampen theorem.

The map ${\pi }_1(\partial W)\to {\pi }_1(W)$ is surjective. One way to see this is to turn the handle decomposition upside down: the relative handles based on $\partial W$ have indices $2,3,4$, and so add no fundamental group generators. Also $\partial W$ is connected, since surgery on embedded circles in a connected three-manifold preserves connectedness.

Take the double of a manifoldIt is a closed connected smooth oriented four-manifold. The Seifert-van Kampen theorem givesBoth maps from ${\pi }_1(\partial W)$ are the same surjection, under the natural identification of the two copies. The pushout identifies the two copies of every element of $G$ and imposes no new relations: the fold homomorphism is inverse to either inclusion. ThusThis is four-manifold realization of finitely presented groups.