= Four-manifold realization of finitely presented groups
Every finitely presented <group> is the <fundamental group> of a closed connected smooth oriented four-manifold. Build a compact four-dimensional <handle decomposition> with one zero-handle, a one-handle for each <generator of a group> and a two-handle for each <relator>. Turning the <handle decomposition> upside down shows that its boundary surjects on the <fundamental group>. Its <double of a manifold> has the same <fundamental group> by the <Seifert-van Kampen theorem>.
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