= Closed-manifold realization of a finitely presented fundamental group
{title2=$\pi_1(B)=T,\quad\dim B\geq4$}
Every <finitely presented group> is the <fundamental group> of a closed connected smooth manifold in dimension at least four. Build a compact zero/one/two-handle manifold $W$ with the generators and relators of the presentation. Disjoint attaching loops can be embedded in its boundary of dimension at least three. The <Seifert-van Kampen theorem> gives $\pi_1(W)=T$, and the dual relative handle structure makes $\pi_1(\partial W)\to T$ surjective. Doubling $W$ along its boundary gives the amalgam of two copies of $T$ along that same surjection, which is again $T$. This construction underlies <binary families of nonhomeomorphic Sunada quotients>.
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