at.algebraic topology - finite generated group realized as fundamental group of manifolds

Theorem. Every finitely presentable group is the fundamental group of a closed 4-manifold.

Sketch proof. Let $langle a_1,ldots,a_mmid r_1,ldots, r_nrangle$ be a presentation. By van Kampen, the connected sum of $m$ copies of $S^1times S^3$ has fundamental group isomorphic to the free group on $a_1,ldots, a_m$. Now we can quotient by each relation $r_j$ as follows. Realise $r_j$ as a simple loop. A tubular neighbourhood of this looks like $S^1times D^3$. Do surgery and replace this tubular neighbourhood with $S^2times D^2$. This kills $r_j$. QED

There are many restrictions on 3-manifold groups. One of the simplest arises from the existence of Heegaard splittings. It follows easily that if $M$ is a closed 3-manifold then $pi_1(M)$ has a balanced presentation, meaning that $nleq m$.

Other obstructions to being a 3-manifold group were discussed in this MO question.

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