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Hypothesis: Let $F$ be a finitely presented group s.t.

$$ F = \left\langle S \mid R \right\rangle $$

Let $X$ be a $4$-manifold.

Question: I've seen it asserted that we can represent each relation in $R$ by a loop in $X$. But in what sense can we represent relations in $R$ by loops in $X$? What precisely does this mean?

user1770201
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See the same question here with a nice answer :

Heading ##https://mathoverflow.net/questions/15411/finite-generated-group-realized-as-fundamental-group-of-manifolds

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Walid Taamallah
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  • Thanks for the link. I made a post asking questions about the outline more or less presented in that MathOverflow link. It might be of interest to you: http://math.stackexchange.com/questions/788097/showing-that-every-finitely-presented-group-has-a-4-manifold-with-it-as-its-fu – user1770201 May 09 '14 at 23:40