How do we represent relations in a free group by loops in a manifold?

Question

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?

Answer 1

See the same question here with a nice answer :

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