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From studiosus' answer to

A 3-manifold with fundamental group isomorphic to a surface group.

a closed 3-manifold cannot have a fundamental group isomorphic to that of a closed surface of genus $\geq 2$.

What about genus 1? I wonder if there exist a closed 3-manifold with fundamental group isomorphic to the fundamental group of the 2-dimensional torus or the Klein bottle.

Thanks in advance

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RRR
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    I found the answer here

    http://mathoverflow.net/questions/31904/closed-3-manifolds-with-free-abelian-fundamental-groups

    Thanks again

     – RRR Dec 11 '14 at 10:10
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    Actually, the proof in my answer only uses the assumption that genus is $\ge 1$. – Moishe Kohan Dec 12 '14 at 20:17

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