Suppose $M$ is a 3-manifold with $\pi_1(M)=\mathbb{Z}/p\mathbb{Z}$. Is it true that $M$ is diffeomorphic to a lens space $L(p,q)$?
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4Take a lens space and take a pointt away from it. The result is a 3-manifold with the same fundamental group. – Mariano Suárez-Álvarez Jun 19 '15 at 22:33
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you probably forgot to add the hypothesis that the manifold is closed, as evinced by my example above. You should then google for «fake lens spaces». – Mariano Suárez-Álvarez Jun 19 '15 at 22:41
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2@MarianoSuárez-Alvarez This is in the setting of 3-manifolds. Fake lens spaces are higher-dimensional phenomena. It is true that a closed 3-manifold with fundamental group $\mathbb Z/p$ is homeomorphic to some $L(p,q)$. – Jun 20 '15 at 04:38
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5Use the elliptization theorem to realize $M$ as a quotient of $S^3$ by an orthogonal matrix of order $p$. Use linear algebra to conjugate this to the desired form and see that $M$ is an $L(p,q)$. There is surely a more classical proof that such a manifold is homeomorphic to $L(p,q) # S$ where $S$ is a homotopy 3-sphere but I do not know the argument. – Jun 20 '15 at 04:55
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@MikeMiller, well, the intention was for the OP to google and find that fact! – Mariano Suárez-Álvarez Jun 20 '15 at 09:01
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@MarianoSuárez-Alvarez Ah, sorry (: – Jun 20 '15 at 15:37