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Prove or disprove that closed 3-manifolds which are not simply connected cannot be embedded in three-dimensional Euclidean spaces. I am not a mathematics major and I am taking introductory topology this semester. But I need to apply this result for my research. Any help is much appreciated.

Thanks, Srikanth.

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    A 3-manifold without boundary embedded in $\mathbb{R}^3$ is embedded as an open subset of $\mathbb{R}^3$, hence is not compact.2010-11-03

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If your manifold is closed you must be using some definition that ensures it's compact? So if $f : M \to \mathbb R^3$ is an embedding, isn't the image simultanously compact and open?

edit in response to your comment: if $f : M \to \mathbb R^3$ is an embedding, let $B \subset M$ be an open subset of $M$ which is homeomorphic to an open ball in $\mathbb R^3$. You need to argue that $f(B)$ is open in $\mathbb R^3$. If your embedding is smooth there's a big theorem from calculus that gives you the result. If you're talking about topological embeddings you're going to need a tool. Have you studied the "invariance of dimension" theorem?

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    Not yet. But I will read up on it. Thanks a lot. This is really helpful.2010-11-03