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Is there a connected compact manifold $M$ of dimension 4 such that $\pi_1 (M) = \mathbb{Z} * (\mathbb{Z} \oplus \mathbb{Z}_3)$? I had this question in a test yesterday. I think that the answer is no, because of the $\mathbb{Z}_3$ (it in general makes things not locally euclidean), however I do not know how to prove it.

Thanks in advance.

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up vote 2 down vote accepted

You can have any finitely generated group as the fundamental group of a compact four manifold.

For your example, take a lens space in 3d, cross it with a circle, then take a 3-sphere crossed with a circle, then attach the two spaces by connected sum.

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Could prove or give me a reference for the statement "You can have any finitely generated group as a fundamental group of a compact four manifold." ? –  user40276 Dec 4 '13 at 22:25
    
See this MO question:mathoverflow.net/q/30238/27933 –  Brian Rushton Dec 4 '13 at 22:28
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The reason is essentially that generators are single dimensional and relators are two dimensional, and two-dimensional things fit nicely into 5-dimensional space. Smoothing it into a manifold makes it 4-d. –  Brian Rushton Dec 4 '13 at 22:39

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