Let $M$ be a manifold , and $\pi_1(M)=\mathbb{Z}$. then can we say, the double covering of $M$ exists and is unique?
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Classification theorem for covering maps says that coverings corresponds to subgroups of fundamental group, and what's more, $n$-fold coverings correspond to subgroups of index $n$. Now, does $\mathbb{Z}$ contain a subgroup of index $2$?
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I had seen in a book , because fundamental group of $GL(n,\mathbb{C})$ is $\mathbb{Z}$ so t has unique double cover but I couldn't undrestand – Jan 16 '14 at 16:50
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$GL(n, {\bf C})$ has the homotopy type of $U(n)$ which has the same fundamental group as $U(1)$ (this isomorphism is induced by $\det: U(n) \to U(1)$ and backwards by the inclusion $U(1) \to U(n)$ as scalars). – Marek Jan 16 '14 at 17:30