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Is the group $\mathcal{U} (n)$ of all $n\times n$ unitary matrices over $\mathbb{C}$ a (local) path-connected space?

If so, what are the connected components of the unitary matrix group $\mathcal{U}(n)$? Is the number of components finite? What is the representative for each component? Is each components closed in norm topology?

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This is explained in en.wikipedia.org/wiki/Unitary_group . I voted to close as "too localized". –  Angelo Apr 10 '13 at 11:43
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closed as too localized by Angelo, Allen Knutson, Misha, S. Carnahan Apr 10 '13 at 14:42

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up vote 5 down vote accepted

Every matrix Lie group is a smooth manifold, hence it is path-connected if and only if it is connected. And $U(n)$ is compact and connected as a topological space (any unitary matrix can be diagonalized by a unitary matrix, this gives a path from it to the identity). It is not simply-connected, though. We have $\pi_1(U(n))\simeq \mathbb{Z}$.

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