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In mathematics, a covering group of a topological group ''H'' is a covering space ''G'' of ''H'' such that ''G'' is a topological group and the covering map ''p'' : ''G'' → ''H'' is a continuous group homomorphism. The map ''p'' is called the covering homomorphism. A frequently occurring case is a double covering group, a topological double cover in which ''H'' has index 2 in ''G;'' examples include the Spin groups, Pin groups, and metaplectic groups. Roughly explained, saying that for example the metaplectic group ''Mp''2''n'' is a ''double cover'' of the symplectic group ''Sp''2''n'' means that there are always two elements in the metaplectic group representing one element in the symplectic group. ==Properties== Let ''G'' be a covering group of ''H''. The kernel ''K'' of the covering homomorphism is just the fiber over the identity in ''H'' and is a discrete normal subgroup of ''G''. The kernel ''K'' is closed in ''G'' if and only if ''G'' is Hausdorff (and if and only if ''H'' is Hausdorff). Going in the other direction, if ''G'' is any topological group and ''K'' is a discrete normal subgroup of ''G'' then the quotient map ''p'' : ''G'' → ''G''/''K'' is a covering homomorphism. If ''G'' is connected then ''K'', being a discrete normal subgroup, necessarily lies in the center of ''G'' and is therefore abelian. In this case, the center of ''H'' = ''G''/''K'' is given by : As with all covering spaces, the fundamental group of ''G'' injects into the fundamental group of ''H''. Since the fundamental group of a topological group is always abelian, every covering group is a normal covering space. In particular, if ''G'' is path-connected then the quotient group is isomorphic to ''K''. The group ''K'' acts simply transitively on the fibers (which are just left cosets) by right multiplication. The group ''G'' is then a principal ''K''-bundle over ''H''. If ''G'' is a covering group of ''H'' then the groups ''G'' and ''H'' are locally isomorphic. Moreover, given any two connected locally isomorphic groups ''H''1 and ''H''2, there exists a topological group ''G'' with discrete normal subgroups ''K''1 and ''K''2 such that ''H''1 is isomorphic to ''G''/''K''1 and ''H''2 is isomorphic to ''G''/''K''2. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「covering group」の詳細全文を読む スポンサード リンク
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