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Transactions of the American Mathematical Society

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The space of conjugacy classes of a topological group


Author: Dennis Daluge
Journal: Trans. Amer. Math. Soc. 200 (1974), 345-353
MSC: Primary 57E99; Secondary 22A05
MathSciNet review: 0368057
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Abstract: The space $ {G^\char93 }$ of conjugacy classes of a topological group $ G$ is the orbit space of the action of $ G$ on itself by inner automorphisms. For a class of connected and locally connected groups which includes all analytic $ [Z]$-groups, the universal covering space of $ {G^\char93 }$ may be obtained as the space of conjugacy classes of a group which is locally isomorphic with $ G$, and the Poincaré group of $ {G^\char93 }$ is found to be isomorphic with that of $ G/G'$, the commutator quotient group. In particular, it is shown that the space $ {G^\char93 }$ of a compact analytic group $ G$ is simply connected if and only if $ G$ is semisimple. The proof of this fact has not appeared in the literature, even though more specialized methods are available for this case.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0368057-3
Keywords: Space of conjugacy classes, homotopy, commutator quotient group, maximal toroid, Weyl group, conjugacy class, covering space, simply connected
Article copyright: © Copyright 1974 American Mathematical Society