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Characterizing maximal compact subgroups

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Abstract

We prove that for a compact subgroup H of a locally compact almost connected group G, the following properties are mutually equivalent: (1) H is a maximal compact subgroup of G, (2) the coset space G/H is \({\mathbb{Q}}\) -acyclic and \({\mathbb{Z}/2\mathbb{Z}}\) -acyclic in Čech cohomology, (3) G/H is contractible, (4) G/H is homeomorphic to a Euclidean space, (5) G/H is an absolute extensor for paracompact spaces, (6) G/H is a G-equivariant absolute extensor for paracompact proper G-spaces having a paracompact orbit space.

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References

  1. Abels H.: Parallelizability of proper actions, global K-slices and maximal compact subgroups. Math. Ann. 212, 1–19 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. S. A. Antonyan Equivariant extension properties of coset spaces of locally compact groups and approximate slices, Topol. Appl., doi:10.1016/j.topol.2011.10.019.

  3. Antonyan S.A., de Neymet S.: Invariant pseudometrics on Palais proper G-spaces, Acta Math. Hung. 98, 41–51 (2003)

    Google Scholar 

  4. Arhangel’skii A., Tkachenko M.: Topological Groups and Related Structures. Atlantis Press/World Scientific, Amsterdam-Paris (2008)

    Book  MATH  Google Scholar 

  5. G. Bredon Introduction to Compact Transformation Groups, Academic Press, 1972.

  6. A. A. George Michael On three classical results about compact groups, Scientiae Mathematicae Japonicae Online, e-2011, no. 24, 271–273.

  7. Hoffmann B.: A compact contractible topological group is trivial. Archiv Math. 32, 585–587 (1979)

    Article  MATH  Google Scholar 

  8. Hofmann K.H., Morris S.A.: The structure of almost connected pro-Lie groups. J. of Lie Theory 21, 347–383 (2011)

    MathSciNet  MATH  Google Scholar 

  9. K. H. Hofmann and P. S. Mostert Elements of compact semigroups Charles E. Marril Books, INC., Columbus, Ohio, 1966.

  10. Hofmann K.H., Terp C.: Compact subgroups of Lie groups and locally compact groups. Proc. AMS 120, 623–634 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. Hatcher Algebraic Topology, Cambridge Univ. Press, 2001.

  12. G. Hochschild The structure of Lie groups, Holden-Day Inc., San Francisco, 1965

  13. Palais R.: On the existence of slices for actions of non-compact Lie groups. Ann. of Math. 73, 295–323 (1961)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Departamento de Matemáticas, Facultad de Ciencias, Universidad Nacional Autónoma de México, 04510, México Distrito Federal, Mexico

    Sergey A. Antonyan

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  1. Sergey A. Antonyan
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Correspondence to Sergey A. Antonyan.

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Dedicated to the memory of Professor E.G. Skljarenko

S. A. Antonyan was supported by grant 165246 from CONACYT (Mexico).

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Antonyan, S.A. Characterizing maximal compact subgroups. Arch. Math. 98, 555–560 (2012). https://doi.org/10.1007/s00013-012-0389-8

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  • DOI: https://doi.org/10.1007/s00013-012-0389-8

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