Opuscula Math. 38, no. 3 (2018), 395-408
https://doi.org/10.7494/OpMath.2018.38.3.395

 
Opuscula Mathematica

On the boundedness of equivariant homeomorphism groups

Jacek Lech
Ilona Michalik
Tomasz Rybicki

Abstract. Given a principal \(G\)-bundle \(\pi:M\to B\), let \(\mathcal{H}_G(M)\) be the identity component of the group of \(G\)-equivariant homeomorphisms on \(M\). The problem of the uniform perfectness and boundedness of \(\mathcal{H}_G(M)\) is studied. It occurs that these properties depend on the structure of \(\mathcal{H}(B)\), the identity component of the group of homeomorphisms of \(B\), and of \(B\) itself. Most of the obtained results still hold in the \(C^r\) category.

Keywords: principal \(G\)-manifold, equivariant homeomorphism, uniformly perfect, bounded, \(C^r\) equivariant diffeomorphism.

Mathematics Subject Classification: 57S05, 58D05, 55R91.

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  • Jacek Lech
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
  • Ilona Michalik
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
  • Tomasz Rybicki
  • AGH University of Science and Technology, Faculty of Applied Mathematics, al. A. Mickiewicza 30, 30-059 Krakow, Poland
  • Communicated by P.A. Cojuhari.
  • Received: 2017-12-19.
  • Revised: 2018-01-23.
  • Accepted: 2018-02-02.
  • Published online: 2018-03-19.
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Cite this article as:
Jacek Lech, Ilona Michalik, Tomasz Rybicki, On the boundedness of equivariant homeomorphism groups, Opuscula Math. 38, no. 3 (2018), 395-408, https://doi.org/10.7494/OpMath.2018.38.3.395

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