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.
