On the boundedness of equivariant homeomorphism groups

Lech, Jacek; Michalik, Ilona; Rybicki, Tomasz

doi:10.7494/opmath.2018.38.3.395

Cited by 4 publications

(3 citation statements)

References 17 publications

(20 reference statements)

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“…The problem of the uniform perfectness and boundedness of groups of equivariant diffeomorphisms, leaf-preserving diffeomorphisms or diffeomorphisms which preserve some submanifolds, etc, has been also studied in [1], [2], [7], [13], [9], etc.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds

Fukui¹

2017

OpMath

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Abstract. We proved in [K. Abe, K. Fukui, On commutators of equivariant diffeomorphisms, Proc. Japan Acad. 54 (1978), [52][53][54] that the identity component Diff r G,c (M )0 of the group of equivariant C r -diffeomorphisms of a principal G bundle M over a manifold B is perfect for a compact connected Lie group G and 1 ≤ r ≤ ∞ (r = dim B + 1). In this paper, we study the uniform perfectness of the group of equivariant C r -diffeomorphisms for a principal G bundle M over a manifold B by relating it to the uniform perfectness of the group of C r -diffeomorphisms of B and show that under a certain condition, Diff r G,c (M )0 is uniformly perfect if B belongs to a certain wide class of manifolds. We characterize the uniform perfectness of the group of equivariant C r -diffeomorphisms for principal G bundles over closed manifolds of dimension less than or equal to 3, and in particular we prove the uniform perfectness of the group for the 3-dimensional case and r = 4.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

On the uniform perfectness of equivariant diffeomorphism groups for principal G manifolds

Fukui¹

2017

OpMath

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“…The boundedness of the diffeomorphism group Diff r (M ) of a smooth manifold M is studied by D. Burago, S. Ivanov and L. Polterovich [3] and T. Tsuboi [18,19,20], et al The case of equivariant diffeomorphism groups under free Lie group actions is studied by K. Fukui [1,7], J. Lech, I. Michalik and T. Rybicki [13] et al, and the case of leaf-preserving diffeomorphism groups is studied in [6,15,17].…”

Section: Introductionmentioning

confidence: 99%

Boundedness of bundle diffeomorphism groups over a circle

Fukui¹,

Yagasaki²

2022

Preprint

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In this paper we study boundedness of bundle diffeomorphism groups over a circle. For a fiber bundle π : M → S 1 with fiber N and structure group Γ and r ∈ Z ≥0 ∪ {∞} we distinguish an integer k = k(π, r) ∈ Z ≥0 and construct a function ν : Diffπ(M )0 → R k . When k ≥ 1, it is shown that the bundle diffeomorphism group Diffπ(M )0 is uniformly perfect and clbπ Diff r π (M )0 ≤ k + 3, if Diffρ,c(E)0 is perfect for the trivial fiber bundle ρ : E → R with fiber N and structure group Γ. On the other hand, when k = 0, it is shown that ν is a unbounded quasimorphism, so that Diffπ(M )0 is unbounded and not uniformly perfect. We also describe the integer k in term of the attaching map φ for a mapping torus π : M φ → S 1 and give some explicit examples of (un)bounded groups.

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“…The problem of the algebraic structure of homeomorphism groups, especially the boundedness and uniform perfectness of them, has drawn much attention. It has been studied among others in [1], [4], [5], [6], [7], [8], [9], [11], [12], [14], [15], [16] (see also references therein).…”

Section: Introductionmentioning

confidence: 99%