On the Space of Piecewise Linear Homeomorphisms of a Manifold

Geoghegan, Ross; Haver, William E.

doi:10.2307/2041861

Cited by 4 publications

(2 citation statements)

References 6 publications

(9 reference statements)

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“…This yields the same property for Hp L (X) c H U ?(X). By uniform local contractibility of Hup(X) 9 the same arguments as in the proof of [GH,Theorem 1] also show that the closure of #PL(ΛQ in H U p(X) is the union of components of Hup(X)…”

Section: Theoremmentioning

confidence: 81%

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On the space of Lipschitz homeomorphisms of a compact polyhedron

Sakai¹,

Wong²

1989

Pacific J. Math.

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Let X be a positive dimensional compact Euclidean polyhedron. Let H(X), HUP{X) and H PL (X) be respectively the space of homeomorphisms, the space of Lipschitz homeomorphisms and the space of piecewise-linear homeomorphisms of X onto itself. In this paper, we establish a homeomorphism taking the triple (H(X), H U p(X), H ?L (X)) onto the triple (H(X) x s, H U p(X) x Σ, H FL (X) X σ), where 5 = (-1, l) ω , Σ = {(*,) e s\sup\Xi\ < 1} and σ = {(*,) e s\x t = 0 except for finitely many /}. As a consequence we prove that when X is a PL manifold with dim c Φ 4 and dX = 0, in case dimZ = 5, (H(X),HUP(X)) is an (s, Σ)-manifold pair if H(X) is an s-manifold. We also prove that if dim* = 1 or 2, then (H(X),H PL (X)) is an (5, σ)-manifold pair and (H(X),H U p(X)) is an (s, Σ)-manifold. 0. Introduction. Let X = (X 9 d) and Y = (Y, p) be metric spaces. A map /: X -> 7 is said to be Lipschitz (resp. bi-Lipschitz) if there is some A: > 0 (resp. A: > 1) such that p{f\x),f{x')) < k rf(x ? x') (resp.

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Section: Theoremmentioning

confidence: 81%

“…Using uniform local contractibility of // PL (X), we can construct such a map (cf. the proof of [GH,Theorem 2]). Thus we have the following version of Theorem 2.3.…”

Section: Theoremmentioning

confidence: 99%

On the space of Lipschitz homeomorphisms of a compact polyhedron

Sakai¹,

Wong²

1989

Pacific J. Math.

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Characterizing the topology of infinite-dimensional 𝜎-compact manifolds

Mogilski¹

1984

Proc. Amer. Math. Soc.

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A metric space ( X , d ) (X,d) , which is a countable union of finite-dimensional compacta, is a manifold modelled on the space l 2 f = { ( x i ) ∈ l 2 l_2^f = \{ ({x_i}) \in {l_2} all but finitely many x i = 0 } {x_i} = 0\} iff X X is an ANR and the following condition holds: given ε > 0 \varepsilon > 0 , a pair of finite-dimensional compacta ( A , B ) (A,B) and a map f : A → X f:A \to X such that f | B f|B is an embedding, there is an embedding g : A → X g:A \to X such that g | B = f | B g\left | {B = f} \right |B and d ( f ( x ) , g ( x ) ) > ε d(f(x),g(x)) > \varepsilon for all x ∈ A x \in A . An analogous condition characterizes manifolds modelled on the space Σ = { ( x i ) ∈ l 2 : ∑ i = 1 ∞ ( i x i ) 2 > ∞ } \Sigma = \{ ({x_i}) \in {l_2}:\sum _{i = 1}^\infty {(i{x_i})^2} > \infty \} .

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