Some Topics in Geometric Topology

Kawamura, Kazuhiro

doi:10.1016/b978-044450980-2/50011-x

Cited by 2 publications

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“…Proposition 2. 27. For every open subset U of R 2n+1 the inclusion of U ∩ ν n into U is a weak n-homotopy equivalence.…”

Section: Proofmentioning

confidence: 99%

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Characterization and topological rigidity of Nöbeling manifolds

Nagórko¹

2012

Memoirs of the AMS

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We develop a theory of Nöbeling manifolds similar to the theory of Hilbert space manifolds. We show that it reflects the theory of Menger manifolds developed by M. Bestvina [8] and is its counterpart in the realm of complete spaces. In particular we prove the Nöbeling manifold characterization conjecture.We define the n-dimensional universal Nöbeling space ν n to be the subset of R 2n+1 consisting of all points with at most n rational coordinates. To enable comparison with the infinite dimensional case we let ν ∞ denote the Hilbert space. We define an ndimensional Nöbeling manifold to be a Polish space locally homeomorphic to ν n . The following theorem for n = ∞ is the characterization theorem of H. Toruńczyk [36]. We establish it for n < ∞, where it was known as the Nöbeling manifold characterization conjecture.Characterization theorem An n-dimensional Polish ANE(n)-space is a Nöbeling manifold if and only if it is strongly universal in dimension n.The following theorem was proved by D. W. Henderson and R. Schori [23] for n = ∞. We establish it in the finite dimensional case.Topological rigidity theorem. Two n-dimensional Nöbeling manifolds are homeomorphic if and only if they are n-homotopy equivalent.We also establish the open embedding theorem, the Z-set unknotting theorem, the local Z-set unknotting theorem and the sum theorem for Nöbeling manifolds.

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“…Proposition 2. 27. For every open subset U of R 2n+1 the inclusion of U ∩ ν n into U is a weak n-homotopy equivalence.…”

Section: Proofmentioning

confidence: 99%

“…The following theorem for n = ∞ is the characterization theorem of H. Toruńczyk [36]. We establish it for n < ∞, where it was known as the Nöbeling manifold characterization conjecture [13,15,27,39].…”

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confidence: 99%