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Characterization and topological rigidity of Nöbeling manifolds


About this Title

Andrzej Nagórko

Publication: Memoirs of the American Mathematical Society
Publication Year 2012: Volume 223, Number 1048
ISBNs: 978-0-8218-5366-5 (print); 978-0-8218-9872-7 (online)
DOI: http://dx.doi.org/10.1090/S0065-9266-2012-00643-5
Published electronically: October 9, 2012
Keywords: Nöbeling manifold characterization, Nöbeling space characterization, -set unknotting theorem, open embedding theorem, carrier, nerve theorem, regular cover, semiregular cover
MSC (2010): Primary 55M10, 54F45; Secondary 54C20

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Table of Contents


Chapters

  • Part 1. Introduction and preliminaries
  • Part 2. Reducing the proof of the main results to the construction of -regular and -semiregular -covers
  • Part 3. Constructing -semiregular and -regular -covers

Abstract


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 and is its counterpart in the realm of complete spaces. In particular we prove the Nöbeling manifold characterization conjecture. We define the -dimensional universal Nöbeling space to be the subset of consisting of all points with at most rational coordinates. To enable comparison with the infinite dimensional case we let denote the Hilbert space. We define an -dimensional Nöbeling manifold to be a Polish space locally homeomorphic to . The following theorem for is the characterization theorem of H. Toruńczyk. We establish it for , where it was known as the Nöbeling manifold characterization conjecture. Characterization theorem. An -dimensional Polish ANE()-space is a Nöbeling manifold if and only if it is strongly universal in dimension . The following theorem was proved by D. W. Henderson and R. Schori for . We establish it in the finite dimensional case. Topological rigidity theorem. Two -dimensional Nöbeling manifolds are homeomorphic if and only if they are -homotopy equivalent. We also establish the open embedding theorem, the -set unknotting theorem, the local -set unknotting theorem and the sum theorem for Nöbeling manifolds.

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