Characterization and topological rigidity of Nöbeling manifolds

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 [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.

A construction of {N}

“…A space is a Nöbeling manifold if it is locally homeomorphic to a Nöbeling space. There is analogous rigidity result for separable Nöbeling manifolds proved in [14]. In particular it implies that a separable Nöbeling manifold is homeomorphic to ν n if and only if it has vanishing homotopy groups in dimensions less than n.…”

“…An abstract n-dimensional Nöbeling space of weight κ is an abstract n-dimensional Nöbeling manifold of weight κ that has vanishing homotopy groups of dimensions less than n. Note that in the separable case, by Open Embedding Theorem [14], every separable n-dimensional Nöbeling manifold is homeomorphic to an open subset of ν n .…”

Asymptotic Dimension

“…In non-compact case there are universal spaces ν n in dimension n called Nöbeling spaces, defined as the set of all points in R 2n+1 with at most n rational coordinates. An analogous topological characterization of the Nöbeling space ν n was given by Nagorko [76] (see also [69] for a different treatment). Since by its construction ν n is a subset of R 2n+1 , we obtain Theorem 15 (Nöbeling-Pontryagin Theorem).…”

A short proof of Toruńczyk’s characterization theorems

“…These results, first proved by Toruńczyk [13,14,15], are widely known and were applied in diverse settings. They also inspired the characterization results of the universal Menger spaces by Bestvina [4] and the recent work on Nöbeling spaces [1,2,3], [12], [8,9].…”