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 [10] 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 n-dimensional Nöbeling manifold to be a Polish space locally homeomorphic to ν n . The following theorem for n = ∞ is the characterization theorem of H. Toruńczyk [41]. 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 [27] 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. © 2012 American Mathematical Society.