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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: 2013; Volume 223, Number 1048
ISBNs: 978-0-8218-5366-5 (print); 978-0-8218-9872-7 (online)
DOI: https://doi.org/10.1090/S0065-9266-2012-00643-5
Published electronically: October 9, 2012
Keywords: Nöbeling manifold characterization, Nöbeling space characterization, $Z$-set unknotting theorem, open embedding theorem, carrier, nerve theorem, regular cover, semiregular cover
MSC: Primary 55M10, 54F45; Secondary 54C20

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Table of Contents

1. Introduction and preliminaries

• 1. Introduction
• 2. Preliminaries

2. Reducing the proof of the main results to the construction of $n$-regular and $n$-semiregular ${\mathcal {N}_{n}}$-covers

• 3. Approximation within an $\mathcal {N}_{n}$-cover
• 4. Constructing closed $\mathcal {N}_{n}$-covers
• 5. Carrier and nerve theorems
• 6. Anticanonical maps and semiregularity
• 7. Extending homeomorphisms by the use of a “brick partitionings” technique
• 8. Proof of the main results

3. Constructing $n$-semiregular and $n$-regular ${\mathcal {N}_{n}}$-covers

• 9. Basic constructions in $\mathcal {N}_{n}$-spaces
• 10. Core of a cover
• 11. Proof of theorem

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 [Mladen Bestvina, Characterizing $k$-dimensional universal Menger compacta, Mem. Amer. Math. Soc. 71 (1988), no. 380] 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 $\nu ^n$ 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 $\nu ^\infty$ denote the Hilbert space. We define an $n$-dimensional Nöbeling manifold to be a Polish space locally homeomorphic to $\nu ^n$. The following theorem for $n = \infty$ is the characterization theorem of H. Toruńczyk [H. Toruńczyk, Characterizing Hilbert space topology, Fund. Math. 111 (1981), no. 3, 247–262]. We establish it for $n < \infty$, 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 [David W. Henderson and R. Schori, Topological classification of infinite dimensional manifolds by homotopy type, Bull. Amer. Math. Soc. 76 (1970), 121–124] for $n = \infty$. 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.