A \(Z\)-set unknotting theorem for Nöbeling spaces. (English) Zbl 1163.55001

Levin, Michael

Fundam. Math. 202, No. 1, 1-41 (2009).

All spaces in this paper are separable metric spaces. A complete \(n\)-dimensional space \(X\) is said to be an \(n\)-dimensional Nöbeling space if the following conditions are satisfied:
(i) \(X\) is an absolute extensor in dimension \(n\);
(ii) For every map \(f:Y\rightarrow X\) from a complete metric space \(Y\) of dimension \(\leq n\) and for every open cover \(\mathcal {U}\) of \(X\) there is a closed embedding \(g:Y\rightarrow X\) such that for each \(y\in Y\) there is \(U \in \mathcal {U}\) that contains both \(f(y)\) and \(g(y)\).
The main result in this paper is the following:
Let \(X_1\) and \(X_2\) be \(n\)-dimensional Nöbeling spaces and let \(A_1\) and \(A_2\) be homeomorphic \(Z\)-sets in \(X_1\) and \(X_2\) such that \(X_1\setminus A_1\) and \(X_2\setminus A_2\) are homeomorphic to Nöbeling spaces modeled on triangulated manifolds. Then every homeomorphism \(f_A:A_1\rightarrow A_2\) extends to a homeomorphism \(f_X:X_1\rightarrow X_2\).

Reviewer: Liljana Babinkostova (Boise)

MSC:

55M10	Dimension theory in algebraic topology
54F45	Dimension theory in general topology

Keywords:

\(Z\)-set; Nöbeling spaces