Extending maps by injective \(\sigma\)-\(Z\)-maps in Hilbert manifolds

Abstract

The aim of the paper is to prove that if \(M\) is a metrizable manifold modelled on a Hilbert space of dimension \(\alpha \geq \aleph_0\) and \(F\) is its \(\sigma\)-\(Z\)-set, then for every completely metrizable space \(X\) of weight no greater than \(\alpha\) and its closed subset \(A\), for any map \(f: X \to M\), each open cover \(\mathcal{U}\) of \(M\) and a sequnce \((A_n)_n\) of closed subsets of \(X\) disjoint from \(A\) there is a map \(g: X \to M\) \(\mathcal{U}\)-homotopic to \(f\) such that \(g\bigr|_A = f\bigr|_A\), \(g\bigr|_{A_n}\) is a closed embedding for each \(n\) and \(g(X \setminus A)\) is a \(\sigma\)-\(Z\)-set in \(M\) disjoint from \(F\). It is shown that if \(f(\partial A)\) is contained in a locally closed \(\sigma\)-\(Z\)-set in \(M\) or \(f(X \setminus A) \cap \bar{f(\partial A)} = \empty\), the map \(g\) may be taken so that \(g\bigr|_{X \setminus A}\) be an embedding. If, in addition, \(X \setminus A\) is a connected manifold modelled on the same Hilbert space as \(M\) and \(\bar{f(\partial A)}\) is a \(Z\)-set in \(M\), then there is a \(\mathcal{U}\)-homotopic to \(f\) map \(h: X \to M\) such that \(h\bigr|_A = f\bigr|_A\) and \(h\bigr|_{X \setminus A}\) is an open embedding.