Extending maps by injective $σ$-$Z$-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 $α\geq \aleph_0$ and $F$ is its $σ$-$Z$-set, then for every completely metrizable space $X$ of weight no greater than $α$ 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 $σ$-$Z$-set in $M$ disjoint from $F$. It is shown that if $f(\partial A)$ is contained in a locally closed $σ$-$Z$-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.