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 $\alpha \ge {\aleph }_0$ 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 $g{|}_A=f{|}_A$, $g{|}_{A_n}$ 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 $f(X\setminus A)\cap \overline{f(\partial A)}=\varnothing$, the map $g$ may be taken so that $g{|}_{X\setminus A}$ be an embedding. If, in addition, $X\setminus A$ is a connected manifold modelled on the same Hilbert space as $M$ and $\overline{f(\partial A)}$ is a $Z$-set in $M$, then there is a $\mathcal{U}$-homotopic to $f$ map $h:X\to M$ such that $h{|}_A=f{|}_A$ and $h{|}_{X\setminus A}$ is an open embedding.

arXiv comment12 pages