Extending Maps in Hilbert Manifolds

Piotr Niemiec. "Extending Maps in Hilbert Manifolds." Bulletin of the Polish Academy of Sciences. Mathematics 60.3 (2012): 295-306. <http://eudml.org/doc/286102>.

@article{PiotrNiemiec2012,
abstract = {Certain results on extending maps taking values in Hilbert manifolds by maps which are close to being embeddings are presented. Sufficient conditions on a map under which it is extendable by an embedding are given. In particular, it is shown that if X is a completely metrizable space of topological weight not greater than α ≥ ℵ₀, A is a closed set in X and f: X → M is a map into a manifold M modelled on a Hilbert space of dimension α such that $f(X∖A) ∩ \overline\{f(∂A)\} = ∅$, then for every open cover of M there is a map g: X → M which is -close to f (on X), coincides with f on A and is an embedding of X∖A into M. If, in addition, X∖A is a connected manifold modelled on the same Hilbert space as M, and $\overline\{f(∂A)\}$ is a Z-set in M, then the above map g may be chosen so that $g|_\{X∖A\}$ be an open embedding.},
author = {Piotr Niemiec},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
keywords = {Hilbert manifold; -set; embedding},
language = {eng},
number = {3},
pages = {295-306},
title = {Extending Maps in Hilbert Manifolds},
url = {http://eudml.org/doc/286102},
volume = {60},
year = {2012},
}

TY - JOUR
AU - Piotr Niemiec
TI - Extending Maps in Hilbert Manifolds
JO - Bulletin of the Polish Academy of Sciences. Mathematics
PY - 2012
VL - 60
IS - 3
SP - 295
EP - 306
AB - Certain results on extending maps taking values in Hilbert manifolds by maps which are close to being embeddings are presented. Sufficient conditions on a map under which it is extendable by an embedding are given. In particular, it is shown that if X is a completely metrizable space of topological weight not greater than α ≥ ℵ₀, A is a closed set in X and f: X → M is a map into a manifold M modelled on a Hilbert space of dimension α such that $f(X∖A) ∩ \overline{f(∂A)} = ∅$, then for every open cover of M there is a map g: X → M which is -close to f (on X), coincides with f on A and is an embedding of X∖A into M. If, in addition, X∖A is a connected manifold modelled on the same Hilbert space as M, and $\overline{f(∂A)}$ is a Z-set in M, then the above map g may be chosen so that $g|_{X∖A}$ be an open embedding.
LA - eng
KW - Hilbert manifold; -set; embedding
UR - http://eudml.org/doc/286102
ER -