Wild imbedding
of a topological space in a topological space
An imbedding which is topologically non-equivalent to an imbedding from a certain class of chosen imbeddings known as tame or nice imbeddings. The cases listed below are the most useful; the -dimensional Euclidean space
is taken as
.
1) Let be a
-dimensional topological manifold (cf. Topology of manifolds). A topological imbedding
(cf. Topology of imbeddings) is called wild if there does not exist a homeomorphism of
onto itself which would convert
into a locally flat submanifold of
.
2) Let be a
-dimensional polyhedron. A topological imbedding
is called wild if there does not exist a homeomorphism of
onto itself which would convert
into a polyhedron (i.e. into a body having a certain triangulation) in
.
3) Let be a
-dimensional locally compact space. A topological imbedding
is called wild if there does not exist a homeomorphism of
onto itself which would convert
into a subset of the
-dimensional Menger compactum
.
If the dimension and if
, then the properties introduced in all three cases are characterized by the following locally homotopic property: An imbedding is wild if and only if
does not satisfy the property
(cf. Topology of imbeddings). The situation is much more complicated for the codimensions
and
: The problem has been solved for manifolds of codimension 1 for
, but has not been fully solved for imbeddings of codimension 2 both for manifolds and for polyhedra. All that has been said is also meaningful if
is an
-dimensional manifold — topological or piecewise linear.
Wild imbedding. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Wild_imbedding&oldid=31602