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.