Wild imbedding

of a topological space $X$ in a topological space $Y$

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 $n$-dimensional Euclidean space $\mathbf R^n$ is taken as $Y$.

1) Let $M$ be a $k$-dimensional topological manifold (cf. Topology of manifolds). A topological imbedding $g\colon M\to\mathbf R^n$ (cf. Topology of imbeddings) is called wild if there does not exist a homeomorphism of $\mathbf R^n$ onto itself which would convert $g(M)$ into a locally flat submanifold of $\mathbf R^n$.

2) Let $P$ be a $k$-dimensional polyhedron. A topological imbedding $g\colon P\to\mathbf R^n$ is called wild if there does not exist a homeomorphism of $\mathbf R^n$ onto itself which would convert $g(P)$ into a polyhedron (i.e. into a body having a certain triangulation) in $\mathbf R^n$.

3) Let $K$ be a $k$-dimensional locally compact space. A topological imbedding $g\colon K\to\mathbf R^n$ is called wild if there does not exist a homeomorphism of $\mathbf R^n$ onto itself which would convert $g(K)$ into a subset of the $k$-dimensional Menger compactum $M_n^k$.

If the dimension $k\leq n-3$ and if $n\geq5$, then the properties introduced in all three cases are characterized by the following locally homotopic property: An imbedding is wild if and only if $g(X)$ does not satisfy the property $1-ULC$ (cf. Topology of imbeddings). The situation is much more complicated for the codimensions $n-k=1$ and $2$: The problem has been solved for manifolds of codimension 1 for $n\geq6$, 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 $Y$ is an $n$-dimensional manifold — topological or piecewise linear.