The embedding of compacta in Euclidean space

M. A. Shtan'ko

Sbornik. Mathematics, Tome 12 (1970) no. 2, pp. 234-254

Résumé

Recently the fundamental importance of the $1-ULC$ property of the complementary space in describing a given embedding in $E^n$ has become clear. “Wild” embeddings in $E^n$ are characterized by the absence of the $1-ULC$ property. In this paper “tame” and “wild” embeddings in $E^n$ of arbitrary compacta in codimension at least 3 are defined. For this purpose the notion of the “dimension of embedding” of compacta in $E^n$ is introduced. The main theorem asserts that an embedding of a compactum in $E^n$, $n\geqslant6$, is “wild” if and only if the complementary space is not $1-ULC$. Bibliography: 23 titles.