# Topology of imbeddings

The branch of topology in which one studies local topological properties of dispositions of closed subsets of a Euclidean space or manifold.

The topology of imbeddings arose in the work of A. Schoenflies, L. Antoine, P.S. Urysohn, and J. Alexander. Imbeddings in $E ^ {3}$ were studied in the 1950s. In particular, it was proved that an imbedding of a surface in $E ^ {3}$ can be topologically approximated by a polyhedral imbedding. The systematic study of the topology of imbeddings in $E ^ {n}$ for $n > 3$ began after the solution of the Schoenflies conjecture. Basically it came about in an environment of accumulation of facts and solutions to a large number of problems of a special character. Relations between methods of the theory of the topology of imbeddings and the geometric topology of manifolds were also clarified. At approximately the middle of the 1970s, the topology of imbeddings was formulated as an independent branch with its own themes, methods and problems. It was used to solve a number of basic problems in the geometric topology of manifolds: the existence was proved of a non-combinatorial triangulation of spheres of dimension $\geq 5$, the characterization of topological manifolds was obtained and the simply-connected four-dimensional manifolds were classified.

A topological imbedding of a space $X$ (as a rule, a manifold, a polyhedron or a compact set) in a Euclidean space $E ^ {n}$ is an arbitrary homeomorphism $f: X \rightarrow E ^ {n}$ from $X$ onto a space $f ( X) \subset E ^ {n}$. Sometimes, a topological imbedding is simply understood to be an inclusion $X \subset E ^ {n}$. Two imbeddings $f _ {1} , f _ {2} : X \rightarrow E ^ {n}$ are said to be equivalent if there exists a homeomorphism $h: E ^ {n} \rightarrow E ^ {n}$ such that $h \circ f _ {1} = f _ {2}$. If $h$ is an isotopy, then the imbeddings are said to be isotopic.

The simplest examples of non-equivalent imbeddings are obtained using knots (see Knot theory); it is much more difficult to construct non-equivalent imbeddings of zero-dimensional compacta or segments in $E ^ {3}$ (see Wild knot).

A Cantor set on a rectilinear segment lying in $E ^ {3}$ and a wild zero-dimensional Antoine compactum in $E ^ {3}$ are non-equivalent.

The fact that the basic problems of the theory of topological imbeddings concentrate on local properties is explained by the existence of so-called wild imbeddings, for which the regularity of the local structure is destroyed. The study of global properties of tame (locally flat) imbeddings is, as a rule, not included in the topology of imbeddings (cf. also Tame imbedding).

The following four theorems may be considered as fundamental in the theory of topological imbeddings.

### Theorem 1 (characterization).

An imbedding $X \subset E ^ {n}$ is tame if and only if the complement $Y = E ^ {n} \setminus X$ has the property 1-ULC (for an arbitrary $\epsilon > 0$ there exists a $\delta > 0$ such that each $\delta$-mapping $S ^ {1} \rightarrow Y$ is $\epsilon$-homotopically zero in $Y$).

### Theorem 2 (on close imbeddings).

Any two close tame imbeddings are isotopic by a small isotopy.

### Theorem 3 (on imbeddings in a trivial-dimensional domain).

If $2 \mathop{\rm dim} X + 2 \leq n$, then any two tame imbeddings are isotopic.

### Theorem 4 (on approximation).

Any imbedding can be approximated by a tame imbedding. With the exception of theorem 3, all these theorems have been proved only under certain restrictions on the dimensions; these restrictions are different for manifolds, polyhedra and compacta.

An imbedding of a manifold $X$ in $E ^ {n}$ is said to be tame (or locally flat) if for an arbitrary point $x \in X$ there exists a neighbourhood $U ( x)$ in $E ^ {n}$ such that the pair $( U ( x), U ( x) \cap X)$ is homeomorphic to the standard pair $( E ^ {n} , E ^ {r} )$ under a homeomorphism transferring the point to the origin of $E ^ {n}$.

Theorem 1 holds if $n \neq 4$ and $r \neq n - 2$ (if $r = n - 2$ and $n \geq 5$ the answer is also known: It is necessary for the complement $Y = E ^ {n} \setminus X$ to be, roughly speaking, homotopically equivalent to a circle).

Theorem 2 holds if $r \neq n - 2$ and $n \geq 5$ (the addition of a small knot shows that for $r = n - 2$, theorem 2 is obviously false; a condition for two imbeddings to be isotopically close is known when $r = n - 2$). In addition, if $X$ is the sphere $S ^ {r}$, it has been proved that an arbitrary tame imbedding $S ^ {r} \rightarrow E ^ {n}$ is isotopically standard if $r \neq n - 2$ (if $r = n - 2$ and $n \neq 4$ this is true if and only if the complement $E ^ {n} \setminus S ^ {n - 2 }$ is homotopically equivalent to a circle).

Theorem 4 holds if $r \neq n - 2$ and $n \neq 4$ (moreover, if $r = n - 2$ and $n \geq 6$ this theorem — as corresponding counterexamples show — is obviously false).

An imbedding of a polyhedron $X$ in $E ^ {n}$ is said to be tame if it is equivalent to a piecewise-linear imbedding. Theorems 1, 2 and 3 hold if $\mathop{\rm dim} X \leq n - 3$.

An imbedding of an $r$-dimensional compactum $X$ in $E ^ {n}$ is said to be tame if it is possible to remove it by an isotopy from an arbitrary rectilinear polyhedron $P \subset E ^ {n}$ of dimension $\leq n - r - 2$. Theorem 1 holds if $r \leq n - 3$ and $n \neq 4$, theorem 2 is, generally speaking, false (if $2r + 2 > n$), while theorem 4 holds for arbitrary $r$.

How to Cite This Entry:
Topology of imbeddings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Topology_of_imbeddings&oldid=51907
This article was adapted from an original article by M.A. Shtan'ko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article