UDC 513.83

MATHEMATICS

M. SHTAN’KO

EMBEDDING COMPACTA IN EUCLIDEAN SPACE

(Presented by Academician P. S. Aleksandrov on 13 XII 1968)

1. An embedding of a space \(X\) in a space \(Y\) is an arbitrary homeomorphic mapping \(X \to Y\); we shall omit the notation of the mapping and by an embedding understand simply the inclusion \(X \subset Y\).

Here we shall obtain some results for embeddings of compacta of dimension \(\le n - 3\) in Euclidean space \(E^n\). The basic concept is the embedding dimension of a compactum \(K\) in the space \(E^n\), \(K \subset E^n\), denoted by \(\operatorname{dem} K\), (\(\operatorname{dem} =\) dimension of embedding), which is an integer-valued invariant, unchanged under homeomorphisms of \(E^n\) onto itself (nonnegative for nonempty compacta), reminiscent of the characterization of the ordinary topological dimension \(\dim K\) of a compactum \(K\) by P. S. Aleksandrov’s method of approximating compacta by polyhedra (see \((^1)\)). In the case \(\dim K \le n - 3\) this invariant depends on the embedding of \(K\) in \(E^n\).

We shall call an embedding \(K \subset E^n\), where \(\dim K \le n - 3\), tame if \(\operatorname{dem} K = \dim K\), and wild otherwise. (We note that always \(\operatorname{dem} K \ge \dim K\).)

In the case of zero-dimensional compacta and \(r\)-dimensional topological polyhedra in \(E^n\) \((2r + 2 \le n)\), the definition of a tame embedding introduced by us coincides with the previously known one (see \((^2)\)). For nontrivial dimensions it is unknown whether the concept of a tame embedding introduced by us coincides with the classical one (see unsolved questions III, III′). Menger’s problem on universal compacta is also considered (see \((^3)\)), which turns out to be connected with embeddings.

Let us recall some simplest notation and concepts. \(U(K,\varepsilon)\) denotes the \(\varepsilon\)-neighborhood of the compactum \(K\) in \(E^n\). A compactum \(K\) lying in \(E^n\) is called cellularly separated if in every neighborhood \(U\) of the compactum \(K\) one can inscribe such a neighborhood \(V\) that \(U \supset \overline V \supset V \supset K\), and \(\overline V\) is homeomorphic to a disjoint union of a finite number of topological cubes. A homotopy \(F_t : X \to X\), \(t \in [0,1]\), is called a pseudoisotopy if for every \(t < 1\), \(F_t\) is a homeomorphism of the space \(X\) onto itself, and an isotopy if \(F_t\) is a homeomorphism for every \(t\).

Definition 1. The embedding dimension of a nonempty compactum \(K \subset E^n\) is the least integer \(\operatorname{dem} K = r \ge 0\) such that for every \(\varepsilon > 0\) there exists an \(\varepsilon\)-pseudoisotopy \(F_t : E^n \to E^n\), \(t \in [0,1]\), of the space \(E^n\) onto itself, fixed outside \(U(K,\varepsilon)\), and such that \(F_1(K)=P\), where \(P\) is a rectilinear polyhedron in \(E^n\), \(\dim P = r\).

The embedding dimension of the empty compactum in \(E^n\) is by definition equal to \(-1\).

Main theorem. If \(K \subset E^n\), \(n \ne 3\), \(\dim K \ge n - 2\), then always \(\operatorname{dem} K = \dim K\); if \(n \ge 5\) and \(\dim K \le n - 3\), then the embedding dimension assumes two (and only two) values: either \(\dim K\), or \(n - 2\), the latter occurring if and only if \(E^n \setminus K\) does not have the property \(1\text{—}ULC\).

2. Definition 2. The dual embedding dimension of a nonempty compactum \(K \subset E^n\) is called the least

an integer $\operatorname{Dem} K=r\geqslant 0$ such that, whatever rectilinear polyhedron $P\subset E^n$, $\dim P\leqslant n-r-1$, for every $\varepsilon>0$ there exists an $\varepsilon$-isotopy $F_t:E^n\to E^n$, $t\in[01]$, fixed outside $U(K\cap P,\varepsilon)$ and such that $F_1(K)\cap P=\Lambda$. The dual embedding dimension of the empty compactum in $E^n$ is, by definition, equal to $-1$.

Let $M^n$ be the geometric cube in $E^n$. Divide the cube $M^n$ into $3^n$ equal cubes of first rank by hyperplanes drawn perpendicular to the edges of the cube $M^n$ at the points dividing the edges into three equal parts; from these $3^n$ cubes select those which intersect the $r$-dimensional skeleton of the cube $M^n$, and denote their union by $M_{r,1}^n$. Each cube entering into $M_{r,1}^n$ is divided in an analogous manner into $3^n$ cubes of second rank; the sum of all the analogously selected cubes of second rank is denoted by $M_{r,2}^n$, and so on. We obtain a decreasing sequence of closed sets

\[ M^n\supset M_{r,1}^n\supset M_{r,2}^n\supset\cdots \]

The compactum $M_r^n=\bigcap_{k=1}^{\infty}M_{r,k}^n$ is the Menger compactum (see (3)).

For a compactum $K\subset E^n$ the following assertions follow from the definitions or were known earlier.

(a) Each of the conditions $\operatorname{dem}K=0$ or $\operatorname{Dem}K=0$, respectively, is equivalent to the conjunction of two conditions: $(1^0)$ $\dim K=0$; $(2^0)$ $K$ is cellularly decomposable.

b) If $n\geqslant 3$ and $A_0$ is a wild zero-dimensional compactum in $E^n$, $A_0\subset E^n$, then $\operatorname{dem}A_0=\operatorname{Dem}A_0=n-2$.

(c) There exists a compactum $K$ in $E^3$ such that $\dim K=1$, whereas $\operatorname{dem}K=\operatorname{Dem}K=2$, for example, the compactum constructed by Bothe (4).

(d) The embedding dimensions $\operatorname{dem}K$, $\operatorname{Dem}K$ possess the properties of invariance and monotonicity, i.e. the corresponding value of the embedding dimension does not change if the space is subjected to a homeomorphism onto itself, and, on passing to a subcompactum, the corresponding value of the embedding dimension does not increase.

(e) $\operatorname{dem}(M_r^n)=\operatorname{Dem}(M_r^n)=\dim(M_r^n)=r$, $M_r^n\subset E^n$.

1. The main theorem can be formulated as follows:
   If a compactum $K$ lies in $E^n$, $n\ne3$, $\dim K\geqslant n-2$, then always $\operatorname{dem}K=\dim K$; if $n\geqslant5$ and $\dim K\leqslant n-3$, then the compactum $K$ is tame in $E^n$ if and only if $E^n\setminus K$ has the property $1\text{-}ULC$; but if $E^n\setminus K$ does not have the property $1\text{-}ULC$, then always $\operatorname{dem}K=n-2$.

Remark. By virtue of Bing’s theorem this assertion is also true for $n=3$ and $\dim K=0$ (5). An analogous assertion was earlier proved by McMillan for zero-dimensional compacta in $E^n$, $n\geqslant5$ (6).

Theorem 1. For every compactum $K$ lying in $E^n$, $\operatorname{dem}K=\operatorname{Dem}K$.

Theorem 2. Let compacta $K$ and $C$ lie in the space $E^n$, $n\geqslant5$, $\dim K=r$, $2r+2\leqslant n$, and let $h:K\to C$ be a homeomorphism; if $E^n\setminus K$ and $E^n\setminus C$ have the property $1\text{-}ULC$, then for every $\varepsilon>\rho(h,1)$ there exists an $\varepsilon$-isotopy $F_t:E^n\to E^n$, $t\in[01]$, fixed outside the prescribed neighborhood $V\supset X(K,h,C)$ and such that $F_1|_K=h$. (Here $X(K,h,C)$ denotes the union of the segments $\overline{xh(x)}$, $x\in K$.)

Let us note a particular case of Theorem 2, obtained earlier by Bryant and Seebeck: let a topological polyhedron $P$ lie in $E^n$, $n\geqslant5$, $\dim P=r$, $2r+2\leqslant n$; if $E^n\setminus P$ has the property $1\text{-}ULC$, then for every $\varepsilon>0$ there exists an $\varepsilon$-isotopy of the space $E^n$ onto itself, fixed outside the $\varepsilon$-neighborhood of the polyhedron $P$ and carrying it into a piecewise-linear one (7).

Theorem 3. If a compactum $K$ lies in $E^n$, then the condition $\operatorname{Dem}K=r$ is equivalent to the fact that $M_r^n\subset E^n$ is a Menger compactum of the least dimension for which there exists an isotopy $F_t:E^n\to E^n$, $t\in[01]$, fixed outside some bounded domain $V$ and such that $F_1(K)\subset M_r^n$.

A somewhat weaker assertion was proved earlier by Bothe \((^8)\).

Corollary 1. The Menger compactum \(M_r^n\), \(M_r^n \subset E^n\), \(n \geqslant 5\), \(r \leqslant n-3\), is universal for all \(r\)-dimensional compacta in \(E^n\), \(K \subset E^n\), such that \(E^n \setminus K\) has the property \(1\text{-}ULC\); moreover, in this case there exists an isotopy \(F_t:E^n \to E^n\), \(t \in [01]\), fixed outside some bounded domain \(V\) and such that \(F_1(K) \subset M_r^n\).

Corollary 2. The Menger compactum \(M_r^n\), \(M_r^n \subset E^n\), \(n \geqslant 5\), \(r \leqslant n-3\), is universal for all \(r\)-dimensional compacta \(K\) lying in the hyperplane \(E^{n-1}\), \(K \subset E^{n-1} \subset E^n\); moreover, in this case there exists an isotopy \(F_t:E^n \to E^n\), \(t \in [01]\), fixed outside some bounded domain \(V\) and such that \(F_1(K) \subset M_r^n\).

4. For the proof of the main theorem the absorption lemma \((^9)\) is used. We briefly describe the idea of the proof.

We shall prove that if \(\dim K \leqslant n-3\) and \(E^n \setminus K\) has the property \(1\text{-}ULC\), then \(\operatorname{Dem} K \leqslant \dim K\)*; hence, taking into account that always \(\operatorname{Dem} K \geqslant \dim K\), we shall have \(\operatorname{Dem} K=\dim K\). All the remaining assertions of the theorem follow easily from the definitions. The proof of the indicated assertion is carried out by induction on the increasing dimension of the polyhedron \(P\), \(\dim P=p\), from which, according to Definition 2, the compactum \(K\) is removed, \(0 \leqslant p \leqslant n-r-1\), where \(r=\dim K\), \(n \geqslant 5\). Since \(\dim K \leqslant n-3\), the inductive assertion is fulfilled for the values \(p=0,1\). Assuming that it is fulfilled for some \(p\), \(0 \leqslant p<n-r-1\), we prove that it is fulfilled for \(p+1\).

Let \(P\) be a rectilinear polyhedron in \(E^n\), \(\dim P=p+1\). By the induction hypothesis we may assume that \(K \cap P^p=\Lambda\), where \(P^p\) is the \(p\)-skeleton of the polyhedron \(P\) in some \(\varepsilon\)-triangulation of the polyhedron \(P\). Let \(\Delta^{p+1}\) be some simplex in the given \(\varepsilon\)-triangulation of the polyhedron \(P\) such that \(\Delta^{p+1}\cap K\ne \Lambda\); since \(\partial(\Delta^{p+1})\cap K=\Lambda\), there exists a simplex \(\Delta_0^{p+1}\), concentric with the simplex \(\Delta^{p+1}\), \(\operatorname{int}\Delta^{p+1}\supset \Delta_0^{p+1}\), and, if \(C=\Delta^{p+1}\setminus \operatorname{int}\Delta_0^{p+1}\), then \(C\cap K=\Lambda\).

Let \(L^n(\Delta^{p+1})=L^n\) be a convex polyhedral \(n\)-cube in \(E^n\) such that \(L^n\cap P=\Delta^{p+1}\), \(\partial\Delta^{p+1}\subset \partial L^n\), \(\operatorname{int}\Delta^{p+1}\subset \operatorname{int}L^n\). Since \(E^n\setminus K\) has the property \(1\text{-}ULC\) and \(p \leqslant n-r-2\), one can show that the sphere \(\partial\Delta_0^{p+1}\) contracts to a point in the set \((\operatorname{int}L^n)\setminus K\); therefore there exists a polyhedral homotopy film \(\Pi_0\) in \((\operatorname{int}L^n)\setminus K\), effecting the contractibility of the \(p\)-dimensional sphere \(\partial\Delta_0^{p+1}\) in \((\operatorname{int}L^n)\setminus K\) to a point; putting it in general position, we may assume that the self-intersection of this film has minimal dimension. If the self-intersection is empty and \(\Pi_0\cap C=\partial\Pi_0=\partial\Delta_0^{p+1}\), then the resulting piecewise-linear disk \(\Pi=\Pi_0\cup C\) forms a piecewise-linear pair of cells \((L^n,\Pi)\); if it is unknotted, then \(\Pi\) is carried onto \(\Delta^{p+1}\) fixed outside \(\operatorname{int}L^n\), whereby the compactum \(K\) is removed from \(\Delta^{p+1}\). If, however, the self-intersection of \(\Pi_0\) is nonempty, then either the absorption lemma is applied in a special way, or Ziman’s device (with cones) is applied directly to eliminate the self-intersection.

5. Questions. (I). Let a compactum \(K\) lie in \(E^n\) and \(\dim K \leqslant n-3\). Does there exist a homeomorphism \(h:K\to E^n\) such that \(E^n\setminus h(K)\) has the property \(1\text{-}ULC\)?

\((\mathrm{I}')\). Let a compactum \(K\) lie in \(E^n\) and \(\dim K \leqslant n-3\). Does there exist, for every \(\varepsilon>0\), a homeomorphic \(\varepsilon\)-shift \(h_\varepsilon:K\to E^n\) such that \(E^n\setminus h(K)\) has the property \(1\text{-}ULC\)?

\((\mathrm{II})\). Let \(P\) be a topological polyhedron in \(E^n\), \(\dim P\leqslant n-3\). Does there exist a piecewise-linear embedding \(h:P\to E^n\)?

\((\mathrm{II}')\). Let \(P\) be a topological polyhedron in \(E^n\), \(\dim P\leqslant n-3\). Does there exist, for every \(\varepsilon>0\), a homeomorphic piecewise-linear \(\varepsilon\)-shift \(h_\varepsilon:P\to E^n\)?

* According to Theorem 1, in the formulation of the main theorem we may replace \(\operatorname{dem} K\) by \(\operatorname{Dem} K\).

(III). Let \(P\) be a topological polyhedron in \(E^n\), \(\dim P \leq n-3\), and suppose that \(E^n \setminus P\) has the property \(1\text{-}ULC\). Will the polyhedron \(P\) be tame in \(E^n\)?

(III′). Let \(P\) be a topological polyhedron in \(E^n\), \(\dim P \leq n-3\), and suppose that \(E^n \setminus P\) has the property \(1\text{-}ULC\). Can the polyhedron \(P\) be carried into a piecewise-linear one by means of an \(\varepsilon\)-isotopy of the space \(E^n\), fixed outside the \(\varepsilon\)-neighborhood of \(P\)?

(We note that questions I—III′ have positive solutions in trivial dimensions.)

(IV). Let a compactum \(K\) lie in \(E^n\), \(K \subset E^n\), \(n=4\). Will the equality \(\dim K=\operatorname{Dem} K\) hold when \(\dim K=0\) or \(1\) and \(E^n \setminus K\) has the property \(1\text{-}ULC\)?

Received
1 XI 1968

REFERENCES

¹ P. S. Aleksandrov, Combinatorial Topology, Moscow, 1947.
² L. V. Keldysh, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 81 (1966).
³ K. Menger, Dimensionstheorie, Leipzig—Berlin, 1928.
⁴ H. Bothe, Fund. Math., 54, No. 3, 251 (1964).
⁵ R. Bing, Pacific J. Math., 11, No. 2, 435 (1961).
⁶ D. McMillan, Bull. Am. Math. Soc., 70, No. 5, 706 (1964).
⁷ J. Bryant, C. Seebeck, Quart. J. Math., 19, No. 75, 257 (1968).
⁸ H. Bothe, Fund. Math., 56, No. 2, 203 (1964).
⁹ E. Zeeman, Seminar on Combinatorial Topology, Ch. 7, Engulfing, Institut des Hautes Etudes Sci., 1963, (Revised 1965), mimeographed.