UDC 513.83

MATHEMATICS

M. A. SHTAN’KO

EMBEDDING OF TREE-LIKE COMPACTA IN \(E^3\)

(Presented by Academician P. S. Aleksandrov on 4 XI 1965)

1. A metric compactum which, for every \(\varepsilon > 0\), has an \(\varepsilon\)-covering whose nerve is a finite number of trees is called tree-like. In this paper an outline is given of the proof that the class of tree-like compacta coincides with the class of those one-dimensional compacta that can be embedded cellularly in three-dimensional Euclidean space. In addition, a simple criterion is given that characterizes cellularly decomposed compacta in \(E^3\) by means of properties of the complementary space.

2. Definition 1. A compactum \(K \subset E^n\) is called cellularly decomposed in \(E^n\) if it is the intersection of a countable number of finite sums of topological cubes:
\[ K=\bigcap_{n=1}^{\infty} L_n, \]
where \(L_{n+1}\subset \operatorname{int} L_n\) \((n=1,2,\ldots)\), and \(L_n\) is a sum of a finite number of nonintersecting topological cubes
\[ L_n=L_n^1\cup L_n^2\cup \ldots \cup L_n^{s_n}. \]

Definition 2. A compactum \(K\subset E^n\) is called uniformly cellularly decomposed in \(E^n\) if it can be represented as the intersection of a countable number of finite sums of topological cubes
\[ K=\bigcap_{n=1}^{\infty} L_n \]
such that \(L_{n+1}\subset \operatorname{int} L_n\) \((n=1,2,\ldots)\), and
\[ L_n=L_n^1\cup L_n^2\cup \ldots \cup L_n^{s_n}, \tag{1} \]
each \(L_n^i\) is a topological cube whose diameter is less than \(\varepsilon_n\), where \(\varepsilon_n>0\) \((n=1,2,\ldots)\) and \(\varepsilon_n\to 0\) as \(n\to\infty\); no three cubes in (1) have common points; any two cubes either do not intersect, or intersect in a disk and have no common interior points; the nerve of the system \((L_n^1; L_n^2; \ldots; L_n^{s_n})\) is a sum of a finite number of nonintersecting trees.

Definition 3. The closure of a bounded domain of the space \(E^n\) is called a simple closed domain if its boundary is a manifold.

Theorem 1. For every tree-like compactum \(K\) in \(E^3\) \((K\subset E^3)\) and for every \(\varepsilon>0\) there exists a homeomorphic \(\varepsilon\)-mapping of the compactum \(K\) onto a uniformly cellularly decomposed compactum \(K_\omega\) in \(E^3\) \((h_\varepsilon: K \xrightarrow{\text{onto}} K_\omega)\).

The proof is based on the following lemmas.

Lemma 1. For every tree-like compactum \(K\subset E^3\) and for every \(\varepsilon>0\) there exists a finite number of simple closed domains of the space \(E^3\): \(Q_1,Q_2,\ldots,Q_s\), such that:

1) \(\operatorname{diam}(Q_i)<\varepsilon\) \((i=1,2,\ldots,s)\);

2) \(Q_i\cap Q_j\cap Q_k=\Lambda\), if all indices are distinct;

3) \(\operatorname{int}(Q_i)\cap \operatorname{int}(Q_j)=\Lambda\), if \(i\ne j\);

4) \(Q_i\cap Q_j\) is either empty or is a finite sum of nonintersecting topological squares;

5) \(\operatorname{int}\left(\bigcup_i Q_i\right)\supset K\);

b) the nerve of the system \((Q_1, Q_2, \ldots, Q_s)\) is the sum of a finite number of nonintersecting trees.

Lemma 2. Let
\[ U=\operatorname{int}\left(\bigcup_{i=1}^{s} Q_i\right) \]
be a neighborhood of a dendroid compactum \(K\) satisfying the condition of Lemma 1; then there exists a simple closed region \(V\) such that \(U \supset V\), and \(V\) is homeomorphic to the sum of a finite number of topological cubes, with \(V \cap Q_i\) a topological cube; moreover, there exists an \(\varepsilon\)-homeomorphism mapping the compactum \(K\) into \(\operatorname{int} V\) (the number \(\varepsilon\) satisfies the conditions of Lemma 1).

Lemma 3. Let \(K_1, K_2, \ldots, K_n, \ldots\) be a sequence of compacta lying in a bounded part of Euclidean space \(E^n\), and suppose that for every \(n\) there exists a homeomorphic mapping
\[ h_n:K_n \xrightarrow[\ ]{\text{onto}} K_{n+1} \]
\((n=1,2,\ldots)\), which is an \(\varepsilon_n\)-shift. If the sequence of positive numbers \(\varepsilon_n>0\) for every \(n\) satisfies the condition
\[ \sum_{k=n+1}^{\infty}\varepsilon_k < \frac{1}{2}\min \rho\bigl(h_n\cdots h_1(x);\ h_n\cdots h_1(y)\bigr) \]
over all \(x,y\in K_1\) such that \(\rho(x,y)\ge \delta_n\), where \(\delta_n\to 0\) as \(n\to\infty\), then the sequence of compacta \(K_1,K_2,\ldots,K_n,\ldots\) converges to the compactum
\[ K_\omega=\operatorname{lt}_{n\to\infty} K_n, \]
and \(K_1\) is homeomorphic to the compactum \(K_\omega\), and this homeomorphism is an \(\varepsilon\)-shift,
\[ \varepsilon=\sum_{n=1}^{\infty}\varepsilon_n. \]

For the proof of Theorem 1, Lemmas 1, 2, 3 are applied successively. Lemma 3 can be applied, since on the basis of Lemma 2 we can homeomorphically \(\varepsilon_n\)-shift the compactum \(K_n\) into the corresponding neighborhood \(V_n\), for arbitrary \(\varepsilon_n\to 0\) \((n\to\infty)\) and independently for different \(n\).

If in the condition of Lemma 3 we take
\[ \sum_{k=1}^{\infty}\varepsilon_n=\varepsilon, \]
then in the limit, as \(n\to\infty\), we obtain a compactum \(K_\omega\), homeomorphic to the compactum \(K=K_1\), the homeomorphism being an \(\varepsilon\)-shift \((h_\varepsilon:K_1\to K_\omega)\).

The compactum \(K_\omega\) is the intersection of simple closed neighborhoods
\[ V_n:K_\omega=\bigcap_{n=1}^{\infty} V_n, \]
satisfying the conditions of Definition 2.

3. Definition 4. For a compactum \(K\subset E^n\), the number \(d_2(K)\) is the exact lower bound of all numbers \(\varepsilon>0\) such that there exists a finite closed \(\varepsilon\)-covering of the compactum \(K\) whose multiplicity is not greater than 2.

Definition 5. For a compactum \(K\subset E^n\), the number \(A_2(K)\) is the exact lower bound of all numbers \(\varepsilon>0\) such that there exists a finite closed \(\varepsilon\)-covering of the compactum \(K\), every component of whose nerve is a tree.

Theorem 2. Every one-dimensional cellularly decomposable compactum in \(E^n\), \(K\subset E^n\), is dendroid.

For the proof the following lemma is needed.

Lemma 4. If \(L^n\subset E^n\) is a topological cube, then
\[ d_2(L^n)=A_2(L^n). \]

It follows from Theorems 1 and 2 that, in order that a one-dimensional compactum \(K\) can be cellularly decomposably embedded in \(E^3\), it is necessary and sufficient that \(K\) be dendroid.

4. Definition 6. A compactum \(K\subset E^n\) is called spherically embedded if for every compactum \(C\subset E^n\setminus K\) that does not separate ...

\(E^n\), there exists a topological sphere \(S^{n-1}\) such that
\(\operatorname{int}(S^{n-1}) \supset K\), \(\operatorname{ext}(S^{n-1}) \supset C\),
\(S^{n-1} \cap (K \cup C) = \Lambda\).

In the case \(n = 3\), by Bing’s approximation theorem \((^1)\), the sphere \(S^2\) may be assumed polyhedral.

Theorem 3. In order that a compactum \(K \subset E^3\) be cellularly separated, it is necessary and sufficient that it be spherically embedded.

We outline the idea of the proof of necessity. If \(U\) is an arbitrary neighborhood of the compactum \(K\), then there exists a finite number of simply connected closed domains \(Q_1, Q_2, \ldots, Q_s\) such that

\[ U \supset \bigcup_{i=1}^{s} Q_i \supset \bigcap_{i=1}^{s} \operatorname{int}(Q_i) \supset K. \]

Consider one of the domains \(Q_i\), denoting it by \(Q\). From the boundary of the domain \(Q\) remove a disk \(D\); then
\(M^2 = \operatorname{fr}(Q) \setminus D\) does not split \(E^3\), and, by the hypothesis, there exists a sphere \(S^2\) such that
\(\operatorname{int}(S^2) \supset K\), \(\operatorname{ext}(S^2) \supset M^2\),
\(S^2 \cap (K \cup M^2) = \Lambda\). Moreover, it may be assumed that \(S^2 \cap D\) consists of a finite number of disjoint circles. To prove the theorem we replace the domain \(Q\) by a finite number of disjoint topological cubes covering the compactum \(\operatorname{int}(Q) \cap K\). This can be done by cutting the cube \(S^2 \cup \operatorname{int}(S^2)\) “into parts,” using the properties of the intersection \(S^2 \cap D\) and observing that, by Alexander’s theorem, \(S^2 \cup \operatorname{int}(S^2)\) is a topological cube.

Theorem 4. Every subcompactum of a one-dimensional cellularly separated compactum in \(E^3\) is also cellularly separated.

In the proof one applies Theorem 2 and the idea of the proof of Theorem 3.

Question 1. Is Theorem 4 true for \(E^n\) when \(n > 3\)? McMillan gives an affirmative answer to this question for embeddings of cellular arcs in \(E^n\) \((n \ne 4)\) \((^2)\).

The following theorem strengthens the result of Theorem 3.

Theorem 5. In order that a compactum \(K \subset E^3\) be cellularly separated, it is necessary and sufficient that it be spherically separable from every one-dimensional polyhedron lying in the complement.

The proof rests on a technical lemma asserting that a two-dimensional polyhedral manifold with boundary lying in \(E^3\) has a complement homeomorphic to the complement of some one-dimensional polyhedron.

Question 2. Is it true that a compactum \(K \subset E^3\) is cellularly separated if and only if it is spherically separable from every polyhedral neighborhood in the complement?

Moscow State University
named after M. V. Lomonosov

Received
15 X 1965

REFERENCES

1. R. H. Bing, Ann. Math., 65, No. 3, 456 (1957).
2. D. R. McMillan, Ann. Math., 79, No. 2, 327 (1964).