Abstract
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Reports of the Academy of Sciences of the USSR
- Volume 136, No. 1
MATHEMATICS
Lyudmila KELDYSH
ON THE EMBEDDING IN \(E^4\) OF CERTAIN MONOTONE IMAGES OF \(E^3\)
(Presented by Academician P. S. Aleksandrov on 8 VII 1960)
- Bing constructed a continuous decomposition \(E_f^3\) of three-dimensional Euclidean space \(E^3\), whose elements are points and tame simple arcs* and whose space \(H\) is not embeddable in \(E^3\) \((^2)\). The set of nondegenerate (to a point) elements of the decomposition \(E_f^3\) is zero-dimensional and compact. Curtis indicated a sufficient condition for the embeddability of a monotone image of \(E^k\) in \(E^n\) and showed \((^5)\), on its basis, that \(H\) is topologically embeddable in \(E^4\), while Bing showed \((^8)\) that \(E^4 = H \times E^1\), where \(E^1\) is a line. Here we indicate a sufficient criterion for the topological embeddability of a monotone image of \(E^3\) in \(E^4\), different from Curtis’s criterion, from which it follows directly that \(H\) is embeddable in \(E^4\).
We consider a continuous decomposition of \(E^3\) for which the closure of the set of all nondegenerate elements of the decomposition is zero-dimensional and compact, and each nondegenerate element \(\xi\) is a smooth continuum; we call a continuum \(K \subset E^3\) smooth if \(E^3 \setminus K\) is homeomorphic to the complement of \(E^3\) to a point. Bing showed that an arbitrary topological sphere \(S^2\) in \(E^3\) can be approximated as closely as desired by a polyhedral sphere \((^3)\). By Aleksander’s theorem \((^1)\), for a polyhedral sphere there exists a homeomorphic mapping of \(E^3\) onto itself under which \(S^2\) is carried into the sphere \(x^2 + y^2 + z^2 = 1\), and by Moise’s theorem \((^7)\), for any two finite sequences of polyhedral spheres in \(E^3\)
\[ S_1 \supset S_2 \supset \ldots \supset S_n;\qquad S'_1 \supset S'_2 \supset \ldots \supset S'_n \]
there exists a homeomorphic mapping \(\varphi : E^3 \to E^3\), under which \(\varphi(S_k) = S'_k,\ k = 1, 2, \ldots, n\). On the basis of these three theorems it is easy to show that the condition of smoothness of a continuum \(K\) in \(E^3\) is equivalent to the condition: for any \(\varepsilon > 0\) there exists a neighborhood of the continuum \(K\) of diameter \(< \varepsilon\), whose closure is a polyhedral ball.
- Denote by \(P=\{\xi\}\) the set of all nondegenerate elements of some continuous decomposition of \(E^3\), and by \(P^*=\bigcup \xi\) the set-theoretic sum of these elements; by \(A_f\) denote the continuous decomposition of the space \(A\), and by \(f(A)\) the space of this decomposition.
Theorem 1. If \(E_f^3\) is a continuous decomposition of \(E^3\) into points and smooth continua such that the closure \(\bar P\) of the set \(P\) of all nondegenerate elements of the decomposition is zero-dimensional and compact, then the space of this decomposition \(f(E^3)\) is topologically embeddable in \(E^4\).
* A simple arc \(l \subset E^3\) is called tame if there exists a homeomorphic mapping \(\varphi : E^3 \to E^3\) under which \(l\) is carried into a segment of a straight line.
The proof of Theorem 1 is based on two lemmas.
Lemma 1. Let \(E^4=E^3\times E^1\) and \(W=\Delta\times [a,b]\), where \(\Delta\) is a tame topological ball in \(E^3\), and \([a,b]\) is a segment of \(E^1\). For any number \(\varepsilon>0\) and segment \([a',b']\), \(a<a'<b'<b\), there exist two tame topological balls in \(E^3\)
\[ \Delta^{*}\subset \tilde{\Delta}\subset \bar{\Delta}\subset \Delta \]
such that*
\[ \operatorname{diam}\Delta^{*}<\varepsilon;\qquad h(\Delta,\tilde{\Delta})<\varepsilon, \]
and a topological mapping
\[ \varphi:\bar{W}\to \bar{W}, \]
fixed on the boundary of \(W\) and carrying \(\tilde{W}=\tilde{\Delta}\times [a',b']\) into \(W^{*}=\Delta^{*}\times [a',b']\), such that
\[ \varphi(M,t)=[M',t], \]
where \(M\in\Delta\), \(M'\in\Delta\), \(t\in E^1\), \(a\leq t\leq b\).
Lemma 2. If \(E_f^3\) is a continuous decomposition of \(E^3\) satisfying the conditions of Theorem 1, then the closure of the sum \(P^{*}\) of all nondegenerate elements of the decomposition can be represented in the form
\[ \text{a) }\ \bar{P}^{*}=\bigcap_{k=1}^{\infty}\bigcup_{n=1}^{N_k} U_n^k;\qquad \text{b) }\ \bar{P}^{*}=\bigcap_{k=1}^{\infty}\bigcup_{m=1}^{M_k}\Delta_m^k, \tag{1} \]
where each \(U_n^k\) is a domain consisting of whole elements of the decomposition \(E_f^3\); the boundary of \(U_n^k\) does not intersect \(\bar{P}^{*}\), and
\[ \bar{U}_n^k\cap \bar{U}_{n'}^k=\Lambda \quad \text{for } n\ne n'. \tag{2} \]
Each \(\bar{\Delta}_m^k\) is a polyhedral ball, and for each \(U_n^k\) there are \(U_r^{k-1}\) and \(\Delta_m^k\) such that
\[ \bar{U}_m^k\subset \Delta_m^k\subset \bar{\Delta}_m^k\subset U_r^{k-1} \tag{3} \]
and any \(\xi\in P\) can be represented in the form
\[ \xi=\bigcap_{k=1}^{\infty}U_{n_k}^k;\qquad \bar{U}_{n_k}^k\subset U_{n_{k-1}}^{k-1}. \tag{4} \]
It follows from condition (2) that the representation (4) is unique. By virtue of (3), \(\xi\) can also be represented as the intersection of the polyhedral balls \(\Delta_m^k\); however, the balls \(\Delta_m^k\) and \(\Delta_{m'}^k\) may intersect, and therefore such a representation of \(\xi\) is not unique.
3. The idea of the proof of Theorem 1. We first construct in \(E^4\) a surface \(L\), given by a continuous function
\[ t=F(M),\qquad M\in E^3,\quad t\in E^1, \]
such that \(F(M)\) is constant on each continuum \(\xi\in\bar{P}\). To this end, to each \(U_n^k\) we assign a segment \(\pi_n^k\subset E^1\) so that
\[ \bar{\pi}_i^k\cap \bar{\pi}_j^k=\Lambda,\quad \text{if } \bar{U}_i^k\cap \bar{U}_j^k=\Lambda; \]
\[ \bar{\pi}_r^{\,k+1}\subset \pi_s^k,\quad \text{if } \bar{U}_r^{\,k+1}\subset U_s^k, \]
and the lengths of \(\pi_n^k\) tend to zero together with \(1/k\).
* By \(h(A,B)\) we denote the Hausdorff distance between the sets \(A\) and \(B\).
Set
\[ F(M)=t_\xi,\qquad \text{if } M\in \xi, \]
where
\[ \xi=\bigcap_{k=1}^{\infty} U_n^k,\qquad t_\xi=\bigcap_{k=1}^{\infty} \pi_n^k. \]
We extend \(F\) to a continuous function on \(E^3\) and denote by \(L\) the graph of \(F\).
Next we choose a sequence of numbers \(\varepsilon_n>0\), \(\varepsilon_n\to 0\), and construct a sequence of homeomorphic \(\varepsilon_n\)-shifts of \(E^4\) onto itself:
\(\varphi_1,\varphi_2,\ldots,\varphi_n,\ldots\), so that for the homeomorphism
\(\Psi_n=\varphi_n\varphi_{n-1}\cdots\varphi_1\) and for each \(\xi\in \overline P\) one has
\[ \operatorname{diam}\Psi_n(\xi\times t_\xi)<\varepsilon_n. \]
Each \(\varphi_n\) does not change the coordinate \(t\) of a point \((M,t)\in E^4\) and is fixed outside
\(\bigcup \Psi_{n-1}(W_i^k)\), where \(W_i^k=\pi_i^k\times \Delta_m^k\), and \(\Delta_m^k\) is some polyhedral ball satisfying condition (3) for \(U_i^k\), chosen from the system constructed on the basis of Lemma 2.
The sequence of homeomorphisms \(\Psi_n\) converges uniformly to a continuous mapping
\(\Phi:E^4\to E^4\); the only nondegenerate elements of the decomposition \(E_\Phi^4\) are the continua \(\xi\times t_\xi\). Therefore \(\Phi(L)\) is homeomorphic to \(f(E^3)\) and \(\Phi(L)\subset E^4\).
Let us note that it is not known whether, in the formulation of Theorem 1, the condition that \(\overline P\) be zero-dimensional can be replaced by the weaker condition that \(P\) be zero-dimensional.
- From the sufficient condition for the embeddability of a monotone image of \(E^3\) in \(E^3\), given by Harrold \((^6)\), it follows directly that if in formula (1) all \(U_n^k\) are topological balls, then \(f(E^3)\cong E^3\).
In Bing’s example \((^2)\) of a decomposition space of \(E^3\) not embeddable in \(E^3\), the set of all nondegenerate elements of the decomposition is representable by formula (1), where each \(U_n^k\) is a ball with two handles. Moreover, this continuous decomposition is completely continuous on \(P^*\), i.e. the corresponding mapping \(f\) is open on \(P^*\).
Bing constructed another continuous decomposition of \(E^3\) into points and a zero-dimensional perfect set \(P\) of tame simple arcs, for which the decomposition space coincides with \(E^3\) \((^4)\). Here each \(\overline U_n^k\) is a solid torus, i.e. a topological ball with one handle; the decomposition is completely continuous on \(P\). The question arises: can one construct a continuous decomposition of \(E^3\) into points and a zero-dimensional perfect set of tame continua so that all \(\overline U_n^k\) are solid tori and the decomposition space is not embeddable in \(E^3\)?
Such an example can be constructed by specifying the set \(P\) so that in each solid torus \(\overline U_n^{k-1}\), \(k\ge 1\), there are contained three solid tori \(\overline U_1^k,\overline U_2^k,\overline U_3^k\), arranged so that each of them lies in a topological cube contained in \(\overline U_n^{k-1}\), but no sum of two distinct \(\overline U_i^k\), \(i=1,2,3\), can be enclosed in a topological cube contained in \(\overline U_n^{k-1}\). Then the decomposition space is not embeddable in \(E^3\). The nondegenerate elements of the decomposition \(\xi\) are tame continua, among which there is an uncountable set of indecomposable continua; the decomposition is completely continuous on \(P^*\).
At the same time the following holds.
Theorem 2. Let \(E_f^3\) be a continuous decomposition of \(E^3\) into points and a zero-dimensional compact set \(P\) of tame locally connected continua. If the sum \(P^*\) of all nondegenerate elements of the decomposition is representable in the form (1a) in such a way that all \(\overline U_n^k\) are solid tori and the decomposition \(E_f^3\) is completely continuous on \(P^*\), then the decomposition space coincides with \(E^3\).
The proof of Theorem 2 is based on the following lemma:
Lemma 3. Under the hypotheses of Theorem 2, whatever the number \(\varepsilon>0\) and the integer \(r>0\), there exists a number \(R>r\) and a homeomorphic mapping \(\Phi\) of the space \(E^3\) onto itself, fixed outside \(\displaystyle \bigcup_{n=1}^{R} U'_n\), and such that for each \(\overline{U_i^R}\)
\[ \operatorname{diam}\Phi(\overline{U_i^R})<\varepsilon. \]
For the proof we first show that among all possible \(\overline{U_n^k}\) in \(\{1a\}\) there is at least one \(\overline{U_{n_0}^{k_0}}\), possessing certain special properties, such that for \(\overline{U_{n_0}^{k_0}}\cap P^*\) Lemma 3 can be proved by the method used by Bing in proving the analogous lemma in \((^4)\). After this, Lemma 3 is proved by means of transfinite induction.
It follows from Theorem 2 that if one requires complete continuity of the decomposition on \(P^*\), as is the case in Bing’s examples, then it is impossible to construct an upper semicontinuous decomposition of \(E^3\) into points and a null perfect set of tame (and even smooth) simple arcs in such a way that all \(\overline{U_n^k}\) are full tori, while the decomposition space is not embeddable in \(E^3\).
It is unknown, however, whether this nevertheless cannot be done if the condition of complete continuity of the decomposition on \(P^*\) is dropped.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
30 VI 1960
REFERENCES
\(^1\) J. W. Alexander, Proc. Nat. Acad. Sci. U. S. A., 10 (1924).
\(^2\) R. H. Bing, Ann. of Math., 65, No. 3, 484 (1957).
\(^3\) R. H. Bing, Ann. of Math., 65, No. 3, 456 (1957).
\(^4\) R. H. Bing, Ann. of Math., 56, No. 2 (1952).
\(^5\) M. L. Curtis jr., Duke Math. J., 24, No. 3 (1957).
\(^6\) O. G. Harrold, Proc. Am. Math. Soc., 9, No. 6 (1958).
\(^7\) E. E. Moise, Ann. Math., 55, No. 1 (1952).
\(^8\) R. H. Bing, Ann. Math., 70, No. 3, 399, 1959).