Abstract
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Reports of the Academy of Sciences of the USSR
- Volume 185, No. 3
MATHEMATICS
L. V. KELDYSH
TOPOLOGICAL EMBEDDINGS IN \(E^3\) OF SIMPLE ARCS AND CLOSED CONTOURS
(Presented by Academician P. S. Aleksandrov on 8 VII 1968)
- In \((^{7,8})\) it was proved that for every two-dimensional manifold \(M^2\) in Euclidean space \(E^3\) (or in a piecewise-linear three-dimensional manifold \(M^3\)) and number \(\varepsilon > 0\) there exist a polyhedral manifold \(N^2 \subset E^3\), homeomorphic to \(M^2\), and an \(\varepsilon\)-pseudoisotopy \(F_t,\ 0 \leq t \leq 1\), of the space \(E^3\), taking \(N^2\) onto \(M^2\). (A pseudoisotopy is a homotopy such that \(F_t\) is a homeomorphism for every \(t < 1\).) \(F_t\) is an \(\varepsilon\)-pseudoisotopy if, for all \(x\) and \(t\), \(\rho(x,F_t(x)) < \varepsilon\), where \(\rho\) is distance. Moreover, \(F_0=1\), \(F_1(N^2)=M^2\), \(F_{1/N^2}\) is a homeomorphism, and the set of nondegenerate inverse images of the mapping \(F_1\) is zero-dimensional and is contained in the set of wild points of \(M^2\).
Krups extended this theorem to embeddings in a piecewise-linear three-dimensional manifold with boundary of a two-dimensional polyhedron having no locally separating points \((^5)\). It remained unknown whether an analogous pseudoisotopy exists for embeddings in \(E^3\) of one-dimensional polyhedra, in particular simple arcs. It is plausible (although not proved) that for an arbitrary simple arc \(l \subset E^3\) one cannot construct a pseudoisotopy of the space \(E^3\) taking a line segment onto \(l\). We give a sufficient condition for the existence of a pseudoisotopy \(F_t:E^3 \to E^3\) taking a line segment onto a given simple arc and some polyhedral closed contour onto a given one.
It is known \((^{6,2,9})\) that in order for a simple arc (closed contour) \(l\) in \(E^3\) to be tame, i.e., for there to exist a homeomorphic mapping of \(E^3\) onto itself taking \(l\) onto a polyhedron, it is necessary and sufficient that two conditions be fulfilled:
1) local peripheral unknottedness, i.e., for every point \(x \in l\) there exists a three-dimensional cell \(Q_x\) such that \(x \in \operatorname{int} Q_x\) and \(\partial Q_x \cap l\)* consists of two points (or one, if \(x\) is an end point);
2) local unknottedness, i.e., for every point \(x \in l\) there exists a neighborhood \(l_x\) of the point \(x\) on \(l\), which lies on a disk (two-dimensional cell): \(x \in l_x \subset d_x\).
There exist examples of wild simple arcs in \(E^3\) for which one of these conditions is fulfilled \((^1)\).
- We prove the following theorems:
Theorem 1. For every locally unknotted simple arc \(l\) in \(E^3\) there exists a pseudoisotopy \(F_t:E^3 \to E^3\) from the identity mapping, taking a line segment \(I^1\) onto \(l\). Moreover \(F_{1/I^1}\) is a homeomorphic mapping of the segment \(I^1\) onto the arc \(l\), and the set of points \(x \in E^3\) whose inverse images under the continuous mapping \(F_1\) are nondegenerate is zero-dimensional and is contained in the set of wild points of the arc \(l\).
If a homeomorphic mapping \(\varphi:I^1 \to l\) is given, then the pseudoisotopy \(F_t\) can be constructed so that \(F_{1/I^1}=\varphi\).
* By \(\operatorname{int} M\) we denote the interior of the manifold \(M\), and by \(\partial M\) its boundary.
Theorem 2. For every locally unknotted closed contour \(C \subset E^3\) and number \(\varepsilon > 0\) there exists a polyhedral closed contour \(P\) and an \(\varepsilon\)-pseudoisotopy \(F_t : E^3 \to E^3\) carrying \(P\) onto \(C\), with \(F_0 = 1\), \(F_1/P\) a homeomorphism, \(F_1(P) = C\), and the set of nondegenerate point-inverses of the mapping \(F_1\) zero-dimensional and contained in the set of wild points of the contour \(C\).
Both of these theorems follow easily from the theorem on pseudoisotopy for two-dimensional manifolds with boundary \((^5)\) in \(E^3\), and from the following theorem and its corollary.
Theorem 3. Every locally unknotted simple arc \(l\) in \(E^3\) lies on a disk \(D \subset E^3\), the set of wild points of which coincides with the set of wild points of \(l\).
Corollary. Every locally unknotted simple closed contour \(C\) in \(E^3\) lies on a two-dimensional manifold with boundary \(M^2 \subset E^3\), and the set of wild points of \(M^2\) coincides with the set of wild points of \(C\).
The proof of Theorem 3 contains many technical details. We shall briefly set forth its idea. A detailed text will be published later.
Let \(l\) be a locally unknotted simple arc in \(E^3\) with endpoints \(a\) and \(b\). Then \(l\) can be represented as a sum of arcs \(l_i\):
\[ l=\bigcup_{i=1}^{n} l_i,\qquad l_1 \ni a,\qquad l_n \ni b; \tag{1} \]
\[ l_i \cap l_j=\Lambda,\qquad \text{if } |i-j|>1; \]
\(l_i \cap l_{i+1}\) is an interval \(l_i'\), and \(l_i\) lie on disks \(l_i \subset d_i\). The disks \(d_i\) may be chosen so that
\[ d_i \cap l=l_i,\qquad i \leq n;\qquad d_i \cap d_j=\Lambda,\qquad \text{if } |i-j|>1, \]
and \(d_i\) is contained in a neighborhood \(U_i\) of the arc \(l_i\), with \(U_i \cap U_j=\Lambda\), \(|i-j|>1\). Moreover, \(l_i\) lies on the boundary of \(d_i\), and by Bing’s approximation theorem \((^3)\) the disks \(d_i\) may be assumed locally polyhedral at the points \(d_i \setminus l\).
We construct successively disks \(D_k\) so that \(D_k \cap l=\bigcup_{i=1}^{k} l_i\) and \(D_k \setminus l\) is locally polyhedral. Put \(D_1=d_1\), and suppose that a disk \(D_k\), \(1 \leq k < n\), has been constructed. Obviously \(D_k \cap d_{k+1} \ne \Lambda\), since \(D_k \cap d_{k+1} \supset l_k \cap l_{k+1}=l_k'\).
Consider the different cases.
1) \(D_k \cap d_{k+1}=l_k'\). Then put \(D_{k+1}=D_k \cup d_{k+1}\).
2) \(D_k \cap d_{k+1} \supset L\), where \(L\) is either a simple arc, one end of which lies on \(l_k'\), or a continuum homeomorphic to the closure of the graph of the function \(\sin 1/x\) with its continuum of condensation on \(l_k'\), and \(L\) separates both \(D_k\) and \(d_{k+1}\):
\[ D_k \setminus L=D_k' \cup D_k'',\qquad d_{k+1}\setminus L=d_{k+1}' \cup d_{k+1}'', \]
where \(D_k' \ni a\) and \(d_{k+1}'' \supset l_{k+1}\setminus l_k'\), and \(D_k' \cap d_k''=L\). Then we obtain:
\[ D_{k+1}=\overline{D}_k' \cup \overline{d}_{k+1}'' * . \tag{2} \]
In the general case the intersection \(D_k \cap d_{k+1}\) may be much more complicated. We transform the disks \(d_{k+1}\) (and sometimes \(D_k\)) inside \(U_k \cap U_{k+1}\) so as to obtain 1) or 2). By a small displacement of \(d_{k+1}\), fixed on \(l_{k+1}\), the intersection \(D_k \cap (d_{k+1}\setminus l)\) is brought into general position. Since \(D_k\setminus l\) and \(d_{k+1}\setminus l\) are polyhedra, \(D_k \cap d_{k+1}\), apart from \(l_k'\) and \(L_i\), if 1) or 2) is not satisfied, may contain, possibly intersecting, polyhedral closed contours \(\gamma_i\) and polygonal lines \(\lambda_i\), both of whose ends lie on \(\partial d_{k+1}\) or on \(\partial D_k\), as well as polygonal lines \(\mu_i\), having one end on \(\partial d_{k+1}\setminus l\) and the other on \(\partial D_k\setminus l\). The number of contours \(\gamma_i\) and polygonal lines \(\lambda_i, \mu_i\) outside an arbitrary neighborhood of \(l_k'\) is finite. By rebuilding \(d_{k+1}\) and \(D_k\) inside \(U_k \cap U_{k+1}\), the contours \(\gamma_i\) and polygonal lines \(\lambda_i\) are successively eliminated. If, after eliminating all
\[ * \ \overline{A} \text{ denotes the closure of the set } A. \]
\(\gamma_i\) and \(\lambda_i\) and a finite number of \(\mu_i\) we do not obtain 1) and 2), then the transformed intersection \(D_k \cap (d_{k+1}\setminus l)\) consists of an infinite number of nonintersecting polygonal lines \(\mu_i\) tending to \(l_k'\) as \(i\) increases. The ends of \(\mu_i\) form two sequences of points \(\{x_i\}\to x_0\) and \(\{x_i'\}\to x_0'\), converging to the ends of \(l_k'\). Joining, by a polygonal line in the strip \(D_k \cap \overline U_k \cap \overline U_{k+1}\), between two neighboring \(\mu_i\) (and between \(\mu_1\) and \(\partial D_k\)) the points of the sequence \(\{x_i\}\), we construct on \(D_k\) a simple arc \(\tau\) with end \(x_0 = l_k' \cap \overline{(D_k\setminus l_k')}\). Similarly, we construct on \(d_{k+1}\) a simple arc \(\nu\) with end \(x_0'\). The arc \(\tau\) cuts off from \(D_k\) a disk \(\widetilde D_k \subset U_k \cap U_{k+1}\), and the arc \(\nu\) cuts off from \(d_{k+1}\) a disk \(\widetilde d_{k+1}\subset U_k\cap U_{k+1}\). The polygonal lines \(\mu_i\) divide \(\widetilde D_k\) and \(\widetilde d_{k+1}\) into strips \(\sigma_i\subset \widetilde D_k\) and \(s_i\subset \widetilde d_{k+1}\), bounded by neighboring polygonal lines \(\mu_i\) and \(\mu_{i+1}\) on \(\widetilde D_k\) (on \(\widetilde d_{k+1}\), other \(\mu_j\) may lie between \(\mu_i\) and \(\mu_{i+1}\)), and \(\sigma_i\cup s_i=\pi_i\) is a cylinder with generator \(\mu_i\) and base \(\tau_i\cup \tau_i'\), where \(\tau_i\) is the segment of \(\tau\) between \(\mu_i\) and \(\mu_{i+1}\), and \(\tau_i'\) is the segment of \(\partial d_{k+1}\setminus l\) from \(x_i\) to \(x_{i+1}\); \(\tau_i'\) may intersect \(\tau\) at some further points \(x_j\). We choose on the cylinders \(\pi_i\) new generators \(\theta_i\), lying on \(\pi_i\setminus \partial\pi_i\), so that the diameters of \(\theta_i\) tend to zero together with \(1/i\). We prove that:
A. For sufficiently large \(i\), the contour \(\theta_i\) bounds a disk contained in \(U_k\cap U_{k+1}\) and intersecting \(\pi_i\) in \(\theta_i\).
Replacing, if necessary, the disk \(d_{k+1}\) by a smaller disk not containing those \(\mu_i\) for which \(\theta_i\) does not bound a disk in \(U_k\cap U_{k+1}\), we may assume that A is true for all \(i\). The diameters of the disks spanned by \(\theta_i\) tend to zero together with \(1/i\).
Two cases are possible:
a) \(\theta_i\) bounds in \(U_k\cap U_{k+1}\) a disk not intersecting \(l\).
b) Every disk in \(U_k\cap U_{k+1}\) bounded by \(\theta_i\) intersects \(l\).
In case a) we rearrange each strip \(s_i\) cut off from \(\widetilde d_{k+1}\) by the polygonal lines \(\mu_i\) and \(\mu_{i+1}\) in such a way that all \(\mu_j\subset s_i\) are replaced by several polygonal lines of the form \(\lambda_p\), the ends of which lie on \(\partial D_k\setminus l\); moreover, the new strip \(s_i'\) contains a disk spanned by \(\theta_i\). After, under such a rearrangement of \(d_{k+1}\), all \(\mu_i\) satisfying a) have been successively replaced by arcs \(\lambda_p\), we eliminate the \(\lambda_p\), rearranging \(D_k\) in the part \(\widetilde D_k\).
When all \(\mu_i\) for which a) holds have been eliminated, the intersection \(D_k\cap d_{k+1}\setminus l\) (we keep the former notation) consists of \(\mu_i\) for which b) holds. If there is a finite number of such \(\mu_i\), then it remains to trim the disk \(d_{k+1}\) in order to obtain 1). Let \(D_k\cap d_{k+1}\) contain an infinite set of \(\mu_i\). It is proved that the order of the \(\mu_i\) on \(D_k\) and on \(d_{k+1}\) is the same and \(s_i\cap \mu_j=\Lambda\), if \(i\ne j\ne i+1\). Replacing the cylinder \(\pi_i\), if necessary, by the cylinder \(\pi_i'\) cut off from \(\pi_i\) by a pair of closed contours close to \(\partial\pi_i\), we show that each \(\pi_i'\) lies on a sphere \(S_i\), close to \(l_k'\) for large \(i\), and moreover
\[ \pi_i'\cap \pi_{i+1}' \subset \mu_{i+1}; \qquad \pi_r'\subset \operatorname{int} S_i,\quad r\ge i+2, \]
\[ S_i\cap S_{i+1}\subset \mu_{i+1}; \qquad \operatorname{int} S_{i+1}\subset \operatorname{int} S_i; \]
and
\[ \bigcap_{i=1}^{\infty}\operatorname{int} S_i=\widetilde l_k \]
is the segment \(l_k'\). Applying the Bing–Kirkor criterion \((^4)\), one can show that the segment \(\widetilde l_k\) is tame. Next we transform the disk \(d_{k+1}\), changing it in the part \(\widetilde d_{k+1}\) so that each polygonal line \(\mu_i\) is replaced by a simple arc \(\mu_i'\), one end of which lies at the end \(y_0\) of the arc \(\widetilde l_k\) near \(x_0\) and which is locally polyhedral, except for the point \(y_0\). After this transformation, the intersection \(D_k\cap d_{k+1}\) is a bundle of arcs \(\widetilde l_k\) and \(\mu_i'\), \(i=1,2,\ldots\), issuing from the point \(y_0\), with \(\mu_i'\cap \mu_j'=y_0\), \(i\ne j\), and \(\mu_{i+1}'\) lying on \(D_k\) and on \(d_{k+1}\) between \(\mu_i'\) and \(\widetilde l_k\).
Consequently, for \(D_k\cap d_{k+1}'\) case 2) holds, and \(D_{k+1}\) is defined by formula (2).
Thus the disks \(D_k\) are successively constructed, and finally \(D=D_n\supset l\). If \(C_1'\) is a locally unknotted closed contour, then the arc
\(l' = C \setminus (l_n \setminus l_{n-1})\) is locally unknotted and lies on the disk \(D\). In order to obtain a manifold with boundary \(M^2\) containing the contour \(C\), it suffices, starting from the disks \(D \supset l'\) and \(d_n \supset l_n\), to carry out the construction described above in neighborhoods of both components \(l_n' \cap l_n\).
The same method proves
Theorem 4. Let \(P\) be a topological polyhedron in \(E^3\), all of whose one-dimensional simplexes are locally unknotted, and at each vertex \(x\) the following condition is satisfied:
C. In \(E^3\) there exists a neighborhood \(U_x\) of the vertex \(x\) such that a sufficiently fine triangulation of \(P \cap \overline{U}_x\) is a subpolyhedron of a polyhedron \(\pi_x \subset \overline{U}_x\) having no locally separating points.
Then some triangulation of the polyhedron \(P\) is a subpolyhedron of a polyhedron \(\Pi \subset E^3\) without locally separating points.
The polyhedron \(\Pi\) satisfies the hypotheses of Craggs’ theorem on pseudoisotopy \((^5)\); consequently Theorem 2 is true for \(\Pi\), and hence also for \(P\).
Question 1. Is the condition of local unknottedness of a simple arc and of a simple closed contour necessary for the existence of a pseudoisotopy satisfying the hypotheses of Theorems 1 and 2?
Question 2. Are condition \(C\) and the local unknottedness of the one-dimensional simplexes of the polyhedron \(P\) necessary for the existence of an analogous pseudoisotopy?
Theorem 3, and consequently Theorems 1, 2, and 4, remain valid for the embedding of a locally unknotted simple arc (closed contour) in a piecewise-linear three-dimensional manifold.
Mathematical Institute named after V. A. Steklov
Academy of Sciences of the USSR
Received
28 VI 1968
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