UDC 513.833

MATHEMATICS

Lyudmila KELDYSH

TOPOLOGICAL EMBEDDINGS AND PSEUDOISOTOPY

(Presented by Academician P. S. Aleksandrov, December 7, 1965)

1. We study here arbitrary topological embeddings, i.e. homeomorphic mappings, of a two-dimensional manifold \(M^2\) into three-dimensional Euclidean space \(E^3\), and embeddings of a polyhedron \(P^k\) of dimension \(k \leq n/2 - 1\) into Euclidean space \(E^n\). A topological polyhedron \(\mathscr P\) in \(E^n\) is called tame if there exists a homeomorphic mapping \(H\) of the space \(E^n\) onto itself such that \(H(\mathscr P)\) is a rectilinear polyhedron. Otherwise the polyhedron \(\mathscr P\) is called wild. A polyhedron \(\mathscr P \subset E^n\) is called locally tame at a point \(x\) if there exist a neighborhood \(U\) of the point \(x\) and a homeomorphic mapping \(\varphi: U \to E^n\) such that \(\varphi(U \cap \mathscr P)\) is a rectilinear polyhedron. If for a point \(x \in \mathscr P\) no such neighborhood exists, then \(x\) is called a point of wildness for \(\mathscr P\). Bing \((^2)\) and Moise \((^8)\) showed that a locally tame polyhedron in \(E^3\) is tame. If \(S^2\) is a locally tame topological sphere in \(E^3\), then there exists a homeomorphic mapping \(H: E^3 \to E^3\) carrying \(S^2\) into a polyhedral sphere and, by Aleksandrov’s theorem \((^1)\), onto the boundary \(\sigma^2\) of a three-dimensional simplex: \(H(\sigma^2)=S^2\). The homeomorphism \(H\) can be chosen so that it does not change orientation and, consequently, it can be joined by an isotopy \(F_t\), \(0 \leq t \leq 1\), with the identity mapping of \(E^3\) \((^9)\): \(F_0=1,\ F_1=H\). G. Gluck \((^6)\) proved that for a locally tame polyhedron \(\mathscr P^k\) of dimension \(k \leq n/2 - 1\) in \(E^n\) there exists an isotopy of \(E^n\) from the identity mapping carrying \(\mathscr P^k\) into a rectilinear polyhedron. A. V. Chernavskii proved an analogous theorem for polyhedra of dimension \(k < {}^2/_3 n - 1\) \((^{10})\).

For wildly embedded polyhedra in \(E^n\), by the very definition, there exists no homeomorphism of \(E^n\) onto itself carrying the given polyhedron into a rectilinear one.

The following questions arise:

1) Does there exist, for a topological polyhedron \(\mathscr P^k\) in \(E^n\), a not necessarily one-to-one mapping \(F\) of the space \(E^n\) onto itself and a rectilinear polyhedron \(P^k\) such that \(F|_{P^k}\) is a homeomorphism and \(F(P^k)=\mathscr P^k\)?

2) Can the mapping \(F\) be chosen so that it is pseudoisotopic to the identity?

Lenning gave a positive answer to question 1) for an embedding of the two-dimensional sphere \(S^2\) in \(E^3\) \((^{11})\).

We give an affirmative answer to both questions for an embedding of a compact two-dimensional manifold in \(E^3\) and for a topological polyhedron of dimension \(k \leq 1/2\,n - 1\) in \(E^n\).

A pseudoisotopy is a homotopy \(F_t,\ 0 \leq t \leq 1\), which for every \(t<1\) is a homeomorphic mapping of the space onto itself. \(F_t\) is called an \(\varepsilon\)-pseudoisotopy if for any \(x,t\) and \(t'\), \(\rho[F_t(x)F_{t'}(x)]<\varepsilon\), where \(\rho\) is distance.

Theorem 1. For an arbitrarily embedded two-dimensional sphere \(S^2\) in \(E^3\) there exists a pseudoisotopy from the identity mapping \(F_t: E^3 \to E^3\), carrying the boundary \(\sigma^2\) of a three-dimensional simplex into \(S^2\): \(F_0=1,\ F_1(\sigma^2)=S^2\), such that \(F_1|_{\sigma^2}\) and \(F_1|_{E^3 \setminus F_1^{-1}(S^2)}\) are homeomorphisms and the set of points \(x\) whose inverse images \(F_1^{-1}(x)\) are degenerate is zero-dimensional and is contained in the set of points of wildness of \(S^2\).

Theorem 2. For an arbitrarily embedded two-dimensional compact manifold \(M^2 \subset E^3\) and a number \(\varepsilon > 0\), there exists a polyhedral manifold \(\mathfrak M^2\) and an \(\varepsilon\)-pseudoisotopy \(F_t\) carrying \(\mathfrak M^2\) onto \(M^2\) and satisfying the conditions of Theorem 1.

Theorem 3. For an arbitrarily embedded in Euclidean space \(E^n\) polyhedron \(\mathfrak P^k\) of dimension \(k \leq n/2 - 1\) and a rectilinear polyhedron \(P^k \subset E^n\), homeomorphic to \(\mathfrak P^k\), there exists a pseudoisotopy \(\Phi_t\) from the identity map carrying \(P^k\) onto \(\mathfrak P^k\): \(\Phi_0 = 1\), \(\Phi(P^k)=\mathfrak P^k\), and moreover \(\Phi_1|P^k\) and \(\Phi_1|E^n \setminus \Phi_1^{-1}(\mathfrak P^k)\) are homeomorphisms, and the set of points \(x\) whose inverse images \(\Phi_1^{-1}(x)\) are degenerate is contained in the set of points of wildness for \(\mathfrak P^k\).

Corollary. For an arbitrarily embedded in \(E^n\) polyhedron \(\mathfrak P^k\) of dimension \(k \leq n/2 - 1\) and a number \(\varepsilon > 0\), there exists a rectilinear polyhedron \(P^k \subset E^n\) and an \(\varepsilon\)-pseudoisotopy carrying \(P^k\) onto \(\mathfrak P^k\) and satisfying the conditions of Theorem 2.

We describe here the idea of the proof of Theorem 1. The proof of Theorem 2 is analogous. The proof of Theorem 3 is based on Gluck’s Theorems 1.2 and 8.2 \((^6)\). The corollary to Theorem 3 follows from the proof of this theorem.

2. Idea of the proof of Theorem 1.

By Bing’s approximation theorem, for every \(\varepsilon > 0\) there exists a homeomorphic \(\varepsilon\)-shift \(h_\varepsilon\) of the sphere \(S^2\) onto a polyhedral sphere \(\Sigma^2\). Let \(\sigma^2\) be the boundary of a three-dimensional simplex lying in \(E^3\), and let \(f\) be a homeomorphic mapping of \(\sigma^2\) onto \(S^2\), chosen so that for sufficiently small \(\varepsilon\), \(h_\varepsilon f:\sigma^2 \to \Sigma^2\) can be extended to an orientation-preserving homeomorphism \(\varphi:E^3 \to E^3\). The isotopy \(F_t\) is constructed so that \(F_1|\sigma^2 \equiv f\).

Choosing a decreasing sequence of numbers \(\varepsilon_n \to 0\), we construct a sequence of tame spheres \(S_n\) converging to \(S^2\) in such a way that the sphere \(S_{n-1}\) is carried by an \(\varepsilon_{n-1}\)-isotopy \(H_t^n:E^3 \to E^3\) onto \(S_n\), and the sequence of homeomorphisms \(\Phi_1^n = H_1^n H_1^{n-1}\ldots H_1^1\), where \(H_1^1(\sigma^2)=S_1\), converges uniformly to a continuous map \(F_1:E^3 \to E^3\) satisfying the conditions of the theorem.

By Bing’s theorems \((^4,^5)\), on the sphere \(S^2\) one can choose an increasing sequence of tame Sierpiński curves \(X_1 \subset X_2 \subset \cdots \subset X_n \subset \cdots\) such that the diameters of the components of \(S^2 \setminus X_n\) are less than \(\varepsilon_n/2\), \(X_{n+1}\setminus X_n\) lies in a finite number of components of \(S^2\setminus X_n\), and \(\overline{X_{n+1}\setminus X_n}\)* contains the boundaries of these components.

Let \(\{\delta_{ni}\}\) be the open disks that are the components of \(S^2\setminus X_n\). If the sphere \(S^2\) is wild, then such disks exist (Theorem 8.2 in \((^5)\)), since \(X_n\) is tame. For each \(\delta_{ni}\) we choose in \(E^3\) a neighborhood \(u_{ni}\) such that
\[ S^2 \cap u_{ni} = \delta_{ni},\quad \overline{u}_{ni}\cap \overline{u}_{nj}=\Lambda,\quad i\ne j;\quad \operatorname{diam} u_{ni}<2\operatorname{diam}\delta_{ni}<\varepsilon_n;\quad \overline{u}_{ni}\subset \overline{u}_{n-1,j}, \]
if \(\overline{\delta}_{ni}\subset \delta_{n-1,j}\); \(u_{ni}=u_{n-1,j}\), if \(\delta_{ni}=\delta_{n-1,j}\).

By Bing’s theorem \((^3)\), for each disk \(\delta_{ni}\) there exists a homeomorphic shift \(g_{ni}\), fixed on the boundary \(-\partial\delta_{ni}\), into a disk \(\Delta_{ni}\subset u_{ni}\), locally polyhedral \(\bmod\, \partial\Delta_{ni}\). We choose the homeomorphisms \(g_{ni}\) so that \(g_{ni}=g_{n-1,j}\), if \(\delta_{ni}=\delta_{n-1,j}\), and set
\[ S_n=S^2\setminus \bigcup \delta_{ni}\cup \bigcup \Delta_{ni};\qquad \Delta_{ni}\subset u_{ni}; \tag{1} \]
\[ g_n:S^2\to S_n;\qquad g_n|\delta_{ni}=g_{ni},\quad g_n|S^2\setminus \bigcup \delta_{ni}=1. \]

Each sphere \(S_n\) is locally tame \(\bmod\, X_n\) and, by Theorem 8.2 \((^5)\), is tame. Then \(g_1 f:\sigma^2\to S_1\), by the choice of \(f\), can be extended to a homeo-

* A Sierpiński curve is a continuum \(X\) obtained from the sphere by deleting a countable everywhere dense set of open disks whose boundaries do not intersect. \(X\) is called tame if there exists a homeomorphism \(\psi:E^3\to E^3\) such that \(\psi(X)\) lies in a plane.

** By \(\overline A\) we denote the closure of the set \(A\).

morphism \(H_1^1:E^3\to E^3\) which does not change orientation, and, consequently, there exists an isotopy \(H_t^1:E^3\to E^3\) such that

\[ H_0^1=1;\qquad H_1^1|_{\sigma^2}=g_1f . \tag{2} \]

The isotopies \(\Phi_t^n:E^3\to E^3\) such that \(\Phi_1^n|_{\sigma^2}=g_nf\) are constructed by induction. \(\Phi_t^n\) is the product of the isotopies \(\Phi_t^{\,n-1}\) and \(\widetilde H_t^{\,n}=H_t^n\Phi_1^{\,n-1}\), \(\Phi_t^n=\Phi_t^{\,n-1}\circ \widetilde H_t^{\,n}\), where \(H_t^n\), \(n\ge 2\), is an \(\varepsilon_{n-1}\)-isotopy and \(H_1^n|_{S_{n-1}}=g_ng_{n-1}^{-1}\). By construction, the sphere \(S_{n+1}\) differs from \(S_n\) in a finite number of regions \(u_{ni}\), \(i=1,2,\ldots,k_n\), and

\[ g_{n+1}\bigg|_{S^2\setminus \bigcup_{i=1}^{k_n}\delta_{ni}}=g_n . \tag{3} \]

For each \(\delta_{ni}\), \(n=1,2,\ldots;\ i=1,2,\ldots,k_n\), a disk \(\delta_{ni}'\subset S^2\) is chosen so that \(\partial\delta_{ni}'\subset X_n\) and \(\partial\delta_{ni}'\) does not meet the boundaries of the components of \(S^2\setminus X_n\), and

\[ \delta_{ni}'\supset \delta_{ni};\qquad \delta_{ni}'\cap \delta_{nj}'=\Lambda,\quad i\ne j; \]

\[ \operatorname{diam}\delta_{ni}'<(1+\varepsilon_n)\operatorname{diam}\delta_{ni}; \]

for \(k<n\): \(\overline{\delta}_{nj}'\subset \delta_{ki}'\), if \(\overline{\delta}_{nj}\subset \delta_{ki}'\); \(\overline{\delta}_{nj}'\cap \overline{\delta}_{ki}'=\Lambda\), if \(\overline{\delta}_{nj}\cap \overline{\delta}_{ki}'=\Lambda\).

For each disk \(\delta_{ni}'\) a neighborhood \(u_{ni}'\) is constructed satisfying the conditions:

\[ u_{ni}'\cap S^2=\delta_{ni}';\qquad \operatorname{diam}u_{ni}'<2\operatorname{diam}\delta_{ni}<\varepsilon_n; \]

\[ \overline u_{nr}\subset u_{ni}',\ \text{if } \delta_{nr}\subset \delta_{ni}';\qquad \overline u_{ni}'\cap \overline u_{kj}'=\Lambda,\ \text{if } \delta_{ni}'\cap \delta_{kj}'=\Lambda \tag{4} \]

\[ \overline u_{ni}'\subset \overline u_{kr}',\ \text{if } k<n \text{ and } \delta_{ni}\subset \delta_{kr}' . \]

We construct the isotopy \(H_t^{\,n+1}\), carrying \(S_n\) into \(S_{n+1}\), so that

\[ H_t^{\,n+1}\bigg|_{E^3\setminus \bigcup_{i=1}^{k_n}u_{ni}'}=1,\quad 0\le t\le 1 . \]

The construction is carried out inside the regions \(u_{ni}'\), and we omit the indices: \(\Delta=g_n(\delta)\subset u\subset u'\). In \(\Delta'=S_n\cap u'\) choose a disk \(\overline\Delta\) with boundary in \(X_n\): \(\overline\Delta\subset \Delta\subset \Delta'\).

\(\overline\Delta\) is tame, since \(S_n\) is tame, and one can construct an isotopy \(\varphi_t\) satisfying the conditions

\[ \varphi_0=1;\qquad \varphi_t|_{\partial u'}=1;\qquad \varphi_1(\overline\Delta)=\widetilde d, \]

where \(\widetilde d\) is a polyhedral disk. In view of (4) and (1), \(S_{n+1}\cap u=g_{n+1}(S^2\cap u)=D\); \(S_{n+1}\cap u'=g_{n+1}(S^2\cap u')\). \(\varphi_1(\overline\Delta\setminus \Delta\cup D)\) is a tame disk lying in \(u'\) and locally polyhedral at the points of its boundary, since \(\overline\Delta\setminus \Delta\cup D\subset S_{n+1}\).

Applying the theorems of Bing \((^2)\) and Kister \((^7)\), we construct an isotopy \(\theta_t:u'\to u'\) such that

\[ \theta_0=1;\qquad \theta_t|_{\partial u'\cup \varphi_1(\Delta'\setminus \overline\Delta)}=1 \tag{5} \]

and \(\widetilde D=\theta_1\varphi_1(\overline\Delta\setminus \Delta\cup D)\) is a polyhedral disk.

Then a piecewise-linear isotopy \(h_t:\overline u'\to \overline u'\) is constructed so that

\[ h_0=1;\qquad h_t|_{\partial u'\cup \varphi_1(\Delta'\setminus \widetilde\Delta)}=1;\qquad h_1\varphi_1(\widetilde\Delta)=\widetilde D \tag{6} \]

Denoting by \(\varphi_t^n\), \(\theta_t^n\), and \(h_t^n\) the isotopies constructed as described in each \(u_{ni}'\), \(i=1,2,\ldots,k_n\), and fixed on \(E^3\setminus \bigcup_{i=1}^{k_n}u_{ni}'\), we put

\[ \Gamma^{n+1}=\varphi_t^n\circ \widetilde h_t^{\,n}\circ \widetilde\theta_t^{\,n}\circ \varphi_t^n, \]

where

\[ \widetilde{h}_{t}^{\,n}=h_{t}^{n}\varphi_{1}^{n};\qquad \widetilde{\theta}_{t}^{\,n}=(\theta_{t}^{n})^{-1}\widetilde{h}_{1}^{\,n};\qquad \widetilde{\varphi}_{t}^{\,n}=(\varphi_{t}^{n})^{-1}\widetilde{\theta}_{1}^{\,n}. \]

\(\Gamma_{1}^{n+1}\) carries each disk \(\widetilde{\Delta}_{ni}\) into the disk \(\widetilde{\Delta}_{ni}\setminus \Delta_{ni}\cup D_{ni}\) and, by (5), (6), \(\Gamma_{1}^{n+1}\bigm|_{S_n\setminus \bigcup \widetilde{\Delta}_{ni}}=1\), hence \(\Gamma_{1}^{n+1}(S_n)=S_{n+1}\).

The homeomorphic mapping
\[ \psi^{n+1}=g_{n+1}g_n^{-1}(\Gamma_{1}^{n+1})^{-1}:S_{n+1}\to S_{n+1} \]
is fixed on
\[ S_{n+1}\setminus \bigcup_{i=1}^{k_n}(\widetilde{\Delta}_{ni}\setminus \Delta_{ni}\cup D_{ni}), \]
therefore there exists an isotopy \(\Psi^{n+1}:E^3\to E^3\) such that

\[ \Psi_{0}^{n+1}=1;\qquad \Psi_{t}^{n+1}\biggm|_{E^3\setminus \bigcup_{i=1}^{k_n}u'_{ni}}=1;\qquad \Psi_{t}^{n+1}\bigm|_{S_{n+1}}=\psi^{n+1}. \]

Put

\[ H_{t}^{n+1}=\Gamma_{t}^{n+1}\circ(\Psi_{t}^{n+1}\Gamma_{1}^{n+1});\qquad \widetilde{H}_{t}^{\,n+1}=H_{t}^{n+1}\Phi_{1}^{n}. \]

From (2), by induction we find

\[ \Phi_{1}^{n+1}\bigm|_{\sigma^{2}}=g_{n+1}f. \tag{7} \]

1. The sequence of homeomorphisms \(\Phi_{1}^{n}:E^3\to E^3\) converges uniformly to a continuous mapping \(F:E^3\to E^3\). Since \(\Phi_{1}^{n}\) is joined to \(\Phi_{1}^{n+1}\) by the \(\varepsilon_n\)-isotopy \(\widetilde{H}_{t}^{\,n+1}\), then, after making for each isotopy \(\widetilde{H}_{t}^{\,n}\) a linear change of the variable \(t\), \(\chi_n:[0,1]\to [1-1/2^{\,n-1},\,1-1/2^n]\), and putting
   \[ F_0=H_0^1=1;\qquad F_t=\widetilde{H}_{\chi_n^{-1}(t)},\quad 1-1/2^{\,n-1}\le t\le 1-1/2^n;\qquad F_1\equiv F=\lim_{n\to\infty}\Phi_{1}^{n}, \]
   we obtain a pseudo-isotopy \(F_t:E^3\to E^3\). The relation \(F_1\bigm|_{\sigma^2}\equiv f\) follows from (7).

It is easily proved that the set \(K\) of points whose inverse images are degenerate is, for any \(n\), contained in
\[ V_n=\bigcup_{k=n}^{\infty}\bigcup_{i=1}^{n_k}u_{ki}. \]
By (4), \(V_n\) is the sum of pairwise nonintersecting regions of diameter \(<\varepsilon_n\), and \(u_{ki}\cap S^2\ne \Lambda\). Therefore \(K\subset S^2\); \(\dim K=0\). Since \(u_{ki}\cap S^2\) is a wild disk, \(K\) is contained in the set of wildness points of \(S^2\), and the conditions of Theorem 1 are satisfied.

Let us note that, for an embedding in \(E^3\) of the one-dimensional sphere \(S^1\), a theorem analogous to Theorem 1 is impossible, since tame spheres \(\Sigma^1\) approximating the wild sphere \(S^1\) may, when approaching \(S^1\), form increasingly complicated knots.

V. A. Steklov Mathematical Institute
Academy of Sciences of the USSR

Received
19 XI 1965

REFERENCES

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