Reports of the Academy of Sciences of the USSR

1. Volume 158, No. 1

MATHEMATICS

A. V. CHERNAVSKII

ISOTOPY OF ELEMENTS AND SPHERES IN \(n\)-DIMENSIONAL SPACE FOR \(k > \frac{2}{3}n - 1\)

(Presented by Academician P. S. Aleksandrov, 3 IV 1964)

1. Here a brief description is given of the method of local rearrangement of homeomorphisms of \(n\)-dimensional Euclidean space \(E^n\), some results obtained with its help are formulated, and an outline is given of the proof of one of them. The method also makes it possible to consider embeddings of polyhedra in \(E^n\).

In \(E^n\) a Cartesian coordinate system with origin \(O\) is fixed. Let \(k\) be a fixed integer, \(k < \frac{2}{3}n - 1\). Denote by \(T\) the \((n-1)\)-dimensional subspace \(Ox_1,\ldots,x_{k-1}\), by \(F\) the subspace spanned by \(T\) and the axis \(Ox_k\), and by \(Y\) the orthogonal complement to \(T\). By \(E^n_+\), \(E^n_-\) and \(F_+\), \(F_-\) are denoted the half-spaces, reckoning along the axis \(Ox_k\); \(B^{k-1}\) is the unit ball in \(T^k\), \(B^k\) in \(F\).

An embedding \(p: M^k \to E^n\) of a manifold \(M^k\) in \(E^n\) is called locally flat if for each point \(x \in pM\) there exists a neighborhood \(H\) in \(E^n\) such that the pair \((H, H \cap pM)\) is homeomorphic to the standard pair \((E^n,F)\) or to the pair \((E^n,F_+)\). It is established that a locally flat embedding in \(E^n\) of a closed simplex of any dimension is topologically equivalent to the standard one. An isotopy in \(E^n\) is a system of homeomorphisms of \(E^n\) onto itself \(g_t(x)\), continuous in the aggregate of the variables \(x \in E^n\), \(t \in [0,1]\); \(g_0(x)\) is always assumed to be the identity. Brown and Gluck called in \((^1)\) a homeomorphism stable if it is representable as a superposition of homeomorphisms each of which is the identity on some region. From the classical result of Alexander it follows that a homeomorphism that is the identity on a region is isotopic to the identity under an isotopy that is the identity on the region. From this follows the equivalence of Theorems 1 and \(1'\). Theorems 2 and 3 are consequences of Theorem \(1'\) and Stallings’ result \((^2)\). Embeddings \(p\) and \(p': X \to E^n\) are called isotopic if there exists an isotopy \(g_t\) in \(E^n\) such that \(p = g_1p'\).

Theorem 1. If \(k < \frac{2}{3}n - 1\), then every homeomorphism \(h: E^n \to E^n\) is representable as a superposition of two homeomorphisms \(h = h_kh_n\), where \(h_n\) is stable, and \(h_k\) is the identity on a \(k\)-dimensional hyperplane.

Theorem \(1'\). If \(k < \frac{2}{3}n - 1\), then every locally flat embedding of a \(k\)-simplex in \(E^n\) is isotopic to the standard rectilinear embedding under an isotopy of \(E^n\) that is the identity outside some bounded region.

Theorem 2. If \(k < \frac{2}{3}n - 1\), then every locally flat embedding of a \(k\)-sphere in \(E^n\) is isotopic to the embedding of the boundary of a straight \((k+1)\)-simplex under an isotopy fixed outside a bounded region.

Theorem 3. If \(k < \frac{2}{3}n - 1\), then two locally flat embeddings of a \(k\)-simplex in \(E^n\), coinciding on the boundary, are isotopic under an isotopy that is the identity on the image of the boundary and outside a bounded region.

2. Horns. Consider surfaces in \(E^n\) that are given by equations of the form
\[ x_k = t\sqrt{x_{k+1}^2+\cdots+x_n^2}, \qquad -\infty \leq t \leq \infty . \]
For infinite values of \(t\) one obtains respectively \(F_-\) and \(F_+\). The surface with coefficient \(t\) is the product of \(T\) and a cone in \(Y\) with vertex \(O\), axis \(Ox_k\), and aperture angle \(\operatorname{arc\,ctg} t\). We shall call these surfaces horns, and denote the horn with coefficient \(t\) by \(R_t\). If the set \(L\) is closed outside any neighborhood of \(B^{k-1}\), then we say that \(L\) touches \(R_t\), if

for every \(\varepsilon>0\) there exists \(\delta>0\) such that in the \(\delta\)-neighborhood of \(B^{k-1}\) the set \(L\) is nonempty and lies between \(R_{t-\varepsilon}\) and \(R_{t+\varepsilon}\).

Curved horns. A surface in \(E^n\) (the topological image of an \((n-1)\)-hyperplane) is called a curved horn \(\mathfrak R_t\) if it is tangent to \(R_t\), contains \(T\), and intersects exactly once each semicircle with center on \(T\), with diameter parallel to the axis \(Ox_k\), and with plane orthogonal to \(T\). We shall regard two curved horns as coinciding if they coincide in some neighborhood of \(B^{k-1}\). The following two propositions are easily proved:

A. If a set \(L\), closed outside every neighborhood \(B^{k-1}\), lies on one side of the horn \(R_t\), then there exists a curved horn \(\mathfrak R_t\) lying between \(R_t\) and \(L\); and if, in addition, \(L\) touches \(R_t\), then there exists a curved horn \(\mathfrak R'_t\) such that \(L\) lies between \(\mathfrak R'_t\) and \(R_t\).

B. If the curved horns \(\mathfrak R_{t_1}\) and \(\mathfrak R_{t_4}\) do not intersect, and \(\mathfrak R_{t_2}\) and \(\mathfrak R_{t_3}\) lie between them, then there exists a homeomorphism of \(E^n\) onto itself, identical on \(T\), outside a prescribed neighborhood \(B^{k-1}\), and outside the region between \(\mathfrak R_{t_1}\) and \(\mathfrak R_{t_4}\), which carries \(\mathfrak R_{t_2}\) onto \(\mathfrak R_{t_3}\).

3. The principal homeomorphism. It is easy to construct a homeomorphic mapping
\(g: E^n\setminus B^{k-1}\to E^n\setminus B^k\) with the following property: a sequence of points then and only then converges to a point \(x\in \operatorname{Int} B^k\) when its inverse image converges to the projection of \(x\) onto \(B^{k-1}\) and at the same time touches the horn \(R_t\), where \(\left|\dfrac{1}{t}\right|\) is equal to the radius of the sphere which in \(F\) passes through the boundary of \(B^{k-1}\) and \(x\), and the sign of \(t\) is equal to the sign of the abscissa of the \(k\)-th coordinate of \(x\).

4. The principal lemma. The proof of all the results is based on this lemma. In particular, the dimensional restriction \(k<{}^2/_3\,n-1\) is connected with the fact that the lemma has not been proved for large \(k\). The lemma is also true for \(n=3,\ k=1\) or \(2\), but here a special proof is required. Thus the first unproved case is \(n=4,\ k=2\). The proof of the lemma follows essentially the proof of the Stallings–Zeeman engulfing lemma \((^3)\). The difference is that the Stallings \(p\)-connectedness conditions are not fulfilled and an infinite polyhedron has to be engulfed.

Lemma. Let \(Q_t\) be the closed region bounded by the horn \(R_t\); let \(h: Q\to E^n\) be a homeomorphic mapping, identical on \(T\), and let \(\varepsilon>0\) and a neighborhood \(H(B^{k-1})\) be given. There exists a homeomorphism \(e: E^n\to E^n\), identical on \(T\cup (E^n\setminus H)\), such that \(ehQ_t\) contains a ball segment which lies in \(F\cap Q_t\) and is based on \(B^{k-1}\). If \(hQ_t\subset Q_t\), then one may require that \(E^n\setminus Q_t\) remain fixed.

It is convenient to take for \(B^{k-1}\) a straight simplex in \(T\), and for the ball segment which is to be engulfed, a \(k\)-simplex \(B^k\) in \(F\cap Q\), for which \(B^{k-1}\) serves as the base and the vertex is sufficiently close to \(B^{k-1}\). Put \(hQ=V\) and \(hB^k=\mathfrak B\). Let \(\Delta^{k+1}\) be the standard \((k+1)\)-simplex, \(\Delta^{k-1}\) its \((k-1)\)-face, and \(\Delta^k\) and \(\Delta^k_1\) two \(k\)-faces adjacent to \(\Delta^{k-1}\).

The proof of the lemma splits into two steps. First one constructs a continuous mapping \(\varphi:\Delta^{k+1}\to E^n\), which maps \(\Delta^k\) linearly onto \(B^k\) and \(\Delta^{k-1}\) linearly onto \(B^{k-1}\), and \(\Delta^k_1\) homeomorphically onto \(\mathfrak B\), with \(\varphi(\Delta^{k+1}\setminus\Delta^{k-1})\subset E^n\setminus T\). The construction of this mapping uses only the homotopy properties of the pair \((E^n,T)\) and can be carried out in all dimensions.

Then \(\varphi\) is put in general position with respect to the triangulation \(\Delta^{k+1}/\Delta^{k-1}\). In doing so, \(\varphi\) must still map \(\Delta^k\) linearly onto \(B^k\), and \(\varphi\Delta^k_1\subset V\). In \(\Delta^{k+1}\) one considers the inverse image of all pairwise intersections of the images of open simplexes. Through each of its points a segment is drawn parallel to the edge opposite \(\Delta^{k-7}\), and the image under \(\varphi\) of the resulting polyhedron is taken. The image is denoted by \(S\). In view of \(k<{}^2/_3\,n-1\): a) \(\dim S<n/3+1\); b) \(E^n\setminus T\) has the homotopy type of an \(s\)-sphere for \(s>n/3+1\).

Therefore the pair \((E^n\setminus T,V)\) is \((n/3+1)\)-connected. Using the proof of the absorption lemma, we can construct a homeomorphism \(e'\), identical on \(T\), and such that \(e'V \supset S\). The remaining part of the image consists of piecewise linear cells, situated “regularly” with respect to \(e'V\) and with respect to one another. Then again, using the absorption lemma, one constructs a homeomorphism \(e''V\) such that \(e''e'V \supset \varphi\Delta^{k+1}\). Put \(e=e''e'\). Although here one has to apply the elementary operations of the proof of the absorption lemma an infinite number of times, it is easily shown that the process converges and gives a homeomorphism in the limit.

5. Scheme of the proof of Theorem 1. For \(k=0\) the theorem is trivial. Suppose that it is true for dimensions smaller than \(k\). Then we may assume that the given homeomorphism is identical on \(T\). We shall show that there exists a homeomorphism \(\bar h\), coinciding with \(h\) outside some neighborhood of \(B^{k-1}\) and identical on \(B^k\). From this it is easy to obtain that there exists a homeomorphism coinciding with \(h\) on some region and identical on all of \(F\). This is what is required. Let us first assume that \(h\) has the following property:

C. The image of each horn \(R_t,\ |t|<1\), touches \(R_t\).

In this case, as the required homeomorphism \(\bar h\) one may take the homeomorphism \(ghg^{-1}\), extended identically on \(B^k\). That such an extension leads to a homeomorphism follows at once from the property of \(g\) (item 4) and condition C. Therefore it remains to show that there exists such a homeomorphism \(e\), fixed on \(T\) and the complement of some bounded neighborhood of \(B^{k-1}\), that \(eh\) has property C.

It is convenient to regard two copies \(E^n\) as given and \(h\) as mapping one of them onto the other. The corresponding sets are denoted in the same way, but in the preimage a bar is added above. Thus the horns in the image are denoted, as before, by \(R_t\), and in the preimage by \(\bar R_t\); put \(h\bar R_t=U_t,\ h^{-1}R_t=\bar U_t\). The crooked horns are denoted by \(\mathfrak R_t\) and \(\bar{\mathfrak R}_t\). Denote by \(Q_t^+\) the positive side of \(R_t\), by \(Q_t^-\) the negative side; correspondingly in the preimage, \(\bar Q_t^{\pm}\); put \(h\bar Q_t^{\pm}=V_t^{\pm},\ h^{-1}Q_t^{\pm}=\bar V_t^{\pm}\). Note that if condition C is satisfied for a countable set of \(t\)’s dense in \((-1,1)\), then it is satisfied for all \(t\in(-1,1)\). We construct \(e\) as an infinite superposition of homeomorphisms, achieving the fulfillment of the condition successively for all dyadic rational \(t\in[-1,1]\). The successive construction of homeomorphisms is based only on the lemma and proposition B. Therefore each homeomorphism may be taken to be an \(\varepsilon\)-homeomorphism, identical outside a neighborhood of \(B^{k-1}\), where \(\varepsilon>0\) and the neighborhood are fixed for the given homeomorphism. From this it follows that the construction can be carried out so that in the limit a homeomorphism is obtained, identical on \(T\) and outside some neighborhood of \(B^{k-1}\).

Let the \(t_i\) run in a definite order through the dyadic rational numbers in \([-1,1]\): \(t_0=0,\ t_1=1,\ t_2=-1,\ t_3=1/2\), and so on. If all numbers of the form \(q/2^s\) for a fixed \(s\) have already been numbered, then next, in decreasing order, the numbers \(q/2^{s+1}\) with odd \(q\) are numbered. The construction is inductive. For convenience we shall assume that each homeomorphism \(e_a\), which is constructed at some step in \(E^n\) or in \(\bar E^n\), changes \(h\), i.e. if \(e_a\) is constructed in \(\bar E^n\), then henceforth we take \(\bar h e_a^{-1}\) for \(h\) and denote by \(\bar U_t\) \(e_a h^{-1}R_t\) and by \(U_t\) \(he_a^{-1}\bar R_t\); while if \(e_a\) is constructed in \(E^n\), then henceforth we take \(e_a h\) for \(h\), denote \(e_a h\bar R_t\) by \(U_t\), and \(h^{-1}e_a^{-1}R_t\) by \(\bar U\).

Zeroth step. 1) Using the lemma, we construct a homeomorphism in \(E^n\) which carries \(V_0^+\) into a position in which it covers a certain spherical segment \(D_0^+\) in \(F_+\), resting on \(B^{k-1}\).

2) Using A and B, we first construct a crooked horn \(\mathfrak R_\infty\) separating \(D_0^+\) from \(U_0\), and carry it to \(R_0\) (i.e. to a crooked horn coinciding with \(R_0\) in some neighborhood of \(B^{k-1}\)).

3) Leaving \(Q_0^{+}\) fixed, we make \(V_0^{-}\) enclose \(D_0^{-}\) (a ball segment in \(F_{-}\) supported on \(B^{k-1}\)).

4) We construct a crooked horn \(\mathfrak R_{-\infty}\) separating \(U_0\) from \(D_0^{-}\), and then transform \(\mathfrak R_{-\infty}\) into some crooked horn \(\mathfrak R_0^{-}\) touching \(R_0\) on the negative side. Now \(U_0\) lies between \(R_0\) and \(\mathfrak R_0^{-}\), hence touches \(R_0\) on the negative side. Consequently, \(\bar U_0 \subset \bar Q_0^{+}\).

5) In the preimage, leaving \(\bar Q_0^{-}\) fixed, we make \(\bar V_0^{+}\) enclose \(\bar D_0^{+}\) (a ball segment in \(\bar F_{+}\) supported on \(B^{k-1}\)).

6) We construct a crooked horn \(\bar{\mathfrak R}_{\infty}^{-}\) separating \(\bar D_0^{+}\) from \(\bar U_0\), and then, leaving \(\bar Q_0^{-}\) fixed, transform it into a crooked horn touching \(\bar R_0\) on the positive side. We have now achieved that \(U_0\) touches \(R_0\) on the negative side, while \(\bar U_0\) touches \(\bar R_0\) on the positive side.

Inductive passage. Suppose that we have already achieved that, for every \(t_i\), \(i<i_0\), \(U_{t_i}\) touches \(R_{t_i}\) on the negative side and, simultaneously, \(\bar U_{t_i}\) touches \(\bar R_{t_i}\) on the positive side.

1) Consider those \(t_i\) for which \(t_i>t_{i_0}\), \(i<i_0\). In the preimage, for each such \(t_i\), we construct, according to A, a crooked horn \(\bar R_{t_i}^{+}\) touching \(\bar R_{t_i}\) on the positive side, so that \(\bar U_{t_i}\) lies between it and \(\bar{\mathfrak R}_{t_i}^{+}\). Then, with \(\bar Q_{t_{i_0}}^{-}\) fixed, we transform each \(\bar{\mathfrak R}_{t_i}^{+}\) into \(\bar R_{t_i}\) and simultaneously transform \(\bar R_{t_i}\) into a crooked horn touching \(\bar R_{t_i}\) on the negative side. Then \(U_{t_i}\) \((t_i>t_{i_0},\, i>i_0)\) lies between \(R_{t_i}\) and the horn adjacent to it on the positive side \(R_{t_i'}\), \(i'<i_0\). At the same time \(U_1 \subset Q_1^{+}\).

2) Denote by \(t_{i'}\) and \(t_{i''}\) the two numbers adjacent to \(t_{i_0}\) among the numbers \(t_i\), where \(i<i_0\). All \(U_{t_i}\) with \(i<i_0\) lie outside the region between \(R_{t_{i'}}\) and \(R_{t_{i''}}\), while \(U_{t_{i_0}}\) lies in it. We construct crooked horns \(\mathfrak R'\) and \(\mathfrak R''\) separating \(U_{t_{i_0}}\), respectively, from \(R_{t_{i'}}\) and \(R_{t_{i''}}\), and then, with \(Q_{t_{i'}}^{+}\cup Q_{t_{i''}}^{-}\) fixed, transform \(\mathfrak R'\) to \(R_{t_{i_0}}\), and \(\mathfrak R''\) to a crooked horn touching \(\mathfrak R_{t_{i_0}}\) on the negative side.

3) Similarly, using the fact that each \(U_{t_i}\), where \(1>t_i>t_{i_0}\), \(i<i_0\), lies between two adjacent horns, we ensure that these \(U_{t_i}\) touch \(R_{t_i}\) on the negative side.

4) Leaving \(Q_1^{-}\) fixed, by the lemma we construct in \(\bar E\) a homeomorphism as a result of which \(\bar V_1^{+}\) encloses \(\bar D_{i_0}^{+}\). Then we separate \(\bar U_1\) from \(\bar D_{t_i}^{+}\) by a crooked horn, which, with \(\bar Q_{t_{i'}}^{-}\) fixed, we transform to \(R_1\) so that \(U_1\) touches \(R_2\) on the negative side, where \(t_{i'}\) is the number adjacent to 1 among the numbers \(t_i\), \(i<i_0\). Now the tangency conditions are satisfied in the image. It remains to carry out constructions analogous to the last three steps in the preimage, and then we shall be ready to pass to \(t_{i_0+1}\). This completes the induction. The fact that, as a result of these constructions, in the limit condition C will be satisfied for all dyadic rational numbers in \((-1,1)\), is verified directly.

1. These results in dimensions \(k\) not exceeding \(\frac12 n-2\) can be derived from results announced by Gluck \((^4)\) and Greathouse \((^5)\). On the other hand, the method set forth here makes it possible to obtain the results of Gluck and Greathouse for \(k<\frac23 n-1\).

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

Received
26 III 1964

CITED LITERATURE

\({}^{1}\) M. Brown, H. Gluck, Ann. Math., 79, 1 (1964).
\({}^{2}\) J. Stallings, Ann. Math., 77, 490 (1963).
\({}^{3}\) J. Stallings, Proc. Cambr. Phil. Soc., 58, 481 (1962).
\({}^{4}\) H. Gluck, Bull. Am. Math. Soc., 69, 843 (1963).
\({}^{5}\) Ch. Greathouse, Bull. Am. Math. Soc., 69, 820 (1963).