MATHEMATICS

A. V. CHERNAVSKII

ON THE ARRANGEMENT OF \((n-1)\)-DIMENSIONAL SPHERES IN AN \(n\)-DIMENSIONAL SPHERE

(Presented by Academician P. S. Aleksandrov on May 5, 1961)

It is proved that two similarly situated systems of \((n-1)\)-dimensional spheres in the \(n\)-dimensional sphere \(\mathfrak S^n\) can be isotopically carried one into the other if each sphere is normally embedded and contains an \((n-1)\)-dimensional simplex of the sphere \(\mathfrak S^n\). Some consequences (see \((^4)\)) from Brown’s work \((^2)\) will be applied. The restriction imposed here on the spheres is analogous to Mazur’s condition \((^1)\).

Definitions and remarks. \(\mathfrak S^n\) is a Euclidean sphere; all simplices considered in it are spherical. \(\mathfrak S^{n-1}\) is the equator of the sphere \(\mathfrak S^n\). By an isotopy of the sphere \(\mathfrak S^n\) is meant a system \(g_t\) of topological transformations of \(\mathfrak S^n\), depending continuously on the parameter \(t\), \(0 \le t \le 1\), with \(g_0\) the identity.

A topological sphere \(S^{n-1}\) is normally embedded in \(\mathfrak S^n\) if there exists a topological mapping
\[ h:\mathfrak S^{n-1}\times I \to \mathfrak S^n, \]
where \(I\) is the interval \([-1,1]\), such that
\[ h(\mathfrak S^{n-1})=S^{n-1}. \]

According to Brown \((^{2,4,5})\), for a normally embedded sphere \(S^{n-1}\) there exists a topological mapping of \(\mathfrak S^n\) onto itself under which \(\mathfrak S^{n-1}\) is mapped onto \(S^{n-1}\).

Lemma. Let \(h\) topologically map \(\mathfrak S^n\) onto itself, and let
\[ S^{n-1}=h(\mathfrak S^{n-1}) \]
contain an \((n-1)\)-dimensional simplex \(D^{n-1}\) of the sphere \(\mathfrak S^n\). There exists an isotopy \(g_t\) of the sphere \(\mathfrak S^n\) carrying \(S^{n-1}\) into the boundary of some \(n\)-dimensional simplex \(d^n\), fixed outside a prescribed neighborhood of one of the regions complementary to \(S^{n-1}\), and inside some simplex \(d_1^n \subset d^n\).

Proof. Denote one of the regions bounded by \(S^{n-1}\) by \(U\), and \(h^{-1}(U)\) by \(V\). Take in \(U\) a simplex \(d^n\), the base \(d^{n-1}\) of which lies strictly inside \(D^{n-1}\) and
\[ S^{n-1}\cap d^n=d^{n-1}. \]
The boundary \(S^{n-2}\) of the simplex \(d^{n-1}\) is, evidently, normally embedded in \(D^{n-1}\), and hence
\[ \sigma^{n-2}=h^{-1}(S^{n-2}) \]
is normally embedded in \(\mathfrak S^{n-1}\).

According to Brown, there exists a topological transformation
\[ f:\mathfrak S^{n-1}\to \mathfrak S^{n-1}, \]
under which \(f(\sigma^{n-2})\) is the boundary of some simplex \(\delta^{n-1}\). Extend \(f\) along meridians to a transformation \(\bar f\) of the whole sphere \(\mathfrak S^n\), and consider the mapping
\[ h_1=h\bar f^{-1}. \]
Observe that
\[ h_1(\mathfrak S^{n-1})=S^{n-1} \]
and
\[ h_1(\delta^{n-1})=d^{n-1}. \]

Let
\[ \delta^n \subset V \]
be a simplex with base \(\delta^{n-1}\). Construct a topological mapping \(\tau\) of the region \(V\) into itself, carrying \(\delta^{n-1}\) into the lateral surface of the simplex \(\delta^n\), and fixed on
\[ \mathfrak S^{n-1}\setminus \delta^{n-1}, \]
and, analogously, a mapping

\(t: U \to U\), taking \(d^{n-1}\) to the lateral surface of \(d^n\) and fixed on \(S^{n-1}\setminus d^{n-1}\).

Consider a new mapping \(\varphi: \mathfrak{S}^n \to \mathfrak{S}^n\), equal to \(h_1\) outside \(V\) and to \(t h_1 \tau^{-1}\) on the image of \(\tau\). It is defined outside the simplex \(\delta^n\). Since \(\tau\) is fixed on \(\mathfrak{S}^{n-1}\setminus \delta^{n-1}\), and \(t\) is fixed on \(S^{n-1}\setminus d^{n-1}\) and \(h_1(\delta^{n-1})=d^{n-1}\), we have
\[ h_1(x)=t h_1 \tau^{-1}(x),\quad \text{if } x\in \mathfrak{S}^{n-1}\setminus \delta^{n-1}. \]
Consequently, \(\varphi\) maps topologically \(\mathfrak{S}^n\setminus \delta^n\) onto \(\mathfrak{S}^n\), and the boundary \(\delta^n\) is mapped onto the boundary \(d^n\). Extend \(\varphi\) inside these simplexes along their radii.

There exists an isotopy \(S_t\) which carries the ball \(V\) into the simplex \(d^n\), and is fixed outside \(\varphi^{-1}(O_\varepsilon(U))\), where \(\varepsilon\) is arbitrary, and on \(\varphi^{-1}(d_1^n)\), \(d_1^n\subset d^n\). The isotopy
\[ g_t=\varphi S_t \varphi^{-1} \]
carries \(S^{n-1}\) to the boundary of the simplex \(d^n\), and is fixed outside \(O_\varepsilon(U)\) and on \(d_1^n\).

Theorem. Let a finite number of normally embedded and pairwise nonintersecting spheres \(S_i^{n-1}\), \(1\leq i\leq k\), be situated in the sphere \(\mathfrak{S}^n\). If each of them contains an \((n-1)\)-dimensional simplex of the sphere \(\mathfrak{S}^n\), then there exists an isotopy of \(\mathfrak{S}^n\) carrying all \(S_i^{n-1}\) into the boundaries of \(n\)-dimensional simplexes.

Proof. For each sphere \(S_i^{n-1}\) there exists a topological mapping
\[ h_i:\mathfrak{S}^n\to \mathfrak{S}^n, \]
under which the equator \(\mathfrak{S}^{n-1}\) is mapped onto \(S_i^{n-1}\). Put \(\psi_{t0}\) identically for all \(t\), and suppose that an isotopy \(\psi_{tj-1}\) has already been constructed which carries the first \(j-1\) \((1\leq j\leq k)\) spheres into the boundaries of simplexes and does not change the conditions for the remaining spheres. Let us retain the former notation for the images under \(\psi_{tj-1}\).

Denote by \(U_j\) one of the domains bounded by the sphere \(S_j^{n-1}\). Let \(\varepsilon\) be so small that \(O_\varepsilon(U_j)\) contains no spheres \(S_i^{n-1}\) lying outside \(U_j\). Take in \(U_j\) two simplexes \(d\) and \(d'\subset d\), and let \(g_t'\) be the isotopy existing by the lemma, which carries \(S_j^{n-1}\) into the boundary \(d\) and is fixed outside \(O_\varepsilon(U_j)\) and on \(d'\).

Put in correspondence with each sphere \(S_i^{n-1}\) lying in \(U_j\) the simplex \(d_i\) whose boundary it is, if \(i<j\), or which it contains by hypothesis, if \(i>j\). Before carrying out the isotopy \(g_t\), isotopically carry the simplexes \(d_i\) inside \(d'\), leaving the complement of \(U_j\) fixed. For this, construct finite polygonal lines \(l_i\), pairwise nonintersecting, such that the beginning of \(l_i\) lies in \(d_i\), and the end in \(d'\). Let \(\varepsilon\) be so small that \(O_\varepsilon(l_i)\) are pairwise nonintersecting, homeomorphic to balls, and the ends of \(l_i\) are farther from the boundary of \(d'\) than \(\varepsilon\). Each simplex \(d_i\), by an isotopy fixed outside a sufficiently small neighborhood of it, is mapped onto a simplex lying in \(O_\varepsilon(l_i)\), and then, fixed outside \(O_\varepsilon(l_i)\), onto a simplex lying in \(d'\). Denote the isotopy thus constructed by \(f_t\). The isotopy
\[ \psi_{tj}=g_t f_t \psi_{tj-1} \]
carries the first \(j\) spheres \(S_i^{n-1}\) into the boundaries of simplexes and preserves the conditions of the theorem for the remaining spheres. After \(k\) steps the proof of the theorem is completed.

Corollary 1. If two similar systems of \((n-1)\)-dimensional pairwise nonintersecting spheres are given, each of which is normally embedded and contains an \((n-1)\)-dimensional simplex of the sphere \(\mathfrak{S}^n\), then there exists an isotopy carrying one of these systems into the other.

This follows from the theorem proved and from the validity of the proposition for systems of boundaries of \(n\)-dimensional simplexes.

Corollary 2. For an arbitrary finite system of pairwise nonintersecting \((n-1)\)-dimensional spheres, polyhedral in \(\mathfrak{S}^n\), there exists an isotopy of \(\mathfrak{S}^n\) carrying these spheres into the boundaries of simplexes.

By virtue of the “\(n\)-dimensional theorem of Alexander” proved by Newman \((^3)\),

according to which the polyhedral sphere bounds regions whose closures are homeomorphic to balls, such a sphere is embedded in \(S^n\) normally. Consequently, for systems of polyhedral spheres the conditions of the theorem proved are satisfied.

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

Received
21 IV 1961

REFERENCES

1. B. Mazur, Bull. Am. Math. Soc., 65, No. 1, 59 (1959).
2. M. Brown, Bull. Am. Math. Soc., 66, No. 2, 374 (1960).
3. M. H. A. Newman, Proc. Roy. Soc., A 257, No. 1288 (1960).
4. A. B. Sosinskii, DAN, 139, No. 6 (1961).
5. M. Brown, Am. Math. Soc. Notices, 7, No. 7, 576 (1960).