A. V. CHERNAVSKII

THE IMPOSSIBILITY OF A STRICTLY TWO-FOLD CONTINUOUS DECOMPOSITION OF A HOMOLOGICAL CUBE

(Presented by Academician P. S. Aleksandrov, 3 I 1962)

1. Theorem. A two-fold continuous decomposition of a homological cube \(Q^n\) contains an element consisting of a single point.

The case \(n=1\) was analyzed in \((^1)\), the case \(n=2\) in \((^2)\), and the Euclidean case \(n=3\) independently in \((^{3-5})\). The arguments are applicable to the case of Euclidean space \(E^n\), which, however, is not considered. We note that the example in \((^5)\) of a strictly two-fold continuous mapping of \(E^n\) does not give rise to a continuous decomposition, since the mapping turns out not to be closed.

2. Points belonging to one element of the decomposition are called conjugate, and the point conjugate to a point \(x\) is denoted by \(\bar{x}\). Continuity of the decomposition means that if a point \(y\) is sufficiently close to a point \(x\), then \(\bar{y}\) lies in a prescribed neighborhood of the pair \(x \cup \bar{x}\).

The closure, open kernel, and boundary of a set \(M\) relative to the space \(X\) are denoted respectively by: \([M]_X\), \(\operatorname{Int}_X M\), \(\operatorname{Fr}_X M\).

3,1. The problem was first posed for Euclidean cubes, and for \(n \leq 3\) there is no need to appeal to homological manifolds. For \(n > 3\) such a need arises because the proof rests on the Smith theorems formulated below. All homology is considered modulo 2. The notion of a generalized homological manifold is taken in Smith’s form \((^6);(^7)\), p. 268. In \((^{8,9})\) it is shown that such manifolds coincide with Wilder’s locally orientable manifolds \((^{10})\), p. 281. In what follows they are called \(ng\)-manifolds. If an \(ng\)-manifold has the homological type of an \(n\)-sphere, it is called here an \(ng\)-sphere.

3,2. The notion of a generalized homological cube, henceforth simply an \(ng\)-cube, is defined in \((^{10})\), p. 287. The following assertion holds: if a \(kg\)-sphere is placed in a \((k+1)g\)-sphere, it decomposes it into two regions whose closures are \((k+1)g\)-cubes, and conversely, the sum of two \(kg\)-cubes glued together topologically along their boundaries is a \(kg\)-sphere \((^{10})\), p. 312.

The boundary of \(Q^n\) is denoted by \(\dot{Q}^n\), and \(Q^n \setminus \dot{Q}^n\) by \(\operatorname{Int} Q\).

3,3. Newman–Smith Theorem \((^{11,12});(^7)\), p. 272). The set of fixed points of a continuous involution defined on an \(ng\)-manifold is nowhere dense or coincides with the whole manifold.

3,4. Smith Theorem \((^6);(^7)\), p. 272). The set of fixed points of a continuous involution defined on an \(ng\)-sphere is a \(kg\)-sphere, where \(k<n\).

We note that for a Euclidean sphere the set of fixed points need not be a Euclidean sphere (see, for example, \((^{14})\)).

4. Preliminary construction. Arguing by contradiction, suppose that on \(Q^n\) a continuous decomposition \(\varphi\) into pairs of points is given. The closed sets

\[ M_n=\{x;\ d(x,\bar{x}) \geq \tfrac{1}{n}\} \]

(\(d\) is the distance function) then together give \(Q^n\). The open set \(H_0=\bigcup \operatorname{Int} M_n\), by Baire’s property, is everywhere dense. \(P_1=Q^n \setminus H_0\) is closed and nowhere dense. Suppose that for all \(\alpha<\beta<\omega_1\) closed sets \(P_\alpha\) have already been constructed, and each

each \(P_\alpha\) is contained and nowhere dense in the preceding ones. If \(\beta\) is a limit ordinal, set \(P_\beta=\bigcap P_\alpha,\ \alpha<\beta\); if \(\beta=(\beta-1)+1\), then \(P_\beta=P_{\beta-1}\setminus H_{\beta-1}\), where
\[ H_{\beta-1}=\bigcup \operatorname{Int}_{\beta-1}(M_n\cap P_{\beta-1}). \]
\(P_\beta\) is closed and nowhere dense in all \(P_\alpha,\ \alpha<\beta\). For some \(\gamma<\omega_1\) the set \(P_\gamma\) is empty, and \(Q^n\) is the sum of pairwise disjoint sets \(H_\alpha,\ 0\leq \alpha<\gamma\).

5. Lemmas

Lemma 1. If \(x\in P_1\), then every neighborhood of \(x\) contains conjugate pairs; if, however, \(x\in H_0\), then there is a neighborhood of \(x\) containing no conjugate pairs.

Lemma 2. If \(x_0\in H_0\), then the mapping \(x\to \bar{x}\) maps some neighborhood of \(x_0\) topologically onto some neighborhood of \(\bar{x}_0\).

Both lemmas follow easily from the continuity of the decomposition \(\varphi\) and the definition of the set \(H_0\).

Lemma 3. If \(x\) and \(\bar{x}\) lie in \(H_0\), then they are either both boundary points or both interior points.

Indeed, at boundary points and at interior points \(Q^n\) has different local homological character, whereas, according to Lemma 2, \(x\) and \(\bar{x}\) have homeomorphic neighborhoods.

Lemma 4. If \(x_0\in P_1\), then every neighborhood of \(x_0\) contains conjugate pairs both of whose points belong to \(H_0\cap \operatorname{Int} Q^n\).

Let \(x_0\in H_1\); since \([H_1]=P_1\), it is enough to consider this case. Let
\[ 0<\varepsilon<\frac12 d(x_0,P_2) \]
and choose \(\delta,\ 0<\delta<\varepsilon\), so that if
\[ x\in O_\delta(x_0)\cap H_1, \]
then
\[ \bar{x}\in O_\varepsilon(\bar{x}_0). \]
In \(O_\delta(x_0)\), according to Lemma 1, there must be a conjugate pair. By the choice of \(\varepsilon\), only points of \(H_0\) and \(H_1\) can enter it. But, by the choice of \(\delta\), points of \(H_1\) are excluded. Since \(H_0\) is open and, by Lemma 3, such a pair may be taken in \(\operatorname{Int} Q^n\).

6. Main lemma

The mapping \(e: Q^n\to Q^n\), defined by
\[ e(x)=\bar{x},\quad \text{if } x\in H_0;\qquad e(x)=x,\quad \text{if } x\in P_1, \]
is a continuous involution of \(Q^n\).

First one proves the continuity of \(e\) at interior points, then at boundary points, and finally that if \(x\in H_0\), then also \(\bar{x}\in H_0\). At points of \(H_0\), \(e\) is continuous by Lemma 2. Suppose that for all \(\alpha<\beta\), \(e\) is continuous at the points of
\[ H_\alpha\cap \operatorname{Int} Q^n \]
and let
\[ x_0\in H_\beta\cap \operatorname{Int} Q^n. \]
We show that \(e\) is also continuous at the point \(x_0\). Let
\[ 0<\varepsilon<\frac13 d(x_0,\bar{x}_0\cup P_{\beta+1}\cup \dot{Q}^{\,n}). \]
There is \(\delta_1\) such that
\[ 0<\delta_1<\varepsilon \]
and
\[ \text{if } x\in O_{\delta_1}(x_0)\cap H_\beta,\quad \text{then } \bar{x}\in O_\varepsilon(\bar{x}_0). \tag{1} \]
By the continuity of \(\varphi\), there is \(\delta,\ 0<\delta<\delta_1\), such that
\[ \text{if } x\in O_\delta(x_0),\quad \text{then } \bar{x}\in O_{\delta_1}(x_0)\cup O_\varepsilon(\bar{x}_0). \tag{2} \]
By virtue of (2), \(H_0\cap O_\delta(x_0)\) splits into the set \(\Phi_1\) of points conjugate to which lie in \(O_{\delta_1}(x_0)\), and the set \(\Phi_2\) of points conjugate to which lie in \(O_\varepsilon(\bar{x}_0)\). Both are open. \(\Phi_1\), in turn, splits into the set \(\Phi_1'\) of points conjugate to which lie in \(O_{\delta_1}(x_0)\cap H_0\), and the set \(\Phi_1''\) of points conjugate to which lie in \(O_{\delta_1}(x_0)\cap P_1\). \(\Phi_1'\) is open by Lemma 2. \(\Phi_1''\) is also open. Otherwise, arbitrarily close to \(x\in \Phi_1''\) there would lie points of \(\Phi_1'\), and \(e\) would have a discontinuity at \(\bar{x}\). But \(e\) is continuous at \(\bar{x}\) by the induction hypothesis, since, by (1), \(\bar{x}\) cannot belong to \(H_\beta\).

Since the sets \(\Phi_1'\), \(\Phi_1''\), \(\Phi_2\) are open,
\[ \operatorname{Fr}_{O_\delta}\Phi_1'\cup \operatorname{Fr}_{O_\delta}\Phi_1''\cup \operatorname{Fr}_{O_\delta}\Phi_2\subset P_1. \tag{3} \]

Define on \(O_\delta(x_0)\) a mapping \(g\) by the equalities:

\[ g(x)=\dot{x}, \qquad \text{if } x\in \Phi_1''\cup \Phi_2, \tag{4} \]

\[ g(x)=e(x), \qquad \text{if } x\in \Phi_1'\cup \Phi_2. \tag{5} \]

It is continuous; in view of (3), it is enough to verify this only at the points \(\operatorname{Fr}_{O_\delta}\Phi_1'\). If \(x\in \operatorname{Fr}_{O_\delta}\Phi_1'\cap H_\alpha\), where \(\alpha<\beta\), then \(g\) is continuous at \(x\) by (5) and the induction hypothesis. If, however, \(x\in \operatorname{Fr}_{O_\delta}\Phi_1'\cap H_\beta\), then \(g\) is continuous at \(x\) by the definition of the set \(\Phi_1'\) and the continuity of the partition \(\varphi\). By (4), \(g\) has period 2 on \(O_\delta(x_0)\cap g(O_\delta(x_0))\). A connected neighborhood \(O\) such that \(g(O)=O\subset O_\delta\cap g(O_\delta)\) satisfies the condition of the Newman–Smith theorem, and, since \(\Phi_1\cap O\ne \Lambda\), in this neighborhood the set of fixed points of \(g\) is nowhere dense. Consequently, \(O\cap(\Phi_1\cup\Phi_2)=\Lambda\). Since \(e\) coincides with \(g\) in \(O\), it is continuous on \(O\), hence at \(x_0\), and hence everywhere inside \(Q^n\).

From the emptiness of \(\Phi_1''\cup\Phi_2\) in \(O\) there follows the following observation, used below.

Remark. If a point \(x\in H_0\) is sufficiently close to \(P_1\cap \operatorname{Int} Q^n\), then \(\bar{x}\) also belongs to \(H_0\).

Now let \(x_0\in P_1\cap \dot{Q}^n\), \(0<\varepsilon<\tfrac13 d(x_0,\bar{x}_0)\), and choose \(\delta\), \(0<\delta<\varepsilon\), so that if \(x\in O_\delta(x_0)\), then \(\bar{x}\in O_\varepsilon(x_0\cup \bar{x}_0)\). \(H_0\cap O_\delta\) decomposes, therefore, into two open sets: \(\Phi_1\) of points whose conjugates lie in \(O_\varepsilon(x_0)\), and \(\Phi_2\) of points whose conjugates lie in \(O_\varepsilon(\bar{x}_0)\). \(\Phi_1\) is nonempty by Lemma 4. If \(e\) has a discontinuity at \(x_0\), then \(\Phi_2\) is also nonempty. In \(O_\delta(x_0)\) the sets \(\Phi_1\) and \(\Phi_2\) are separated by points of \(P_1\). But at a point \(y\in \operatorname{Fr}_{O_\delta}\Phi_2\cap P_1\cap \operatorname{Int} Q^n\) the mapping \(e\) would be discontinuous, which, as proved, is impossible at an interior point. Thus \(e\) is continuous everywhere in \(Q^n\).

It remains to show that if \(x\in H_0\), then also \(\bar{x}\in H_0\). Assuming the contrary, consider the set \(\Phi\) of points \(x\) such that \(x\in H_0\), and \(\bar{x}\notin H_0\). Let \(x_0\in \Phi\). Then an entire neighborhood of \(x_0\) lies in \(\Phi\); otherwise \(e\) would have a discontinuity at \(\bar{x}_0\). Hence \(\Phi\) is open. Using the observation made above, we find that \(\operatorname{Fr}\Phi\cap P_1\cap \operatorname{Int} Q^n\) is empty. Consequently, \(\Phi\supset \operatorname{Int} Q^n\). But this contradicts Lemma 4. Therefore \(\Phi\) is empty and \(e\) is a continuous involution of \(Q^n\).

7. Proof of the theorem. Let \(C=\dot{Q}^n\setminus p\) be the cone over the boundary of \(Q^n\). Lemma 3 permits one to extend \(e\), in the following way, to an involution on \(S=Q^n\cup C\). If \(x\in \dot{Q}^n\cap P_1\), the whole segment \([px]\) remains fixed; if \(x\in \dot{Q}^n\cap H_0\), then the segment \([px]\) is mapped linearly onto the segment \([p\bar{x}]\). In \((^{13})\) it is shown that the cone over an \((n-1)\) \(g\)-sphere is an \(ng\)-cube, and, according to 3.2, \(S\) is an \(ng\)-sphere. By 3.4 the set of fixed points \(K\) is a \(kg\)-sphere, and the following cases are possible: \(k=-1,0,1,\ldots,n-1\). The first case \(k=-1\) is impossible, since \(p\) is fixed. The second case \(k=0\), a pair of points, means that \(P_1\) consists of a single point, which therefore has no conjugate, contrary to the supposition. The remaining cases \(k=1,\ldots,n-1\) are reduced to the preceding one as follows. \(C_1=C\cap K\) is a cone over its boundary in \(K\). Using Theorem 2 from \((^{13})\), we conclude that this boundary is a \((k-1)\) \(g\)-sphere, and, consequently, according to 3.2, \(P_1=K\setminus C_1\) is a \(kg\)-cube. By the main lemma, \(\varphi\) induces on \(P_1\) a strictly twofold continuous partition. Therefore all the arguments carried out for \(Q^n\) are applicable to \(P_1\). Since \(k<n\), after a finite number of steps we arrive at the case \(k=0\), and hence at a contradiction.

8. Remark 1. The theorem is valid, in particular, for the Euclidean cube, and also remains true for \(S^n\setminus p\), where \(S^n\) is an \(ng\)-sphere and \(p\in S^n\), in particular for \(E^n\).

Remark 2. The arguments carried out in the proof of the main lemma show that every strictly twofold continuous decomposition \(\varphi\) defined on an \(ng\)-sphere \(S^n\) is determined by a sequence of embedded \(kg\)-spheres
\[ S^n \supset S^{k_1} \supset \cdots \supset S^{k_m}, \]
where \(n>k_1>\cdots>k_m\ge 0\), and by involutions \(\varphi_0, \varphi_1,\ldots,\varphi_m\), defined respectively on the spheres \(S^{k_i}\), with \(S^{k_i+1}\) serving as the set of fixed points for \(\varphi_{k_i}\). The elements of the decomposition \(\varphi\) are the pairs of points corresponding to one another under a certain involution.

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

Received
26 XII 1961

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