MATHEMATICS

A. P. SAVIN

ON DECOMPOSITIONS OF COMPLETELY NORMAL SPACES

(Presented by Academician P. S. Aleksandrov on 16 II 1961)

In the present paper the concept of a decomposition of a given space \(X\) by its closed subset \(A\) is generalized, and an equivalence of generalized decompositions is introduced. On the set of the resulting equivalence classes a group operation is introduced. It is proved that the groups thus constructed are naturally isomorphic to the cohomotopy groups of the space \(X\).

1. Decompositions of order \(k\). Let \(X\) be a topological space. An elementary decomposition of the space \(X\) will mean an ordered triple of subsets of this space
\[ \alpha=(A,U,V), \]
consisting of a closed set \(A\) and open sets \(U\) and \(V\), possessing the property that
\[ U\cap V=\varnothing,\qquad U\cup V=X\setminus A. \]
An ordered pair \(\sigma^0=(B,C)\) of closed subsets of some space \(Y\) will be called a decomposition of order zero of the space \(Y\), if
\[ B\cap C=\varnothing,\qquad B\cup C=Y. \]

Suppose \(k\) elementary decompositions
\[ \alpha_i=(A_i,U_i,V_i),\quad i=1,2,\ldots,k, \]
of the space \(X\) are given, and a decomposition \((B,C)\) of order zero of the set
\[ \bigcap_{i=1}^{k} A_i. \]
In this case the ordered collection
\[ \sigma^k=(\alpha_1,\ldots,\alpha_k,B,C) \]
will be called a decomposition of order \(k\) of the space \(X\).

2. Equivalence classes. Two decompositions of order \(k\) of the space \(X\) will be called equivalent if there exists a decomposition of order \(k\) of the space \(X\times I\), where \(I\) is the unit interval \([0,1]\), such that on the upper base \(X\times 1\) it coincides with one of the given decompositions, and on the lower base \(X\times 0\) with the other. The concept of equivalence introduced is, evidently, reflexive, symmetric, and transitive. The set of all equivalence classes of decompositions of order \(k\) of the space \(X\) will be denoted by \(P^k(X)\).

3. Functions and decompositions. In the present paper the space \(X\) is assumed to be completely normal, i.e. to have the property that every one of its closed subsets is of type \(G_\delta\). The basic properties of a completely normal space used in the present paper are the following:

Property 3.1. Let \(A\) and \(B\) be closed sets of the space \(X\) and
\[ A\cap B=\varnothing; \]
then there exists on the space \(X\) a real continuous function \(f(x)\) satisfying the conditions:
\[ f(x)=0,\quad \text{if } x\in A;\qquad f(x)=1,\quad \text{if } x\in B;\qquad 0<f(x)<1,\quad \text{if } x\in X\setminus(A\cup B). \]

Property 3.2. In order that the space \(X\) be completely normal, it is necessary and sufficient that every mapping of an arbitrary closed set \(A\subset X\) into the sphere \(S^{k-1}\), which is the boundary of the ball \(T^k\), can be extended to a mapping
\[ F:X\to T^k, \]
having the property that
\[ F^{-1}(S^{k-1})=A. \]

Proof. Sufficiency follows from the fact that the preimage under a continuous mapping \(F\) of a set of type \(G_\delta\) is in turn a set of type \(G_\delta\) in the mapped space.

We prove necessity. Let \(F_1: X \to T^k\) be some extension of the mapping \(f: A \to S^{k-1}\) (such an extension exists, since the space \(X\) is, obviously, normal). Further, let \(\varphi(x)\) \((0 \leq \varphi(x) \leq 1)\) be a real continuous function on the space \(X\) possessing the property that \(\varphi(x)=0\) if and only if \(x\in A\). Let \(O\) be the center of the ball \(T^k\), whose radius we shall assume to be equal to one. Denote by \(\Gamma_\lambda\) the homothety of the ball \(T^k\) with center \(O\) and coefficient \(\lambda\). Then the mapping \(F: K \to T^k\), defined by the formula \(F(x)=\Gamma_{1-\varphi(x)}(F_1)(x)\), \(x\in X\), is, obviously, continuous and satisfies the condition.

Let \(f(x)\) be a continuous function on the space \(X\); consider the subsets \(A_f=(x\mid f(x)=0)\), \(U_f=(x\mid f(x)>0)\), \(V_f=(x\mid f(x)<0)\) of the space \(X\). As is easy to see, the triple \(\alpha_f=(A_f,U_f,V_f)\) is an elementary decomposition of the space \(X\), which we shall call the elementary decomposition of the space \(X\) generated by the function \(f\).

4. Properties of decompositions. If a certain collection of decompositions \(\{\sigma_z^k\}\) of order \(k\) of the space \(X\) is given, where \(z\) is some index corresponding to the fixed decomposition, then we shall agree to write the decomposition \(\sigma_z^k\) in the form \((\alpha_1^z,\ldots,\alpha_k^z,B^z,C^z)\), where \(\alpha_i^z=(A_i^z,U_i^z,V_i^z)\).

Property 4.1. Let \(\sigma_1^k\) and \(\sigma_2^k\) be two decompositions of order \(k\) of the space \(X\), and let \(A_i^1\subset A_i^2\), \(U_i^1\supset U_i^2\), \(V_i^1\supset V_i^2\), \(i=1,2,\ldots,k\), \(B^1\subset B^2\), \(C^1\subset C^2\); then the decompositions \(\sigma_1^k\) and \(\sigma_2^k\) are equivalent.

Indeed, consider in the space \(X\times I\) the closed sets:
\(A_i=(A_i^1\times I)\cup A_i^2\), \(B=(B^1\times I)\cup B^2\), \(C=(C^1\times I)\cup C^2\), and the open sets: \(U_i=(U_i^1\times I)\setminus A_i\), \(V_i=(V_i^1\times I)\setminus A_i\). It is easy to see that \(\alpha_i=(A_i,U_i,V_i)\) are elementary decompositions of the space \(X\times I\), while \(\sigma^k=(\alpha_1,\ldots,\alpha_k,B,C)\) is a decomposition of order \(k\) of the space \(X\times I\), establishing the equivalence of the decompositions \(\sigma_1^k\) and \(\sigma_2^k\).

Property 4.2. Let a decomposition \(\sigma_1^k\) of order \(k\) of the space \(X\) be given, and let a neighborhood \(O(B^1)\) of the set \(B^1\) be given; then there exists a decomposition \(\sigma^k\sim\sigma_1^k\) such that
\[ \bigcup_{i=1}^{k}(A_i\cup V_i)\subset O(B^1). \]

Indeed, let \(W(B^1)\) be a neighborhood of the set \(B^1\) lying together with its closure \([W(B^1)]\) in \(O(B^1)\) (such a neighborhood exists by virtue of the normality of the space \(X\)). Define a decomposition \(\sigma_2^k\) of order \(k\) of the space \(X\) by putting \(A_i^2=A_i^1\cup G\), \(U_i^2=U_i^1\setminus G\), \(V_i^2=V_i^1\setminus G\), \(B^2=B^1\), \(C^2=C^1\cup G\), where \(G=(X\setminus W(B^1))\). By Property 4.1, \(\sigma_2^k\sim\sigma_1^k\). The decomposition \(\sigma^k\), in which \(A_i=A_i^2\setminus\Phi\), \(U_i=U_i^2\cup\Phi\), \(V_i=V_i^2\), \(B=B^2\), \(C=C^2\setminus\Phi\), \(\Phi=X\setminus[W(B^1)]\), is, by Property 4.1, equivalent to the decomposition \(\sigma_2^k\); consequently, by transitivity, \(\sigma^k\sim\sigma_1^k\) and, by construction, satisfies the required condition.

Property 4.3. Let \(\sigma^k\) be some decomposition of order \(k\) of the space \(X\); let \(F\) be an arbitrary closed set in \(X\), and \(1\leq i\leq k\); then there exists a decomposition \(\sigma_{(i,F)}^k\) possessing the property that \(\sigma_{(i,F)}^k\sim\sigma^k\), \(\operatorname{Ind}(A^{(i,F)}\cap F)\leq \operatorname{Ind}F-1\), \(\alpha_j^{(i,F)}=\alpha_j\), if \(j\ne i\).

Indeed, it is easy to see that, without loss of generality, one may assume that \([U_i]\cap[V_i]=\varnothing\); then the closed sets \(F\cap[U_i]\) and \(F\cap[V_i]\) can be separated in \(F\) by a closed set \(D\), with \(\operatorname{Ind}D\leq \operatorname{Ind}F-1\). Construct on \(F\cup[U_i]\cup[V_i]\) a real function \(\varphi(x)\) possessing the property that \(\varphi(x)=0\) if and only if \(x\in D\), \(\varphi(x)=1\) for \(x\in[U_i]\), and \(\varphi(x)=-1\) for \(x\in[V_i]\); this is possible by Property 3.1. Let \(f(x)\) be an extension of the function \(\varphi(x)\) to the whole space \(X\); then the elementary decomposition \(\alpha_f\), generated by the function \(f(x)\), possesses the property that \(A_f\subset A_i\), \(U_f\supset U_i\), \(V_f\supset V_i\), \(A_f\cap F=D\). Putting \(\alpha_j^{(i,F)}=\alpha_j\)

if \(j\ne i\), \(\alpha_i^{(i,F)}=\alpha_f\), \(B^{(i,F)}=B\cap A_f\), \(C^{(i,F)}=C\cap A_f\), we obtain a partition \(\sigma_{(i,F)}^k\), which, by property 4.1, is equivalent to \(\sigma^k\) and is the required one.

5. Group operation. The group operation is introduced in the set \(p^k(X)\) on the basis of the following lemma:

Lemma 1. Let \(\operatorname{Ind} X=n\), and let \(s_1^k\) and \(s_2^k\) be two elements of the set \(p^k(X)\), with \(k\ge (n+1)/2\); then in the classes \(s_1^k\) and \(s_2^k\) there exist partitions \(\bar\sigma_1^k\) and \(\bar\sigma_2^k\) possessing the property that
\[ \bigcup_{i=1}^{k}(\bar A_i^1\cup \bar V_i^1)\cap \bigcup_{i=1}^{k}(\bar A_i^2\cup \bar V_i^2)=\varnothing . \]

Proof. Let \(\sigma_1^k\) and \(\sigma_2^k\) be arbitrary partitions from the classes \(s_1^k\) and \(s_2^k\), respectively. Successively applying property 4.3, it is easy to obtain partitions \(\bar\sigma_1^k\) and \(\bar\sigma_2^k\) possessing the property that
\[ \operatorname{Ind}\bar A_1^1\le n-1,\quad \operatorname{Ind}\bar A_1^1\cap \bar A_2^2\le n-2,\ldots,\quad \operatorname{Ind}\bigcap_{i=1}^{k}\bar A_i^1\le n-k, \]
\[ \operatorname{Ind}\Bigl(\bigcap_{i=1}^{k}\bar A_i^1\Bigr)\cap \bar A_1^2\le n-k-1,\ldots,\quad \operatorname{Ind}\Bigl(\bigcap_{i=1}^{k}\bar A_i^1\Bigr)\cap\Bigl(\bigcap_{i=1}^{k}\bar A_i^2\Bigr)\le n-2k\le -1. \]
Consequently, \((\bar B^1\cup \bar C^1)\cap(\bar B^2\cup \bar C^2)=\varnothing\). Moreover, \(\bar B^1\cap \bar B^2=\varnothing\), and, since the space \(X\) is normal, there exist disjoint neighborhoods \(O(\bar B^1)\) and \(O(\bar B^2)\) of the sets \(\bar B^1\) and \(\bar B^2\). From property 4.2 there follows the existence of partitions \(\bar{\bar\sigma}_1^k\sim \bar\sigma_1^k\) and \(\bar{\bar\sigma}_2^k\sim \bar\sigma_2^k\) possessing the property that
\[ \bigcup_{i=1}^{k}(\bar{\bar A}_i^1\cup \bar{\bar V}_i^1)\subset O(\bar B^1), \]
\[ \bigcup_{i=1}^{k}(\bar{\bar A}_i^2\cup \bar{\bar V}_i^2)\subset O(\bar B^2), \]
i.e. satisfying the conditions of the lemma.

Now let \(s_1^k\) and \(s_2^k\) be arbitrary elements of the set \(p^k(X)\). Choose in the classes \(s_1^k\) and \(s_2^k\) elements \(\sigma_1^k\) and \(\sigma_2^k\) satisfying the condition of Lemma 1. The sequence \((\sigma_1^k+\sigma_2^k)=(\alpha_1,\ldots,\alpha_k,B,C)\), where \(A_i=A_i^1\cup A_i^2\), \(U_i=U_i^1\cap U_i^2\), \(V_i=V_i^1\cup V_i^2\), \(B=B^1\cup B^2\), \(C=C^1\cup C^2\), is a partition of order \(k\) of the space \(X\) by virtue of the choice of the partitions \(\sigma_1^k\) and \(\sigma_2^k\). The class containing the partition \((\sigma_1^k+\sigma_2^k)\) will be denoted by \((s_1^k+s_2^k)\). For \(k>(n+1)/2\), it follows easily from Lemma 1 that \((s_1^k+s_2^k)\) does not depend on the accidental choice of \(\sigma_1^k\) and \(\sigma_2^k\) in the classes \(s_1^k\) and \(s_2^k\).

6. The group \(p^k(X)\). As is easy to see, the addition operation introduced in the set \(p^k(X)\) is commutative. One can show that the zero class of the set \(p^k(X)\) with respect to the introduced operation will be the class containing the partition \(\sigma_0^k=(\alpha_1,\ldots,\alpha_k,\varnothing,\varnothing)\), where \(U_i=X\), \(A_i=V_i=\varnothing\), \(i=1,\ldots,k\), and that the class inverse to the class containing the partition \(\sigma^k=(\alpha_1,\ldots,\alpha_k,B,C)\) will be the class containing the partition \(-\sigma^k=(\alpha_1,\ldots,\alpha_k,C,B)\). The associativity of the introduced operation follows directly from the definition. Thus, the following holds:

Theorem 1. The set \(p^k(X)\) is a group if \(\operatorname{Ind} X<2k-1\).

7. Connection with cohomotopy groups. We introduce the following notation for subsets of the \((k+1)\)-dimensional coordinate space \(E^{k+1}\): \(S_k\) is the unit \(k\)-dimensional sphere with center at the origin; \(S_i^{k-1}\) is the intersection of \(S^k\) with the plane \(x_i=0\); \(E_i^{\delta k}\) is the intersection of \(S^k\) with the half-space \(\delta x_i>0\); \(\delta=+1\) or \(-1\); \(a^\delta=(0,\ldots,0,\delta)\).

Let \(f\) be an arbitrary continuous mapping of the space \(X\) into \(S^k\); then \(\alpha_i^f=(f^{-1}(S_i^{k-1}), f^{-1}(E_i^{-k}), f^{-1}(E_i^{+k}))\) will be elementary partitions of the space \(X\), and \(\sigma_f^k=(\alpha_1^f,\ldots,\alpha_k^f, f^{-1}(a^+), f^{-1}(a^-))\) will be a partition of order \(k\) of the space \(X\).

Theorem 2. Let \(f\) and \(g\) be mappings of the space \(X\) into \(S^k\) that are homotopic to each other; then the partitions \(\sigma_f^k\) and \(\sigma_g^k\) are equivalent.

Proof. Since \(f\) and \(g\) are homotopic, there exists a mapping
\(F:(X\times I)\to S^k\) such that on the upper base it coincides with \(f\), and on the lower—with \(g\). Consider the decomposition of order \(k\) of the space \(X\times I\) generated by the mapping \(F\). It is easy to see that this decomposition realizes the equivalence of the decompositions \(\sigma_f^k\) and \(\sigma_g^k\).

Theorem 3. Let \(X\) be a perfectly normal space and let \(\sigma^k\) be its decomposition of order \(k\); then there exists a mapping \(F:X\to S^k\) (unique up to homotopy) having the property that \(\sigma_F^k\) coincides with \(\sigma^k\).

The proof of Theorem 3 is based entirely on Lemma 2.

Lemma 2. Let \(\sigma^k\) be a decomposition of order \(k\) of the space \(X\), and let

\[ f:\Xi\to \bigcup_{i=1}^{k} S_i^{k-1} \]

be a mapping having the property that

\[ f^{-1}\bigl(S_i^{k-1}\cap E_j^{\delta k}\bigr)= \begin{cases} A_i\cap U_j, & \delta=+1,\\ A_i\cap V_j, & \delta=-1. \end{cases} \tag{*} \]

Then there exists a mapping \(F:X\to S^k\) such that \(\sigma_F\) coincides with \(\sigma^k\).

The proof of Lemma 2 is carried out by a direct construction of the required mapping \(F\).

Theorem 4. Let \(\sigma_1^k\) and \(\sigma_2^k\) be decompositions of order \(k\) of the space \(X\), chosen according to Lemma 1 in the classes \(s_1^k\) and \(s_2^k\). Let, further, \(f\) and \(g\) be mappings of the space \(X\) into \(S^k\) corresponding, according to Theorem 3, to the decompositions \(\sigma_1^k\) and \(\sigma_2^k\). Then the mapping corresponding, according to Theorem 3, to the decomposition \(\sigma^k\) constructed in Lemma 1 is homotopic to the mapping
\((f+g):X\to S^k\), where the sum \((f+g)\) is taken in the sense of Borsuk–Spanier \((^1)\).

Proof. Since the decompositions \(\sigma_1^k\) and \(\sigma_2^k\) satisfy the conditions of Lemma 1, there exists an elementary decomposition of the space

\[ X:\alpha=(A,U,V), \]

satisfying the condition:

\[ U\supset X\setminus\bigcap_{i=1}^{k}U_i^1,\qquad V\supset X\setminus\bigcap_{i=1}^{k}U_i^2. \]

Choose mappings \(f\) and \(g\) corresponding to the decompositions \(\sigma_1^k\) and \(\sigma_2^k\) and having the property that

\[ f(A\cup U)=g(A\cup V)=\left(\frac{1}{\sqrt{k}},\ldots,\frac{1}{\sqrt{k}},0\right)=c. \]

Consider the mapping \(F:X\to S_1^k\times S_2^k\), defined by the formula
\(F(x)=(f(x),g(x))\). It is easy to see that in the present case \(F(x)\in S_1^k\times c\) if \(x\in A\cup U\), and \(F(x)\in S_2^k\times c\) if \(x\in A\cup V\). Consequently, the mapping \(F\) is a mapping into the bouquet of spheres \(S_1^k\vee S_2^k\). If we identify the spheres \(S_1^k\) and \(S_2^k\), then we obtain the mapping \((f+g):X\to S^k\) in the sense of Borsuk–Spanier. From the construction it is clear that \(\sigma_{(f+g)}^k=\sigma^k\), and Theorem 4 is proved.

Theorem 5. The group \(P^k(X)\) is isomorphic to the \(k\)-th cohomotopy group \(\pi^k(X)\) of the space \(X\).

The proof follows directly from Theorems 2, 3, and 4.

Moscow State University
named after M. V. Lomonosov

Received
15 II 1961

CITED LITERATURE

1. E. Spanier, Ann. of Math., 50, 203 (1949).