V. I. KUZMINOV

TEST SPACES

(Presented by Academician P. S. Aleksandrov on 27 IV 1963)

We use the following notation: \(\dim_G X\) is the cohomological dimension of the compactum \(X\) with respect to the coefficient group \(G\); \(Z\) is the infinite cyclic group; \(Z_p\) is the cyclic group of order \(p\); \(Q\) is the additive group of rational numbers; \(R_p\) is the additive group of those rational numbers which, when written as an irreducible fraction, have denominator relatively prime to \(p\); \(Q_p\) is the additive group of rational numbers of the form \(n/p^\alpha\), reduced modulo 1. By definition, the system of groups \(\sigma\) consists of the groups \(Z_p, Q_p, R_p, Q\), and the index \(p\) runs through all prime values. All compacta in what follows are assumed to be finite-dimensional.

M. F. Bokshtein \((^2)\) indicated a method for computing the cohomological dimension of a compactum with respect to an arbitrary coefficient group, if its dimensions with respect to the groups of the system \(\sigma\) are known.

We consider functions defined on the set \(\sigma\) and taking values in the set of nonnegative integers. A function \(D\) is called a dimension function if, for every prime number \(p\), it satisfies the following conditions:

\[ \begin{aligned} &1.\quad D(Q_p) \leq D(Z_p) \leq D(Q_p)+1.\\ &2.\quad D(R_p) \leq \max(D(Q),D(Q_p)+1).\\ &3.\quad D(Q_p) \leq \max(D(Q),D(R_p)-1).\\ &4.\quad D(Z_p) \leq D(R_p).\\ &5.\quad D(Q) \leq D(R_p).\\ &6.\quad \text{If } D(G_0)=0 \text{ for some group } G_0 \in \sigma,\text{ then } D(G)=0 \text{ for every group } G\in\sigma. \end{aligned} \tag{1} \]

The number \(\dim D=\max_{G\in\sigma}D(G)\) is called the dimension of the function \(D\). The function

\[ D(G)=\dim_G X \]

is called the dimension function of the compactum \(X\).

The dimension function of an arbitrary \(n\)-dimensional compactum \(X\) is an \(n\)-dimensional dimension function. This assertion is a combination of known results of P. S. Aleksandrov and M. F. Bokshtein. The following theorem gives a partial answer to the question of whether the converse assertion is true.

Theorem 1. An arbitrary \(n\)-dimensional dimension function \(D\) satisfying the condition \(D(G)\geq [n/2]\) for every group \(G\in\sigma\) is the dimension function of some \(n\)-dimensional compactum.

Corollary 1. An arbitrary \(n\)-dimensional dimension function for \(n\leq 3\) is the dimension function of some \(n\)-dimensional compactum.

Corollary 2. For each group \(G\in\sigma\) and each integer \(n\geq 1\) there exists a \(2n\)-dimensional compactum \(X_G^n\), whose dimensions are determined by the following table:

(In this table \(q\) is an arbitrary prime number distinct from \(p\).)

Examples of compacta having the dimensions indicated in the table are the following: \(X_{\mathbb Z_p}^n\)—the \(n\)-th topological power of the two-dimensional compactum of L. S. Pontryagin \((^3)\), \(X_{R_p}^n\) and \(X_Q^n\)—the \(n\)-th topological powers of the two-dimensional compacta of Kodama \((^6)\), \(X_{Q_p}^1\)—the two-dimensional compactum of V. G. Boltyanskii. The compacta \(X_{Q_p}^n\) for \(n>1\) cannot be obtained as products of two-dimensional compacta. They can be constructed by modifying the construction indicated in the note \((^8)\). The compacta \(X_G^n\) are called test spaces. For \(n=1\) they were considered by Kodama \((^6)\).

Lemma. For an arbitrary compactum \(X\) and any group \(G \in \sigma\), the equality
\[ \dim (X \times X_G^n)=\max(\dim X+n,\ \dim_G X+\dim X_G^n) \]
holds.

Proof. The lemma is proved by direct computations using Bockstein’s formulas, which express the dimensions of a product of compacta in terms of the dimensions of the factors *.

Theorem 2. If \(X\) is an arbitrary finite-dimensional compactum, \(G\) is a group from the system of groups \(\sigma\), and \(d\) is a nonnegative integer, then the inequality \(\dim_G X \geq \dim X-d\) is equivalent to the inequality
\[ \dim (X \times X_G^{d+1}) \geq \dim X+\dim X_G^{d+1}-d. \]

Proof. Let \(\dim_G X \geq \dim X-d\). Then, by the preceding lemma:
\[ \dim (X \times X_G^{d+1}) =\max(\dim X+d+1,\ \dim_G X+\dim X_G^{d+1}) \geq \dim X+\dim X_G^{d+1}-d. \]

Conversely, suppose
\[ \dim (X \times X_G^{d+1})=\dim X+\dim X_G^{d+1}-d. \]
Then
\[ \dim (X \times X_G^{d+1})=\dim_G X+\dim X_G^{d+1} \]
and \(\dim_G X \geq \dim X-d\). The theorem is proved.

For \(d=0\) this theorem becomes Kodama’s theorem \((^6)\).

Theorem 2 makes it possible to compute the cohomological dimensions of a compactum from the Urysohn dimensions of its products with test spaces. Therefore, knowing the dimensions \(\dim\) of the products of compacta \(X\) and \(Y\) with test spaces, one can determine the dimension \(\dim (X\times Y)\). Often, in order to solve problems connected with computing the dimension of a product, it is sufficient to know the dimensions of the products not with all test spaces, but only with some of them. For example, in order to establish whether, for given compacta \(X\) and \(Y\) and an integer \(d\), the inequality
\[ \dim (X\times Y)\geq \dim X+\dim Y-d \]
holds, it is sufficient to know the dimensions of the products of the compacta \(X\) and \(Y\) with the compacta \(X_G^{d+1}\). (For \(d=0\) this result was established by Kodama \((^5)\).) Other examples are contained in the following theorems.

Generalizing P. S. Aleksandrov’s definition of dimensional completeness of a compactum, let us call the dimensional defect of a compactum \(X\) the integer \(d(X)\) such that there exists a compactum \(Y\) for which \(\dim (X\times Y)=\dim X+\dim Y-\)

\[ \dim_{\mathbb Z_p}(X\times Y)=\dim_{\mathbb Z_p}X+\dim_{\mathbb Z_p}Y, \]
\[ \dim_Q(X\times Y)=\dim_Q X+\dim_Q Y, \]
\[ \dim_{Q_p}(X\times Y)=\max[\dim_{\mathbb Z_p}(X\times Y)-1,\ \dim_{Q_p}X+\dim_{Q_p}Y], \]
\[ \dim_{R_p}(X\times Y)=\min\{\max[\dim_Q(X\times Y),\ \dim_{\mathbb Z_p}(X\times Y),\ \dim_{Q_p}X+\dim_{Q_p}Y+1], \]
\[ \max[\dim_Q(X\times Y),\ \dim_{\mathbb Z_p}(X\times Y),\ \dim_{R_p}X+\dim_{Q_p}Y,\ \dim_{Q_p}X+\dim_{R_p}Y]\}. \]

— \(d(X)\), and there do not exist compacta \(Y\) for which

\[ \dim (X \times Y) < \dim X + \dim Y - d(X). \]

Theorem 3. \(d(X) \leq d\) if and only if the following conditions are satisfied: 1) \(\dim (X \times X_{Q_p}^{d+1}) \geq \dim X + \dim X_{Q_p}^{d+1} - d\) for every prime number \(p\); 2) \(\dim (X \times X_Q^{d+1}) \geq \dim X + \dim X_Q^{d+1} - d\). In the case \(d=0\), it is sufficient that condition 1) alone be satisfied.

Proof. First of all, note that for \(d=0\) this theorem becomes the well-known theorem of V. G. Boltyanskii \((^4)\).

The necessity of the conditions is obvious.

Let us verify sufficiency. By Theorem 2, \(\dim_{Q_p} X \geq \dim X - d\) and \(\dim_Q X \geq \dim X - d\). Hence, and from inequalities (1), it follows that \(\dim_G X \geq \dim X - d\) for any group \(G \in \sigma\). By Bockstein’s formulas, for any compactum \(Y\) we obtain \(\dim (X \times Y) \geq \dim X + \dim Y - d\). Consequently, \(d(X) \leq d\). The sufficiency of condition 1) for \(d=0\) follows from the fact that in this case, for any group \(G \in \sigma\), we have \(\dim_G X_{Q_p}^1 \leq \dim_G X_Q^1\). The theorem is proved.

We shall say that a compactum \(X\) is a compactum of ordinary type if \(\dim X^n = n \dim X\) for every \(n>0\). If \(\dim X^n = (n-1)\dim X + 1\) for every \(n>0\), then we shall say that the compactum \(X\) is a compactum of special type.

Theorem 4. Every compactum \(X\) is either a compactum of ordinary type or a compactum of special type. A compactum \(X\) is a compactum of special type if and only if, for every prime number \(p\),

\[ \dim (X \times X_{\mathbb Z_p}^1) < \dim X + \dim X_{\mathbb Z_p}^1, \]

\[ \dim (X \times X_Q^1) < \dim X + \dim X_Q^1. \tag{2} \]

Proof. Suppose that inequalities (2) hold for the compactum \(X\). By Theorem 2, \(\dim_{\mathbb Z_p} X \leq \dim X - 1\), \(\dim_Q X \leq \dim X - 1\). For each \(p\), by Bockstein’s formulas, \(\dim_{R_p} X^n \leq n(\dim X - 1)+1\). For those \(p\) for which \(\dim_{R_p} X = \dim X\), in view of inequalities (1) we have \(\dim_{Q_p} X = \dim_{R_p} X - 1\). Therefore, for such \(p\) we have \(\dim_{R_p} X^n = n(\dim X - 1)+1\). Consequently, the compactum \(X\) is a compactum of special type.

Now let \(X\) be an arbitrary compactum for which inequalities (2) are not satisfied. By Theorem 2, either \(\dim_Q X = \dim X\), or for some prime number \(p\), \(\dim_{\mathbb Z_p} X = \dim X\). By Bockstein’s formulas, \(\dim X^n = \max_p \dim_{R_p} X^n = n \dim X\). The compactum \(X\) is a compactum of ordinary type. The theorem is proved.

The following assertion was first proved by Fary \((^7)\).

Corollary. For an arbitrary compactum \(X\), the inequality \(\dim (X \times X) \geq 2\dim X - 1\) holds.

Note that the Boltyanskii compactum is a two-dimensional compactum of special type. The product of a compactum of special type with a dimensionally full-valued compactum is a compactum of special type.

Institute of Mathematics with Computing Center
Siberian Branch of the Academy of Sciences of the USSR

Received
27 IV 1963

REFERENCES

1. P. S. Aleksandrov, UMN, 4, issue 6 (1949).
2. M. F. Bockstein, Tr. Mosk. Matem. Obshch., 5, 3 (1956); 6, 3 (1957).
3. V. G. Boltyanskii, UMN, 6, issue 3 (1951).
4. V. G. Boltyanskii, DAN, 67, No. 5 (1949).
5. Y. Kodama, Proc. Japan Academy, 36, No. 7 (1960).
6. Y. Kodama, Duke Math. J., 29, No. 1 (1962).
7. I. Fary, Bull. Am. Math. Soc., 67, No. 1 (1961).
8. V. Kuzminov, DAN, 141, No. 2 (1961).

\[ \overline{\phantom{xxxx}} \]

* A compactum \(X\) is dimensionally full-valued if and only if \(d(X)=0\). Always \(d(X) \geq 0\).