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ON THE DIMENSION OF GROUPS WITH A LEFT-INVARIANT TOPOLOGY
A. Mishchenko
(Presented by Academician P. S. Aleksandrov on 11 VI 1964)
The article studies the construction of topological groups proposed by A. A. Markov \((^2)\) and Kakutani \((^1)\), in order to clarify the relation between the dimensions of homogeneous spaces—groups with a non-invariant topology—under different definitions of dimension.
Let \(X\) be an arbitrary set. By \(G(X)\) we denote the free algebraic group generated by the set \(X\), and by \(A(X)\) the free algebraic abelian group generated by the set \(X\).
Lemma 1. Let \(\rho\) be a pseudometric on the set \(X\). There exist invariant pseudometrics \(\bar\rho\) on the groups \(G(X)\) and \(A(X)\), satisfying the following conditions: if \(x,y\in X\), and \(e\) is the neutral element of the group, then \(\bar\rho(x,e)=1\), \(\bar\rho(x,y)=\rho(x,y)\). Among all such invariant pseudometrics there exists a largest one (i.e., one taking the largest value). Among all left-invariant pseudometrics satisfying the same conditions there exists a largest one. The indicated largest pseudometrics are constructed effectively. If \(\rho\) is a metric, then the indicated pseudometrics are also metrics.
We shall carry out the proof for left-invariant pseudometrics. The remaining cases are considered in the paper of M. I. Graev \((^3)\).
Every element \(z\in G(X)\) has an irreducible representation
\(z=(x_1^{\varepsilon_1}, x_2^{\varepsilon_2}, \ldots, x_n^{\varepsilon_n})\), \(x_i\in X\), \(\varepsilon_i=\pm 1\). From two words
\(Z_1=(x_1^{\varepsilon_1}, \ldots, x_n^{\varepsilon_n})\) and
\(z_2=(y_1^{\eta_1}, \ldots, y_m^{\eta_m})\) one can form a third
\(Z_3=(x_1^{\varepsilon_1}, \ldots, x_n^{\varepsilon_n}, y_1^{\eta_1}, \ldots, y_m^{\eta_m})\).
Then we write \(Z_3=Z_1Z_2\). If \(Z=Z_1Z_2\ldots Z_n\), then we shall say that a decomposition \(\tau\) of the word \(Z\) is constructed. Let a decomposition \(\tau\) of the word \(Z=Z_1Z_2\ldots Z_n\) be given, where each \(Z_i\) consists of one letter or has the form \(Z_i=x^{-1}y^1\). Such a decomposition will be called proper, and the words \(Z_i\) will be called elementary words. If \(Z_i\) is an elementary word, then set \(\|Z_i\|=\Delta\) if \(Z_i\) consists of one letter, and \(\|Z_i\|=\rho(x,y)\) if \(Z_i=x^{-1}y^1\). Let \(\tau\) be a proper decomposition of the word \(Z\),
\[ Z=Z_1\ldots Z_n. \]
Set
\[ \|Z\|_\tau=\sum_{i=1}^{n}\|Z_i\|. \]
Define the pseudometric on \(G(X)\) by the equalities: \(\bar\rho(z,e)=\min \{\|Z\|_\tau\}\), where \(Z\) is the irreducible representation of the element \(z\), and \(\tau\) runs through all proper decompositions of the word \(Z\);
\[ \bar\rho(z_1,z_2)=\bar\rho(z_2^{-1}z_1,e). \]
The pseudometric \(\bar\rho\) constructed is the largest left-invariant metric compatible with the pseudometric \(\rho\). Further, we shall denote this pseudometric by \(\rho_\ell\), and the largest pseudometric among invariant pseudometrics will be denoted by \(\rho_i\).
Let \(Y_i=X\cup\{e_i\}\cup X^{-1}\) be a copy of the discrete union of three sets: \(X\), a copy \(X^{-1}\), and the point \(\{e_i\}\). If a pseudometric is given on \(X\), then on \(Y_i\) we also define a pseudometric by setting the distances between points from different sets equal to one, and on \(X\) and \(X^{-1}\) equal to the distance in the original pseudometric. By \(Z\) denote the direct sum of the spaces \(Y_i\) with fixed point \(\{e_i\}\). In \(Z\) define a pseudometric by the equality
\[ \rho(z',z'')=\max_i \rho(z_i',z_i'')\, i^2. \]
There exist natural mappings of the space \(Z\) onto the groups \(G(X)\) and \(A(X)\).
Theorem 1. Let \(\rho\) be a pseudometric in \(X\). The mappings
\(f: Z \to G(X)\), \(g: Z \to A(X)\) are uniformly continuous if in \(G(X)\) and \(A(X)\) one defines the pseudometrics \(\rho_{\mathrm{и}}\). These mappings are uniformly open if in \(G(X)\) and \(A(X)\) one defines the pseudometrics \((\sqrt{\rho})_{\mathrm{и}}\). If on \(X\) a system of pseudometrics \(R\) is given, then on \(G(X)\) we construct the corresponding systems of pseudometrics \(R_{\mathrm{и}}\) and \(R_{\mathrm{л}}\).
Lemma 2. Let \(G_n \subset G(X)\) be the set of all elements having a representation of length \(\leq n\). If the system of pseudometrics \(R\) generates a separated topology, then \(G_n\) are closed with respect to the topologies generated by the systems \(R_{\mathrm{и}}\).
Lemma 3. Under the same assumptions \(G_n\) is topologically equivalent to a discrete union of sets of the form
\[ X \times X \times \cdots \times X, \]
where there are \(n\) factors, in which, possibly, some diagonals are contracted to points.
Lemma 4. If in \(G(X)\) one introduces the metrics \(R_{\mathrm{л}}\), then instead of spaces of the form \(X \times X \times \cdots \times X\) in Lemma 3 one should take the sets \(X \times X \cdots \times X\) with a finer topology.
This topology is defined as follows: \(X\) is either in its usual topology or in the discrete one; suppose that on \(X \times \cdots \times X\), with \(n-1\) factors, the topology has already been defined; then in \(X \times \cdots \times X\), with \(n\) factors, layers of the form \(X \times \cdots \times X \times \{x\}\) are glued along the diagonal, and on the diagonal the topology of \(X\) is taken (usual or discrete).
Lemma 5. If \(X\) is paracompact, then the spaces \(X \times \cdots \times X\) described in Lemma 4 are paracompact.
Theorem 2. If \(X\) is paracompact in the system of pseudometrics \(R\), then \(G(X)\) is paracompact in the system of pseudometrics \(R_{\mathrm{л}}\).
Theorem 3. If the topological products \(X \times \cdots \times X\) are paracompact, then \(G(X)\) is paracompact in the system of pseudometrics \(R_{\mathrm{и}}\).
Theorem 4. Let a system of pseudometrics \(R\) be given in \(X\), and a system \(R_{\mathrm{л}}\) be given in \(G(X)\). If \(X\) is normal, then the equality
\[ \dim X = \dim G(X) \]
holds.
Dimension is to be understood as the dimension of the Čech extension of the space.
Corollary. For every metric space \(X\) there exists a homogeneous metric space of the same dimension \(\dim\), containing the space \(X\).
Theorem 5. Let a system of pseudometrics \(R\) be given in \(X\), and a system \(R_{\mathrm{л}}\) be given in \(G(X)\). The inequality
\[ \operatorname{ind} G(X) \leq \operatorname{ind} X + 1 \]
is valid.
Theorem 6. Let \(X\) be metric and \(\operatorname{ind} X = 0\). \(X\) is embedded in a metric group \(G\), zero-dimensional in the sense of \(\operatorname{ind}\), if and only if in \(X \times X\) there is a base of neighborhoods with empty boundary.
The following questions remain open. Does there exist a metric group whose dimensions \(\dim\) and \(\operatorname{ind}\) do not coincide? Does there exist a metric space \(X\) for which \(\dim X - \operatorname{ind} X > 1\)? Can a complete metric space be embedded in a complete metric group with preservation of dimension?
Moscow State University
named after M. V. Lomonosov
Received
22 IV 1964
CITED LITERATURE
- S. Kakutani, Proc. Imp. Acad. Tokyo, 20, 595 (1944).
- A. A. Markov, Izv. AN SSSR, ser. matem., 9, No. 1 (1945).
- M. I. Graev, Izv. AN SSSR, ser. matem., 12, No. 3 (1948).