UDC 513.83

MATHEMATICS

B. A. PASYNKOV

ON TOPOLOGICAL GROUPS

(Presented by Academician P. S. Aleksandrov on 10 I 1969)

All spaces considered are completely regular. We assume
\(\dim X=\dim \beta X\), where \(\beta X\) is the maximal bicompact extension of \(X\).

I. The final solution of Hilbert’s fifth problem (see \((^{1-3})\)) can be formulated as follows:

A topological group is a Lie group if and only if it is locally bicompact, locally connected, and finite-dimensional.

In the class of normal spaces the property of countable compactness is considerably weaker than the property of bicompactness, and in the class of completely regular spaces an even weaker property is that of pseudocompactness (every continuous function is bounded).

Theorem 1. A topological group \(G\) is a Lie group if and only if it is locally pseudocompact (in particular, countably compact), locally connected, and finite-dimensional.

Proof. In one direction the assertion is obvious. Suppose that the group \(G\) is locally pseudocompact, locally connected, and
\(\dim G=\dim \beta G<\infty\). The completion \(\overline G\) of the group \(G\) with respect to its two-sided uniformity \(U\) is a locally bicompact group, since the group \(G\) is locally completely bounded (\((^4)\), Theorem 1.1). The group \(\overline G\), obviously, is locally connected and, if we prove that \(\dim \overline G<\infty\), then we shall prove that \(\overline G\) is a Lie group, and then (by the metrizability of \(\overline G\) and \(G\)) \(\overline G\equiv G\). Thus it remains to show that the following is true.

Lemma 1. \(\dim \overline G<\infty\), if \(\dim G<\infty\).*

Proof. Take in \(\overline G\) a neighborhood of the identity \(V\), the closure of which in \(\overline G\) is pseudocompact. Then the set
\(\overline V=\overline G\setminus [G\setminus V]_{\overline G}\) is a neighborhood of the identity in \(\overline G\), whose closure in \(\overline G\) is bicompact (since \(\overline V\subseteq [V]_{\overline G}\)). Since every locally bicompact group contains an open projective-Lie subgroup \((^3)\), without restricting the generality of the argument the group \(\overline G\) may be assumed to be projective-Lie. Then in \(\overline V\) there is such a bicompact normal divisor \(N\) that \(H=\overline G/N\) is a Lie group. The set

\[ N_1=N\cap [V]_G \tag{1} \]

is pseudocompact and

\[ [N_1]_{\overline G}=N. \tag{2} \]

Let us first prove (2). If \(g_0\in N\setminus [N_1]_{\overline G}\), then the layer \([[V]_G]_{\overline G}\setminus [V]_G=[V]_{\overline G}\setminus [V]_G\) contains a nonempty bicompact set \(B\ni g_0\) of type \(G_\delta\) in \([V]_{\overline G}\) (\(N\) has type \(G_\delta\) in \([V]_{\overline G}\), and the bicompacts \(g_0\) and \([N_1]_{\overline G}\) have disjoint neighborhoods), but this contradicts the pseudocompactness of \([V]_G\). Thus, \([N_1]_{\overline G}=N\), consequently, \(\overline{N}_1\equiv N\).** We show that every nonempty set \(\Gamma\) of type \(G_\delta\) in \(N\) intersects \(N_1\). Let \(\Gamma\cap N_1=\Lambda\). Then, since the set \(\Gamma\) has type \(G_\delta\) also in \([V]_G\), in \(\Gamma\) there is a bicompact set of type \(G_\delta\) in \([V]_{\overline G}\), not intersecting \([V]_G\), and this cannot be by virtue of

* M. Choban independently proved the stronger assertion: \(\dim \overline G=\dim G\).

** \(N_1\) denotes the completion of \(N_1\).

pseudocompactness \([V]_G\). By Theorem 1.2 (items (a) and (d)) from \((^4)\), the group \(N_1\) is pseudocompact; consequently, \(\overline{N}_1 \equiv N = \beta N_1\) (\((^4)\), Theorem 4.1), whence \(\dim N = \dim N_1\). Since already \([N_1]_{\beta \overline{G}} = \beta N_1\), a fortiori \([N_1]_{\beta G} = \beta N_1\), whence \(\dim N = \dim \beta N_1 \leq \dim \beta G < \infty\). But then \(\dim \overline{G} \leq \dim N + \dim \overline{G}/N < \infty\) (5). Lemma 1 (and hence Theorem 1) is proved.

II. Definition 1. A uniformly continuous mapping \(p\) of a uniform space \(G\) with uniformity \(U\) onto a uniform space \(X\) with uniformity \(U_p\) will be called, respectively: a) twice uniformly continuous, if for every uniform cover \(\omega\) of the space \(G\) there exists a uniform cover \(v = \{O_\alpha,\ \alpha \in \mathfrak A\}\) of the space \(X\) such that, for every \(\alpha\), the inverse image \(p^{-1}O_\alpha\) can be covered by a finite system of elements of the cover \(\omega\); b) uniformly perfect, if the mapping \(p\) is closed and all sets \(p^{-1}x,\ x \in X\), are complete with respect to the uniformity \(U\).

Lemma 2. If a mapping \(p: G \to X\) is twice uniformly continuous, and a system of closed sets \(F_\alpha \subseteq X,\ \alpha \in \mathfrak A\), is centered (i.e. the intersection of the elements of every finite subsystem of this system is nonempty) and fine with respect to the uniformity \(U_p\), then the maximal centered system \(\mu\) of closed sets \(\Phi_\beta \subseteq G,\ \beta \in \mathfrak B\), containing as a subsystem the system \(\{p^{-1}F_\alpha,\ \alpha \in \mathfrak A\}\), is fine with respect to the uniformity \(U\).

Proof. Take a uniform cover \(\omega = \{V_\gamma,\ \gamma \in \Gamma\}\) of the space \(G\). In accordance with Definition 1, take a uniform cover \(v = \{O_\delta,\ \delta \in D\}\) of the space \(X\). Then for some \(\alpha_0\) the set \(F_{\alpha_0}\) is contained in some element \(O_{\delta_0}\) of the cover \(v\).

Let the inverse image \(p^{-1}O_{\delta_0}\) be covered by the elements \(V_{\gamma_i},\ i = 1,\ldots,s\), of the cover \(\omega\). The intersection of the sets \(\Psi_i = p^{-1}F_{\alpha_0} \setminus V_{\gamma_i},\ i = 1,\ldots,s\), is empty; therefore at least one of them, for example \(\Psi_1\), does not belong to the system \(\mu\). Then \(\Psi_1 \cap \Phi_{\beta_0} = \Lambda\) for at least one \(\beta_0 \in \mathfrak B\); consequently, \(p^{-1}F_{\alpha_0} \cap \Phi_{\beta_0} \subseteq V_{\gamma_1}\), but \(p^{-1}F_{\alpha_0} \cap \Phi_{\beta_0} \in \mu\). The lemma is proved.

From Lemma 1 it follows that

Proposition 1. If a mapping \(p: G \to X\) is twice uniformly continuous, then the completeness of the uniform space \(G\) implies the completeness of the uniform space \(X\).

Proposition 2. If a mapping \(p: G \to X\) is uniformly perfect, then the completeness of the uniform space \(X\) implies the completeness of the uniform space \(G\). *

Corollary 1. If a mapping \(p: G \to X\) of uniform spaces \(G\) and \(X\) is twice uniform and uniformly perfect, then for the completeness of each of the spaces \(G\) and \(X\) it is sufficient (and necessary) that one of them be complete.

III. Let a group \(G\) be given, and in it a bicompact subgroup \(H\). By \(\Sigma_l\), \(\Sigma_r\), and \(\Sigma\) denote, respectively, the set of all covers \(\omega_{vl} = \{gV\}\), \(\omega_{vr} = \{Vg\}\), and \(\omega_v = \{gV \cap Vg\},\ g \in G\), of the group \(G\), where \(V\) is an arbitrary neighborhood of the identity of the group. As is known, the systems \(\Sigma_l\), \(\Sigma_r\), and \(\Sigma\) are bases of the uniformities \(L\), \(R\), and \(U\) on the group \(G\).

By \(X_l\), respectively \(X_r\), denote the space of left, respectively right, cosets of the group \(G\) modulo the subgroup \(H\). The natural projections \(p_l: G \to X_l\) and \(p_r: G \to X_r\) are not only open, but also, by virtue of the bicompactness of \(H\), perfect (i.e. closed and bicompact). The mappings \(p_l\) and \(p_r\) determine a mapping \(p_l \times p_r: G \to X_l \times X_r\). Denote the set \(p_l \times p_r(G)\) by \(X\), and denote the mapping \(p_l \times p_r: G \to X\) by \(p\). Obviously, the mapping \(p\) is perfect. It is not hard to show that, by virtue of the perfectness of the mappings \(p_l\) and \(p_r\), the set \(X\) is closed in \(X_l \times X_r\).

* For proximity spaces an analogous assertion was obtained earlier by V. Z. Polyakov in \((^{10})\).

It is clear that \(X\) is the quotient space of the space \(G\) with respect to the decomposition into bicompacts of the form \(gH\cap Hg,\ g\in G\).

By \(\omega_{pVl}, \omega_{pVr}\), and \(\omega_{pV}\) we denote the open covers \(\{p_l(gV)=p_l(gVH)\}\), \(\{p_r(Vg)=p_r(HVg)\}\), and \(\{p(gVH\cap HVg)\},\ g\in G\)* of the spaces \(X_l, X_r\), and \(X\), respectively.

Proposition 3. The systems \(\Sigma_p, \Sigma_{pl}\), and \(\Sigma_{pr}\) of all possible covers \(\omega_{pV}, \omega_{pVl}, \omega_{pVr}\), respectively, are bases of uniformities (we denote them by \(U_p, L_p\), and \(R_p\)) on the spaces \(X, X_l\), and \(X_r\), respectively, compatible with the topology of these spaces.

Lemma 3. The mappings \(p:G\to X,\ p_l:G\to X_l\), and \(p_r:G\to X_r\) are twice uniformly continuous with respect to the uniformities \(U\) and \(U_p,\ L\) and \(L_p,\ R\) and \(R_p\), respectively.

Corollary 2. a) If the group \(G\) is complete with respect to the uniformity \(U\) (complete in the sense of Raikov), then the space \(X\) is complete with respect to the uniformity \(U_p\). b) If the group \(G\) is complete with respect to the uniformity \(L\), and hence also \(R\) (complete in the sense of Weil), then the spaces \(X_l\) and \(X_r\) are complete with respect to the uniformities \(L_p\) and \(R_p\), respectively.

Moreover: c) If a subset \(\Gamma\) of the group \(G\) is complete with respect to the uniformity \(U\), respectively \(L, R\), then the set \(p\Gamma\), respectively \(p_l\Gamma, p_r\Gamma\), is complete with respect to the uniformity \(U_p\), respectively \(L_p, R_p\).

Let now the group \(G\) also be given a subgroup \(\Gamma\equiv H\) that is complete: a) with respect to the uniformity \(U\), respectively b) with respect to the uniformities \(L\) and \(R\), and hence closed.

By \(Y_l\) and \(Y_r\) we denote the spaces of left and right cosets of \(G\) modulo \(\Gamma\), and by \(q_l\) and \(q_r\) the natural projections of \(G\) onto \(Y_l\) and \(Y_r\), respectively. The mapping \(q_l\times q_r:G\to q_l\times q_r(G)\subseteq Y_l\times Y_r\) will be denoted by \(q\), and the set \(q_l\times q_r(G)\) by \(Y\). The natural projections \(X_l\) onto \(Y_l\) and \(X_r\) onto \(Y_r\) will be denoted by \(\pi_l\) and \(\pi_r\), respectively. The naturally arising mapping of \(X\) onto \(Y\) will be denoted by \(\pi\) \((\pi(x_l,x_r)=(\pi_l x_l,\pi_r x_r))\).

Lemma 4. a) All sets \(X_y=\pi^{-1}y,\ y\in Y\), are complete with respect to the uniformity \(U_p\), respectively b) all sets \(X_{y_l}=\pi_l^{-1}y_l,\ y_l\in Y_l\), and \(X_{y_r}=\pi_r^{-1}y_r,\ y_r\in Y_r\), are complete with respect to the uniformity \(L_p\) and \(R_p\), respectively.

Proposition 4. If the group \(H\) has countable character in the group \(\Gamma\), then: a) for each \(y\in Y\), respectively b) \(y_l\in Y_l\) and \(y_r\in Y_r\), the uniformity \(U_p\) on \(X_y\), respectively \(L_p\) on \(X_{y_l}\) and \(R_p\) on \(X_{y_r}\), is metrizable and complete.**

Theorem 2. If a subgroup \(\Gamma\) of the group \(G\) is complete in the sense of Weil and almost metrizable (7), then for any paracompact \(A\) contained in the quotient space \(Y_l\) (respectively in \(Y_r\)) there exists in \(G\) a closed subset \(B\) that is mapped perfectly onto \(A\) by the projection \(q_l\) (respectively \(q_r\)). If moreover \(\dim A=0\), then the mapping \(q_l:B\to A\) may be assumed to be a homeomorphism. If \(\Delta X\le n\), (8), then the mapping \(q_l:B\to A\) may be assumed to be at most \((n+1)\)-to-one.

Corollary 3. If a subgroup \(\Gamma\) of the group \(G\) is complete in the sense of Weil and almost metrizable, and the quotient space \(Y_l\) (respectively \(Y_r\)) is paracompact and \(\dim Y_l=0\), then the space of the group \(G\) is homeomorphic to the product \(Y_l\times\Gamma\).

Theorem 3. If a normal divisor \(\Gamma\) of the group \(G\) is complete in the sense of Čech, then for any paracompact \(A\subseteq G/\Gamma\) there exists in the group \(G\) a set \(B\) that is mapped perfectly onto \(A\) by the projection \(q\);

* It is not difficult to show that the set \(gVH\cap HVg\) is the preimage of the set \(p(gVH\cap HVg)\) under the mapping \(p\), since
\(p(gVH\cap HVg)=p(gVH)\cap p(HVg)\).

** Thus, all sets \(p^{-1}X_y\) are complete in the sense of Čech, and for \(\Gamma\equiv G\) we obtain M. Choban’s result on the Čech-completeness of a Raikov-complete almost metrizable group.

if $\dim A = 0$, then the mapping $q: B \to A$ may be regarded as a homeomorphism; if, however, $\Delta A \leq n$, then this mapping may be regarded as no more than $(n+1)$-fold.

Corollary 4. If a (quasi-)component $\Gamma$ of a group $G$ is complete in the sense of Čech, then for any locally bicompact set $A \subseteq G/\Gamma$ the mapping $q: q^{-1}A \to A$ is a locally trivial fibration; if the set $A$ is also paracompact, for example bicompact, then the mapping $q: q^{-1}A \to A$ is a trivial fibration.

Corollary 5. If a group $G$ is strongly paracompact, its (quasi-)component $\Gamma$ is complete in the sense of Čech, and $\operatorname{ind} G/\Gamma = 0$, then the space of the group $G$ is homeomorphic to the product of the spaces $\Gamma$ and $G/\Gamma$.

This corollary generalizes Mostert’s theorem stating that the space of a locally bicompact group $G$ is homeomorphic to the product of its component $\Gamma$ and the quotient space $G/\Gamma$ ($^6$).

Mechanical-Mathematical Faculty
M. V. Lomonosov Moscow State University

Received
20 XII 1968

REFERENCES

$^1$ A. Gleason, Ann. Math., 56, No. 2, 193 (1952).
$^2$ D. Montgomery, L. Zipin, Ann. Math., 56, No. 2, 213 (1952).
$^3$ V. M. Glushkov, UMN, 12, No. 2, 3 (1957).
$^4$ W. W. Comfort, K. A. Ross, Pacific J. Math., 16, No. 3, 483 (1966).
$^5$ E. G. Sklyarenko, Bull. Acad. Polon. Sci., Ser. Math., 10, No. 8, 429 (1962).
$^6$ P. S. Mostert, Duke Math. J., 23, No. 1, 57 (1956).
$^7$ B. A. Pasynkov, DAN, 161, No. 2, 281 (1965).
$^8$ V. I. Ponomarev, Matem. sborn., 60, No. 1, 89 (1963).
$^9$ E. Michael, Duke Math. J., 26, No. 4, 647 (1959).
$^{10}$ V. Z. Polyakov, DAN, 155, No. 5, 1014 (1964).