MATHEMATICS

L. N. IVANOVSKII

ON A CONJECTURE OF P. S. ALEXANDROV

(Presented by Academician P. S. Alexandrov on 21 VII 1958)

In the present note the following proposition of P. S. Alexandrov is proved:

Every bicompact topological group is a dyadic bicompactum.

Let \(\theta\) be an arbitrary limit transfinite number. The inverse spectrum
\(S_\theta=\{G_\alpha,\pi_\beta^\alpha\}\) of bicompact groups over the directed set of all transfinite numbers less than \(\theta\) will be called a \(\theta\)-spectrum if for every limit transfinite number \(\alpha<\theta\) the intersection
\(\bigcap_{\beta<\alpha}\operatorname{Ker}\pi_\beta^\alpha\) contains only the identity element of the group \(G_\alpha\). For any limit transfinite number \(\alpha<\theta\) the segment \(S_{\theta,\alpha}\) of the \(\theta\)-spectrum \(S_\theta\), consisting of the groups \(G_\beta\), \(\beta<\alpha\), and the homomorphisms \(\pi_\beta^\gamma\), \(\beta<\gamma<\alpha\), is an \(\alpha\)-spectrum, whose limit group is, obviously, naturally isomorphic to the group \(G_\alpha\). The natural isomorphism \(G_\alpha\approx \lim S_{\theta,\alpha}\) will be denoted by \(\varphi_\alpha\).

For each \(\alpha<\theta\) consider the bicompactum
\[ X_\alpha=\prod_{0\leq\mu<\alpha} K_\mu, \]
where \(K_\mu\) is a space homeomorphic to the Cantor perfect set (for each \(\mu\)). It is clear that for any \(\beta<\alpha<\theta\) there is defined a natural mapping \(\tilde\pi_\beta^\alpha:X_\alpha\to X_\beta\). The bicompactum \(X_\alpha\) is a zero-dimensional topological group, and the mappings \(\tilde\pi_\beta^\alpha\) are continuous homomorphisms. The groups \(X_\alpha\), \(\alpha<\theta\), and the homomorphisms \(\tilde\pi_\beta^\alpha\), \(\beta<\alpha<\theta\), form, obviously, a \(\theta\)-spectrum \(K(\theta)\). The limit group of this spectrum is the group \(D_\tau\), where \(\tau\) is the cardinality of the number \(\theta\).

Following L. S. Pontryagin \((^2)\), we shall call a \(\theta\)-spectrum \(S_\theta\) a Li row of length \(\theta\) if the group \(G_1\) is a Li group, and for every transfinite number \(\alpha<\theta\) the homomorphism
\[ \pi_\alpha^{\alpha+1}:G_{\alpha+1}\to G_\alpha \]
is an epimorphism whose kernel is some Li group.

A system of continuous (generally speaking, nonhomomorphic!) mappings
\[ f^\beta:X_\beta\to G_\beta,\quad \beta<\alpha\leq\theta, \]
will be called an \(\alpha\)-special mapping of the \(\theta\)-spectrum \(K(\theta)\) into the Li row \(S_\theta\) if: a) the image of the mapping \(f^1:X_1\to G_1\) coincides with the whole group \(G_1\); b)
\[ \pi_\beta^\gamma f^\gamma=f^\beta \tilde\pi_\beta^\gamma,\quad \beta<\gamma<\alpha; \]
c) if
\[ f^\beta(z)=\pi_\beta^{\beta+1}(y)=x, \]
where \(x\in G_\beta\), \(y\in G_{\beta+1}\), \(z\in X_\beta\), \(\beta+1<\alpha\), then there exists such an element \(\vartheta\in X_{\beta+1}\) that
\[ \tilde\pi_\beta^{\beta+1}(\vartheta)=z \]
and
\[ f^{\beta+1}(\vartheta)=y. \]

A \(\beta\)-special mapping \(g:K(\theta)\to S_\theta\) is called an extension of an \(\alpha\)-special mapping \(f:K(\theta)\to S_\theta\), \(\alpha<\beta\), if \(g^\delta=f^\delta\) for \(\delta<\alpha\).

Lemma 1. For any \(\theta\)-special mapping \(f:K(\theta)\to S_\theta\), the image of the limit mapping
\[ f^*:\lim K(\theta)\to \lim S_\theta \]
coincides with the whole group \(\lim S_\theta\).

For the proof, consider an arbitrary thread \(\{x_\alpha\}\) of the Li row \(S_\theta\) and suppose that for each \(\alpha'\), less than some \(\alpha<\theta\), such elements
\[ y_1,y_2,\ldots,y_{\alpha'},\ldots,\ y_{\alpha'}\in X_{\alpha'} \]
have already been defined that
\[ f^{\alpha'}(y_{\alpha'})=x_{\alpha'}. \]

\(\alpha' < \alpha,\ \pi_{\alpha''}^{\alpha'}(y_{\alpha'}) = y_{\alpha''},\ \alpha'' < \alpha' < \alpha,\ y_1 \in (f^1)^{-1}(x_1)\). If \(\alpha\) is a limit transfinite number, then the elements \(y_1, y_2, \ldots, y_{\alpha'}, \ldots,\ \alpha' < \alpha\), form a thread \(\omega\) of the \(\alpha\)-spectrum \(K(\theta)_\alpha\), and therefore the element \(y_\alpha = \tilde{\varphi}_{\alpha}^{-1}(\omega) \in X_\alpha\) is defined. If \(\alpha\) is a non-limit transfinite number, then by property c) there is an element \(y_\alpha \in X_\alpha\) such that \(\pi_{\alpha-1}^{\alpha}(y_\alpha) = y_{\alpha-1}\) and \(f^\alpha(y_\alpha)=x_\alpha\). Continuing the process, we obtain a thread \(\{y_\beta\}\) of the \(\theta\)-spectrum \(K(\theta)\), which is carried under the mapping \(f^*:\lim K(\theta)\to \lim S_\theta\) into the thread \(\{x_\beta\}\) of the Li series \(S_\theta\).

Lemma 2. Let \(f:G\to H\) be a homomorphic mapping of the group \(G\) onto the group \(H\), whose kernel \(\operatorname{Ker} f\) is a compact Lie group, and let \(g:N\to H\) be a continuous mapping of a zero-dimensional bicompactum \(N\) into the group \(H\). Then there exists a continuous mapping \(g':N\to G\) such that \(fg'=g\).

For the proof, note that \((G,H,f,\operatorname{Ker} f)\) is, by Gleason’s theorem \((^3)\), a fiber bundle, and therefore there exists an open covering \(\{u_i\}\) of the group \(H\) and continuous mappings \(\varphi_i:u_i\to G\) such that \(f\varphi_i=1_{u_i}\). Into the open covering \(\{g^{-1}(u_i)\}\) of the zero-dimensional bicompactum \(N\) we now inscribe a finite open covering \(\{V_1,\ldots,V_r\}\) consisting of pairwise disjoint sets, and define the mapping \(g':N\to G\) by setting \(g'(x)=\varphi_i g(x)\) for all \(x\in V_j\subset g^{-1}(u_i)\).

Lemma 3. For every \(\alpha\)-special \((\alpha<\theta)\) mapping \(f:K(\theta)\to S_\theta\) of the \(\theta\)-spectrum \(K(\theta)\) into the Li series \(S_\theta\), there exists an \((\alpha+1)\)-special mapping \(g:K(\theta)\to S_\theta\) extending the mapping \(f\).

If \(\alpha\) is a limit transfinite number, then, setting \(f^\alpha=\tilde{\varphi}_{\alpha}^{-1} f^* \varphi_\alpha\), where \(f^*:\lim K(\theta)_\alpha\to \lim S_{\theta,\alpha}\) is the mapping induced by the mapping \(f:K(\theta)\to S_\theta\), we obtain a certain mapping \(f^\alpha:X_\alpha\to G_\alpha\). If \(\alpha\) is a non-limit transfinite number, then, using Lemma 2, we can construct a continuous mapping \(f':X_{\alpha-1}\to G_\alpha\) such that \(\pi_{\alpha-1}^{\alpha} f'=f^{\alpha-1}\). Consider, in addition, the natural isomorphism \(q:X_\alpha \approx X_{\alpha-1}\times K_{\alpha-1}\) and some continuous mapping \(p:K_{\alpha-1}\to \operatorname{Ker}\pi_{\alpha-1}^{\alpha}\) of the Cantor perfect set \(K_{\alpha-1}\) onto the compact Lie group \(\operatorname{Ker}\pi_{\alpha-1}^{\alpha}\) (see (1)). We now define the mapping \(q':X_{\alpha-1}\times K_{\alpha-1}\to G_\alpha\) by the formula \(q'(x,a)=f'(x)\cdot p(a)\), where \(x\in X_{\alpha-1},\ a\in K_{\alpha-1}\), and put \(f^\alpha=q'q:X_\alpha\to G_\alpha\). It is easy to see that the mapping \(f^\alpha\), together with the mappings \(f^\beta\), where \(\beta<\alpha\), defines an \((\alpha+1)\)-special mapping \(g:K(\theta)\to S_\theta\) extending the mapping \(f\).

Corollary. There exists a \(\theta\)-special mapping \(f:K(\theta)\to S_\theta\) of the \(\theta\)-spectrum \(K(\theta)\) into the Li series \(S_\theta\).

Theorem. For every bicompact group \(G\) of weight \(\tau\) there exists a continuous mapping \(f:D_\tau\to G\) of the group \(D_\tau\) onto the whole group \(G\).

For the proof it suffices to note that, by a theorem of L. S. Pontryagin \((^2)\), there exists a Li series \(S_\theta\) of length \(\theta\), where \(\theta\) is the first transfinite number of cardinality \(\tau\), having as its limit group the group \(G\).

Moscow State University
named after M. V. Lomonosov

Received
5 V 1958

REFERENCES

\(^1\) P. S. Aleksandrov, Introduction to the General Theory of Sets and Functions, Moscow–Leningrad, 1948. \(^2\) L. S. Pontryagin, Continuous Groups, Moscow, 1954. \(^3\) A. M. Gleason, Proc. Am. Math. Soc., 1, 35 (1950).