On the homogeneity of dyadic compact Hausdorff spaces (Q1337901)

\textit{B. A. Efimov} [Tr. Mosk. Mat. O.-va 14, 211-247 (1965; Zbl 0163.17202), p. 233] posed the following question: Does there exist a zero-dimensional homogeneous dyadic compact Hausdorff space that is not homeomorphic to \(D^\tau\)? \textit{V. V. Pashenkov} [Sov. Math., Dokl. 15, 43-47 (1974); translation from Dokl. Akad. Nauk SSSR 214, 44-47 (1974; Zbl 0301.54024)] constructed for each uncountable cardinal \(\tau\) a zero-dimensional homogeneous dyadic compact Hausdorff space of weight \(2^\tau\) that is not homeomorphic to the Cantor cube. But Efimov's question for the space of weight \(\omega_1\) remained open. \textit{M. Bell} had established under the assumption of CH that a zero-dimensional homogeneous dyadic compact Hausdorff space of weight \(\omega_1\) is homeomorphic to \(D^{\omega_1}\). Under the assumption of \(\text{MA}+ \neg\text{CH}\), the author has cited the proof of the same result and expressed the hope of the existence of a ``naive'' solution. This solution was soon obtained and presented at the Aleksandrov Seminar at the Moscow State University. The present paper is devoted to the presentation of this result and those similar to it. (During preparation of this article, the author heard from M. Bell that he also had obtained a ``naive'' proof of the above statement.) The fundamental results results of the paper are Theorem 4. Let \(X\) be a homogeneous dyadic compact Hausdorff space of weight \(\omega_1\). Then \(X\) is a Dugundji space. Theorem 5. Let \(X\) be a dyadic compact Hausdorff space. If the space \(\mathbb{P} (X)\) of probability measures on \(X\) is homogeneous, then \(\mathbb{P} (X)\) is homeomorphic either to \(\mathbb{I}^{\omega_0}\) or to \(\mathbb{I}^{\omega_1}\). These results are obtained using the concept of a point of bicommutativity (Def. 2) and an existence theorem for such points in limit diagrams of morphisms of spectra (Theorem 1).