Spaces which do not embed in topological groups of the same dimension (Q1210362)

The paper fills in several blanks of dimension theory of topological groups. For each integer \(n>0\), there is given a simple example of an \(n\)-dimensional compact metric space \(X_ n\) which does not embed in any \(n\)-dimensional topological group, this answers some questions of Bel'nov and Shakhmatov. Another strong result of Kulesza: the hedgehog \(J(\aleph_ 1)\) with \(\aleph_ 1\) spines embeds in no finite- dimensional topological group, though \(\dim J(\aleph_ 1)= 1\). The third result of the paper answers a question of Kimura: the hedgehog \(J(3)\) with three spines does not embed in any one-dimensional topological group. Note that by the earlier result of \textit{D. B. Shakhmatov} [Topology Appl. 36, 181-204 (1990; Zbl 0709.22001)] every zero- dimensional Tikhonov space embeds in a zero-dimensional topological group.