Transitive actions of compact groups and topological dimension (Q1841833)

The weight \(w(X)\) of a topological space \(X\) is the smallest of all cardinals of bases for the topology. Using the Lebesgue covering dimension \(\text{cdim} X\), a new dimension function is defined: \(\dim{X}=\text{cdim} X\) if the latter is finite, and \(\dim X\) is the supremum over the weights of the connected components of \(X\), otherwise. For any weakly complete topological vector space \(V\) over the reals, this topological dimension of \(V\) equals the linear dimension of the vector space dual of \(V\). For any compact group \(G\), let \(L(G)\) denote the Lie algebra of \(G\); this is a weakly complete space [same authors, The structure of compact groups (Berlin 1998; Zbl 0919.22001)]. The authors prove the following. If a compact group \(G\) acts transitively on \(X\), with isotropy subgroup \(H\), then \(\dim X=\dim(L(G)/L(H))\). Moreover, the space \(X\) contains a cube \([0,1]^{\dim X}\), and the dimension of cubes in \(X\) is bounded by \(\dim X\). Let \(G\) be a locally compact group. Then every neighborhood \(U\) of the identity of \(G\) contains a compact subgroup \(N\) such that there exist a simply connected Lie group \(L\) and a continuous open homomorphism \(\varphi \colon N\times L\to G\) with discrete kernel such that \(\varphi (n,1)=n\) for all \(n\in N\). This generalizes a famous theorem due to \textit{K. Iwasawa} [Ann. Math., Princeton, II. Ser. 50, 507-558 (1949; Zbl 0034.01803)]. Let \(G\) be a locally compact group and \(H\) a closed subgroup of \(G\). Then \(\dim(G/H)=\dim(L(G)/L(H))\), and \(\dim (G/H)\) is maximal among the dimensions of cubes contained in \(G/H\).