Some aspects of dimension theory for topological groups

Abstract

We discuss dimension theory in the class of all topological groups. For locally compact topological groups there are many classical results in the literature. Dimension theory for non-locally compact topological groups is mysterious. It is for example unknown whether every connected (hence at least 1-dimensional) Polish group contains a homeomorphic copy of [0,1]. And it is unknown whether there is a homogeneous metrizable compact space the homeomorphism group of which is 2-dimensional. Other classical open problems are the following ones. Let G be a topological group with a countable network. Does it follow that dimG=indG=IndG? The same question if X is a compact coset space. We also do not know whether the inequality dim(G×H)≤dimG+dimH holds for arbitrary topological groups G and H which are subgroups of σ-compact topological groups. The aim of this paper is to discuss such and related problems. But we do not attempt to survey the literature.