Examples of topological groups homeomorphic to 𝑙^{𝑓}₂

Abstract

We prove that the following spaces are homeomorphic to l 2 f l_2^f : (1) the group of piecewise continuous autotransformations of [ − 1 , 1 ] [ - 1,1] preserving Lebesgue measure, and (2) certain subgroups obtained as group spans of linearly independent arcs in linear spaces. These are consequences of our discussion of the problem whether σ \sigma -fd-compact locally contractible metric groups must be either finite-dimensional or locally homeomorphic to l 2 f l_2^f .