Spaces of Maps into Topological Group with the Whiteny Topology

Let X be a separable locally compact metrizable space and G a separable completely metrizable ANR topological group without isolated points. By C(X, G), we denote the topological group of all continuous maps f : X → G endowed with the Whitney (graph) topology and by Cc(X, G) the subgroup consisting of all maps with compact support. It is known that if X is compact and non-discrete then the space C(X, G) is an l2-manifold. In this article we show that if X is non-compact and not end-discrete then Cc(X, G) is an (R∞× l2)-manifold, and moreover the pair (C(X, G), Cc(X, G)) is locally homeomorphic to the pair of the box and the small box powers of l2.