Spaces of maps into topological group with the Whitney topology

Let X be a locally compact Polish space and G a non-discrete Polish ANR group. By C ( X , G ) , we denote the topological group of all continuous maps f : X → G endowed with the Whitney (graph) topology and by C c ( 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 l 2 -manifold. In this article we show that if X is non-compact and not end-discrete then C c ( X , G ) is an ( R ∞ × l 2 ) -manifold, and moreover the pair ( C ( X , G ) , C c ( X , G ) ) is locally homeomorphic to the pair of the box and the small box powers of l 2 .