Spaces of maps into topological group with the Whitney topology

TL;DR

This paper investigates the topological structure of spaces of continuous maps into a topological group, revealing manifold structures in non-compact cases and describing local homeomorphisms to box power pairs.

Contribution

It extends known results by characterizing the manifold structure of C_c(X,G) and the pair (C(X,G), C_c(X,G)) for non-compact, non-end-discrete spaces.

Findings

C_c(X,G) is an (R^    l_2)-manifold.

The pair (C(X,G), C_c(X,G)) is locally homeomorphic to box and small box powers of l_2.

Manifold structures depend on the non-compactness and end-discreteness of X.

Abstract

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 \to 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^\infty \times 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.