Compactifications of the homeomorphism group of a graph

摘要

Let Γ be a countable locally finite graph and let H(Γ) and H_+(Γ) denote the homeomorphism group of Γ with the compact-open topology and its identity component. These groups can be embedded into the space Cld_F~*(Γ × Γ) of all closed sets of Γ × Γ with the Fell topology, which is compact. Taking closure, we have natural compactifications H(Γ) and H_+(Γ). In this paper, we completely determine the topological type of the pair (H_+(Γ), H_+(Γ)) and give a necessary and sufficient condition for this pair to be a (Q,s)-manifold. The pair (H(Γ), H(Γ)) is also considered for simple examples, and in particular, we find that H(T) has homotopy type of RP~3. In this investigation we point out a certain inaccuracy in Sakai-Uehara's preceding results on (H(Γ), H(Γ)) for finite graphs Γ.