Groups of measure-preserving homeomorphisms of noncompact 2-manifolds

Suppose M is a noncompact connected 2-manifold and μ is a good Radon measure of M with μ(∂M)=0. Let H(M) denote the group of homeomorphisms of M equipped with the compact-open topology and H(M)0 denote the identity component of H(M). Let H(M;μ) denote the subgroup of H(M) consisting of μ-preserving homeomorphisms of M and H(M;μ)0 denote the identity component of H(M;μ). We use results of A. Fathi and R. Berlanga to show that H(M;μ)0 is a strong deformation retract of H(M)0 and classify the topological type of H(M;μ)0.