Measure-preserving homeomorphisms of noncompact manifolds and mass flow toward ends
Volume 197 / 2007
Abstract
Suppose $M$ is a noncompact connected $n$-manifold and $\omega$ is a good Radon measure of $M$ with $\omega(\partial M) = 0$. Let ${\cal H}(M, \omega)$ denote the group of $\omega$-preserving homeomorphisms of $M$ equipped with the compact-open topology, and ${\cal H}_E(M, \omega)$ the subgroup consisting of all $h \in {\cal H}(M, \omega)$ which fix the ends of $M$. S. R. Alpern and V. S. Prasad introduced the topological vector space ${\cal S}(M, \omega)$ of end charges of $M$ and the end charge homomorphism $c^\omega : {\cal H}_E(M, \omega) \to {\cal S}(M, \omega)$, which measures for each $h \in {\cal H}_E(M, \omega)$ the mass flow toward ends induced by $h$. We show that the map $c^\omega$ has a continuous section. This induces the factorization ${\cal H}_E(M, \omega) \cong {\rm Ker}\,c^\omega \times {\cal S}(M, \omega)$ and implies that ${\rm Ker}\,c^\omega$ is a strong deformation retract of ${\cal H}_E(M, \omega)$.
