PL approximations of fiber preserving homeomorphisms of vector bundles

Abstract

We investigate the group of f.p. homeomorphisms of an $n$-dimensional vector bundle $\xi$. In the case $n\geq 5$ and the base space of $\xi$ is countable dimensional, we show that every f.p. stable homeomorphism of $\xi$ can be approximated by f.p. $PL$ homeomorphisms with respect to the majorant topology. As an application we can show that if the base space is compact, then the group of f.p. $PL$ homeomorphisms of $\xi$ with the uniform topology has the mapping absorption property for maps from countable dimensional metric spaces into the group of f.p. homeomorphisms of $\xi$ which are PL on the unit open ball.