Piecewise linear fibrations

Abstract

By an ANR fibration we will mean a Hurewicz fibraton p: E -> B, where E is a compact ANR and B is a compact polyhedron. In case E is also a polyhedron and p is a piecewise linear (PL) map, we say that E is a PL fibration. An important special case of this is the notion of a PL manifold bundle, which is a PL locally trivial bundle for which the fibers are compact PL manifolds (with boundary). It is known that any ANR fibration E -> B is "homotopic" to a PL manifold bundle <f -> B in the sense that there exists a path through ANR fibrations from E to . This takes the form of an ANR fibration over B X [0,1] whose 0-level is E and whose 1-level is <f. The purpose of this paper is to prove that if E is additionally assumed to be a PL fibration, then the ANR fibration over B X [0,1] can be chosen to be a PL fibration.