By an ANR fibration we will
mean a Hurewicz fibration 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 ℰ→ B in the sense that there
exists a path through ANR fibrations from E to ℰ. This takes the form of an
ANR fibration over B × [0,1] whose 0-level is E and whose 1-level is ℰ. 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 × [0,1] can be chosen to be a PL
fibration.
