  We show that the group of piecewise linear homeomorphisms of any compact PL
manifold does not admit a Polish group topology. This uses a) new results on
the relationship between topologies on groups of homeomorphisms, their
algebraic structure, and the topology of the underlying manifold, and b) new
results on the structure of certain subgroups of PL(M). The proof also shows
that the group of piecewise projective homeomorphisms of the circle has no
Polish group topology.
