PL(M) admits no Polish group topology

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.