PL(M) admits no Polish group topology

TL;DR

The paper proves that the group of piecewise linear homeomorphisms of any compact PL manifold cannot have a Polish group topology, revealing fundamental limitations on their topological structure.

Contribution

It introduces new results linking topologies on homeomorphism groups, their algebraic structures, and the topology of the manifolds, specifically for PL and projective cases.

Findings

PL(M) groups admit no Polish group topology

The group of piecewise projective homeomorphisms of the circle has no Polish topology

New structural results on subgroups of PL(M)

Abstract

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.