$\operatorname{PL}(M)$ admits no Polish group topology
Volume 238 / 2017
Fundamenta Mathematicae 238 (2017), 285-295
MSC: Primary 57S05; Secondary 54H15.
DOI: 10.4064/fm285-10-2016
Published online: 3 March 2017
Abstract
We show that the group of piecewise linear homeomorphisms of any compact PL manifold does not admit a Polish group topology, using both general results on topologies on groups of homeomorphisms, and results on the algebraic structure of PL homeomorphism groups. The proof also shows that the group of piecewise projective homeomorphisms of $S^1$ has no Polish topology.
