$\mathop{\rm PL}_+(I)$ is not a Polish group

TL;DR

The paper proves that certain groups of homeomorphisms and diffeomorphisms of the interval cannot be given a Polish group topology, highlighting limitations in their topological structure.

Contribution

It demonstrates that groups of increasing piecewise linear, bi-Lipschitz, and certain smooth diffeomorphisms of the interval cannot be endowed with a compatible Polish topology.

Findings

$ ext{PL}_+(I)$ is not a Polish group under any topology.

Similar non-Polish results for $ ext{Homeo}_+^{Lip}(I)$ and $ ext{Diff}_+^{1+ ext{Hölder}}(I)$.

Highlights limitations in the topological structure of these transformation groups.

Abstract

The group $\mathop{\rm PL}_+(I)$ of increasing piecewise linear self-homeomorphisms of the interval $I=[0,1]$ may not be assigned a topology in such a way that it becomes a Polish group. The same statement holds for the groups $\mathop{\rm Homeo}_+^{Lip}(I)$ of bi-Lipschitz homeomorphisms of $I$, and $\mathop{\rm Diff}_+^{1+\epsilon}(I)$ of diffeomorphisms of $I$ whose derivatives are H\"older continuous with exponent $\epsilon$.