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

Abstract

The group ${\mathrm{PL}}_{+}(I)$ 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 ${\mathrm{Homeo}}_{+}^{Lip}(I)$ of bi-Lipschitz homeomorphisms of $I$, and ${\mathrm{Diff}}_{+}^{1+\epsilon }(I)$ of diffeomorphisms of $I$ whose derivatives are Hölder continuous with exponent $\epsilon$.

19 pages, 2 figures