$\operatorname{PL}(M)$ admits no Polish group topology

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 S 1 has no Polish topology.

Polishability of some groups of interval and circle diffeomorphisms

“…In [3], Kallman and the author investigated the Polishability problem for several of the groups mentioned above, and it turns out that Homeo Lip + (M), Diff 1+ǫ + (M), and Diff 1+Lip + (M) are all examples of a peculiar phenomenon: they are continuumcardinality groups whose algebraic structure precludes the existence of any Polish group topology whatsoever 2 . This puts them in somewhat curious company, alongside free groups and free abelian groups on continuum-many generators ( [4]); the automorphism group of the category algebra of R ( [6]); the homeomorphism groups of the rationals and the irrationals ( [15]); and very few other known examples (although this list seems to be growing recently, see [7], [10], [13]).…”

Topological Transformation Monoids

“…There are many further examples of groups with a unique Polish group topology, such as the group of isometries of Minkowski spacetime (the usual framework for special relativity) [14]; see also [9,13]. Furthermore, there is a wealth of examples of infinite groups with no Polish group topologies [6,19]. Additional references include [5,7,11].…”