"PL homeomorphisms of surfaces and codimension 2 PL foliations"
Nariman, SamHaefliger-Thurston's conjecture predicts that Haefliger's classifying space for codimension $n$ smooth foliations whose normal bundles are trivial is $2n$-connected. In this talk, we discuss how one can use Greenberg's model to confirm this conjecture for codimension $2$ PL foliations. As a consequence, we use our version of Mather-Thurston's theorem for PL homeomorphisms to derive new homological properties for PL surface homeomorphisms. In particular, we answer a question of Epstein regarding the simplicity of the identity component of PL surface homeomorphisms.