PL homeomorphisms of surfaces and codimension $2$ PL foliations

Haefliger-Thurston’s conjecture predicts that Haefliger’s classifying space for $C^r$-foliations of codimension $n$ whose normal bundles are trivial is $2n$-connected. In this paper, we confirm this conjecture for PL foliations of codimension $2$. As a consequence, we use a version of Mather-Thurston’s theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer a question of Epstein in dimension $2$ and prove the simplicity of the identity component of PL surface homeomorphisms.