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PL homeomorphisms of surfaces and codimension 2 PL foliations
- Part of
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- Journal:
- Compositio Mathematica / Volume 160 / Issue 11 / November 2024
- Published online by Cambridge University Press:
- 23 December 2024, pp. 2684-2703
- Print publication:
- November 2024
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- Article
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The Haefliger–Thurston 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 piecewise linear (PL) foliations of codimension 
$2$. Using this, we use a version of the Mather–Thurston theorem for PL homeomorphisms due to the author to derive new homological properties for PL surface homeomorphisms. In particular, we answer the question of Epstein in dimension 
$2$ and prove the simplicity of the identity component of PL surface homeomorphisms.
