Thurston's fragmentation and c-principles

TL;DR

This paper extends Thurston's fragmentation and c-principles to PL and contactomorphism settings, providing new variants of the Mather-Thurston theorem and connecting to non-abelian Poincaré duality.

Contribution

It generalizes Thurston's techniques to PL and contactomorphisms, answering longstanding questions and linking to non-abelian Poincaré duality via fragmentation.

Findings

New variants of Mather-Thurston theorem for PL and contactomorphisms

Thurston's fragmentation implies non-abelian Poincaré duality

Provides a compactly supported c-principle without local open ball statements

Abstract

In this paper, we generalize the original idea of Thurston for the so called Mather-Thurston's theorem for foliated bundles to prove new variants of this theorem for PL homeomorphisms, contactormorphisms. These versions answer questions posed by Gelfand -Fuks and Greenberg on PL foliations and Rybicki on contactomorphisms. The interesting point about the original Thurston's technique compared to the better known Segal-McDuff's proof of the Mather-Thurston theorem is that it gives a compactly supported c-principle theorem without knowing the relevant local statement on open balls. In the appendix, we show that Thurston's fragmentation implies the non-abelian Poincare duality theorem and its generalization using blob complexes.