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In this paper, we generalize the original idea of Thurston for the so calledMather-Thurston's theorem for foliated bundles to prove new variants of thistheorem for PL homeomorphisms, contactormorphisms. These versions answerquestions posed by Gelfand -Fuks and Greenberg on PL foliations and Rybicki oncontactomorphisms. The interesting point about the original Thurston'stechnique compared to the better known Segal-McDuff's proof of theMather-Thurston theorem is that it gives a compactly supported c-principletheorem without knowing the relevant local statement on open balls. In theappendix, we show that Thurston's fragmentation implies the non-abelianPoincare duality theorem and its generalization using blob complexes.

Thurston's fragmentation and c-principles