# Cubical Pachner moves and the Hauptvermutung for cubulations

Pachner proved that any two PL homeomorphic simplicial manifolds are related by a sequence of rather simple moves, called Pachner moves. Motivated by finite element methods and geometric topology, Thurston, Habegger and many others considered the question under which conditions two cubulations $X$, $Y$ of a manifold $M$ are equivalent under Pachner moves. I will discuss the solution to this problem, and the interesting relation to Pontryagin—Thom constructions.