All triangulations have a common stellar subdivision

We address two longstanding open problems, one originating in PL topology, another in birational geometry. We prove the weighted version of Oda’s strong factorization conjecture (1978): we prove that any two toric varieties whose fans have the same support admit a common toric modification induced by iterated stellar subdivisions. This implies that every two PL homeomorphic polyhedra have a common stellar subdivision, which was a conjecture going back to Tietze’s formulation of the Hauptvermutung in 1908, and often attributed to Alexander.