Already in April a paper was put on the arXiv (2404.05930) by Karim Adiprasito and Igor Pak proving the following result: *Every two triangulations of a geometric complex in Euclidean space have a common stellar subdivision.*

At first I was a bit confused, because my mind connected this to the Hauptvermutung, which was disproven a few years ago. But Karim explained to me the difference: The Hauptvermutung claims that every two triangulations of the same manifold are PL-homeomorphic, whereas their new result besically already assumes the existence of a PL homeomorphism and then refines this in a certain way.

Originally I wanted to explain a bit their result in this post, as Karim explained to me. But then I saw that both of the authors have their own blogs and you can find a nice explanation on Igor’s: link. (And here is the link to the corresponding one of Karim: link.)