# Hauptvermutung for non-manifolds

The Hauptvermutung proposes the following: if two finite simplicial complexes are homeomorphic then they are PL-homeomorphic, meaning that they have a common refinement.

People are mostly interested in the manifold case but I am asking about the general case. According to Ranicki's Hauptvermutung book the 2-dimensional case was proven by Papakyriakopoulos in a paper written in Greek in Bull. Soc. Math. Grèce. My first question is: is there another more accessible reference for this?

In this paper Edward Brown proves the 3-dimensional case, but the paper is not mentioned in Ranicki's book (and hardly at all in general). My second question is: is there a particular reason for that?

Context: the Hauptvermutung for combinatorial manifolds is known to hold in dimension $$\le 3$$ by work of Radó and Moise. The Hauptvermutung was shown to be false by Milnor. Today obstructions to being PL-homeomorphic are well understood.

• Incidentally, Dugundji's MathSciNet report (MR0024619) does not help to gain confidence in Papakyriakopoulos's paper: "the lemma [...] cannot be correct", "the proof [...] contains a serious gap", "this leads him [...] to hastily assume, incorrectly". Jan 15, 2020 at 16:09
• That's a great question. The seemingly inaccessible 153 page paper by Papakyriakopoulos (or does anybody have a digital copy?) is a pretty scary reference. I don't know about the status of Brown's paper. We really need a system for flagging papers, otherwise we have no idea which of the older papers are reliable and which ones are not. Perhaps people back in the 1960s knew which papers are shaky, but how we now supposed to know? Jan 20, 2020 at 22:48