# Are triangulations of compact manifolds PL homeomorphic?

I have frequently come across the statement "Any two triangulations of a compact n-manifold are related by bistellar moves" attributed to Pachner via Lickorish's paper 'Simplicial moves on complexes and manifolds'. The theorem this refer to is the following: Closed combinatorial n-manifolds are PL homeomorphic if and only if they are bistellar equivalent.

My question is: Considering that Hauptvermutung is not true for manifolds of dimension more than 3, how can we justify this statement?

## 1 Answer

Pachner‘s Theorem is about PL-homeomorphic manifolds, while the Hauptvermutung is asking for a PL-homeomorphism between manifolds which a priori are only homeomorphic.

• Thanks ThiKu for your reply. Perhaps I should reframe my question: How do I show that any two simplicial triangulations of a compact manifold have a common refinement? As simplicial complexes in some R^n the two triangulations may have different supports. – user136604 Apr 30 '19 at 17:19
• If you are asking for a proof of Pachner‘s Theorem, the reference is sciencedirect.com/science/article/pii/S0195669813800807 – ThiKu Apr 30 '19 at 19:19
• If you want to know why Pachner moves yield triangulations with common refinements, that should hopefully be some elementary exercise, though I‘ve checked it only in dimension 3. – ThiKu Apr 30 '19 at 19:24
• They are not. Pachner's Theorem is only true für PL-homeomorphic manifolds. – ThiKu May 1 '19 at 4:54
• Perhaps those papers were referring to special dimensions. For 2- and 3-dimensional manifolds, every homeomorphism class contains a unique PL-structure. (Moise‘s Theorem, see springer.com/de/book/9781461299080 ). Thus in that case triangulation of homeomorphic manifolds are related via Pachner moves. – ThiKu May 1 '19 at 6:14