Simplicial moves on complexes and manifolds

TL;DR

This paper revisits fundamental theorems in combinatorial topology, demonstrating that stellar and Pachner moves suffice to relate triangulations of simplicial complexes and manifolds, with simplified proofs of these classic results.

Contribution

The paper provides streamlined proofs showing that stellar moves relate PL homeomorphic complexes and that a finite set of Pachner moves suffices for triangulation equivalence.

Findings

Stellar moves connect PL homeomorphic complexes.

A finite set of Pachner moves relates triangulations of PL manifolds.

Simplified proofs of classic theorems in combinatorial topology.

Abstract

Here are versions of the proofs of two classic theorems of combinatorial topology. The first is the result that piecewise linearly homeomorphic simplicial complexes are related by stellar moves. This is used in the proof, modelled on that of Pachner, of the second theorem. This states that moves from only a finite collection are needed to relate two triangulations of a piecewise linear manifold.