A theorem of Pachner's states that if two triangulated PL-manifolds are PL-homeomorphic, the two triangulations are related via a finite sequence of moves, nowadays called "Pachner moves".

A Pachner move has a very simple picture. If $N$ is a triangulated $n$-manifold and $C \subset N$ is a co-dimension zero subcomplex of $N$ which is equivalent to a subcomplex $D'$ of $\partial \Delta_{n+1}$ where $\Delta_{n+1}$ is an $(n+1)$-simplex equipped with its standard triangulation, then you can replace $N'$ in $N$ by $(\partial \Delta_{n+1}) \setminus D'$ from $\partial \Delta_{n+1}$, the gluing maps being the only ones available to you.

One way to say Pachner's theorem is that there is a ''graph'' of triangulations of $N$, and it is connected. Specifically, the vertices of this graph consist of triangulations of $N$ taken up to the equivalence that two triangulations are equivalent if there is an automorphism of $N$ sending one triangulation to the other. The edges of this graph are the Pachner moves.

This formulation hints at an idea. Is there a useful notion of "Pachner complex"?

Of course this would lead to many further questions such as what geometric / topological properties does such a complex have, is it contractible for instance, are there "short" paths connecting any two points in the complex, and so on.

I'm curious if people have a sense for what such a Pachner complex should be. For example, some Pachner moves commute in the sense that the subcomplexes $C_1, C_2 \subset N$ are disjoint, so you can apply the Pachner moves independantly of each other. Presumably this should give rise to a "square" in any reasonable Pachner complex.

This feels related to the kind of complexes that Waldhausen and Hatcher used to use in the 70's but it's also a little different.

- Udo Pachner, P.L. homeomorphic manifolds are equivalent by elementary shellings, European J. Combin. 12 (1991), 129–145.