TITLE: A Pachner complex for triangulated manifolds
QUESTION [31 upvotes]: 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.

REPLY [2 votes]: I discussed this complex with a few folks, a while back (perhaps including Ben Burton and Jonathan Spreer?).  We decided that its "coarse geometry" is not very nice.
Suppose that $M$ is a surface.  Let $P(M)$ be its Pachner graph.  We perform Pachner moves to ensure that there are $k$ disjoint disks in the current triangulation.  We now subdivide independently inside of these disks.  We deduce that $P(M)$ contains "quasi-$k$-orthants": subgraphs quasi-isometric to $\mathbb{N}^k$.  This is true for all $k$.
If we fix the number of zero-cells in dimension two (that is, disallow (3,1) and (1,3) moves) then the "flip graph" is very interesting, and appears in Teichmuller theory.  Fixing the number of zero-cells does not save us in dimension three - the graph again has quasi-orthants of all dimensions.  This suggests the following question:

What if we restrict our attention to a compact, triangulated, four-manifold $X^4$, and only allow $(3, 3)$ moves?

Can we hope that this graph has some sensible connection to the mapping class group of $X^4$?