# What is a sufficient set of links in a simplicial complex to represent any PL manifold?

The link of a vertex in a $$n$$-dimensional simplicial complex is the $$(n-1)$$-dimensional simplicial complex formed by the $$(n-1)$$-simplices that together with the vertex span a $$n$$-simplex. A combinatorial PL manifold is a simplicial complex where the links of all vertices are topologically equivalent to a PL sphere.

It is known that equivalence classes of combinatorial PL manifolds under Pachner moves are in one-to-one correspondence with equivalence classes of PL manifolds under homeomorphism.

Instead of allowing all links with PL topology of a sphere, we can restrict to a finite set of allowed links. This includes restricting to Pachner moves that transform a combinatorial PL manifold with allowed links in another one with allowed links (note that a Pachner move in the manifold acts also with Pachner moves on the links of the involved vertices).

If this set of allowed links is big enough then the equivalence in the second paragraph will still hold. Note that this includes 1) we still have to be able to represent all PL manifolds and 2) we still have to be able to transform any equivalent combinatorial manifolds into each other with Pachner moves, without going via any non-allowed links.

E.g. in 2 dimensions, it is easy to show that 5-, 6-, and 7-gons as links are enough to represent any 2-manifold, and it seems highly plausible that any two such restricted combinatorial manifolds can be transformed by Pachner moves if also 3- to, say, 10-gons are allowed.

Is there a general scheme to construct a sufficiently large set of allowed links that is still rather small? Optimally, this would work for any dimension, but I'm also fine with dimensions 2 and 3. I'm not searching for the strictly smallest possible set of allowed links, but I also would want something better than "provably, taking links with $$<10000$$ vertices is sufficient in $$3$$ dimension".

Not sure about Pachner moves, but as for links, three suffice, see the math review of:

Cooper, D.; Thurston, W. P., Triangulating 3-manifolds using 5 vertex link types, Topology 27, No. 1, 23-25 (1988). ZBL0656.57004.

Kevin Walker walks among us, so he may comment further.

• Thanks, that looks promising that small sufficient sets of links exist. However, there is not a single Pachner move transforming a link in this set to into another link in the set, so one won't be able to go between equivalent combinatorial manifolds via simplicial complexes with only those links. The latter constraint seems to be the hard bit of the problem. Without it, it should not be that difficult to find small sufficient sets of links by refining and "smoothing" a given triangulation. – Andi Bauer Feb 8 '20 at 12:13