Let $C$ be a $d$-dimensional (abstract) polytopal complex. Most of what I say below could be asked in this general setting, but for a start, let's further restrict to simple polytopal spheres, that is, $C$ is homeomorphic to the $d$-sphere, and at each vertex meet exactly $d+1$ facets.

My goal is to formalize and then prove a certain kind of statement. For a warmup, here are two examples of what I am after:

  • Assume that each vertex of $C$ is incident to $d+1$ facets, each one combinatorially equivalent to the $d$-cube. From that I can already deduce that $C$ is combinatorially equivalent to the (boundary of the) $(d+1)$-cube.
  • Assume that $d=2$ and each vertex is incident to a 4-gonal cell and two 6-gonal cells. This already suffices to conclude that $C$ is combinatorially equivalent to the (boundary of the) permutahedron.

In general, I want to prove something like this:

If each vertex of a simple polytopal sphere $C$ looks the same locally (that is, is incident to the same type of facets), then $C$ is already uniquely determined.

So how to prove this?

Intuitively, this is easy. Take, for example, the second example from above (and see the image below): if you try to draw its edge graph, you will start with a single vertex (the white dot) surrounded by a 4-gon and two 6-gons. You then see that there are several vertices for which only a single face is missing (the black dots), and you add these. You go on until you realize that now all the vertices satisfy the constraint. You never made a choice, and so the result must be unique.

My question is the following:

Question: How can this proof be formalized?

There seem to be some subtleties, some are hinting at certain generalizations:

  • You need to use that $C$ is a polytopal sphere, otherwise the complex might not be unique. For example, there are infinitely many possibilities to realize a polytopal torus in which every vertex is incident to exactly three 6-gons. Maybe "sphere" can be replaced by simply connected, though.
  • We do not necessarily need simple, but it is the "simplest" of several conditions that ensure that we never have to make a choice when placing the next facet. Here is a case where above reasoning fails because we dropped simplicity: if we want every vertex of a 2-dimensional complex to be surrounded by a 3-gon and three 4-gons, there are two solutions: this one and this one.
  • How do I know that there is always a vertex at which the next facet is determined?
  • How can one be sure that the order in which I add new facets (which are forced on me) does not make a difference. Or does it?
  • 1
    $\begingroup$ Re: the last question - "How can one be sure that the order in which I add new facets (which are forced on me) does not make a difference. Or does it?" - since a facet can never stop being forced once it is forced, I think a routine application of the 'diamond lemma' proves confluence, i.e., that it does not matter which order you carry out the forced facet additions. $\endgroup$ – Sam Hopkins Mar 27 '20 at 14:09
  • $\begingroup$ @SamHopkins This was already a very helpful hint. I wasn't aware of this lemma. Just a note: one might want to apply above reasoning to tilings of $\Bbb R^n$ too. In that case, the process of adding further facets isn't finite. If I understood correctly, this finiteness is an important condition in the diamond lemma. Is there away around this? $\endgroup$ – M. Winter Mar 27 '20 at 14:50
  • $\begingroup$ In general there is no way around some kind of finiteness condition to apply the diamond lemma. $\endgroup$ – Sam Hopkins Mar 27 '20 at 15:21

Sorry, but there are counterexamples to your local to global quest:

Consider the uniform small rhombicuboctahedron versus the elongated square gyrobicupola (Miller's solid, one of the Johnson solids). - Both show up the same vertex figures all over, still they are globally different (i.e. non-unique) outcomes (polyhedra).

--- rk

  • $\begingroup$ This counterexample is already contained in my post in the third to last bullet point. This is why I need the rule for adding one more tile that the placement must be unique, as it is in the simple case. $\endgroup$ – M. Winter Dec 18 '20 at 12:53
  • $\begingroup$ Just want to let you know that similar finds to Miller's solid lately were found within 4D, cf. polytope.net/hedrondude/polychora.htm#pseudouniform $\endgroup$ – Dr. Richard Klitzing Jul 18 at 20:28

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