Define a piecewise-linear sphere (PL sphere) as an abstract simplicial complex, whose gemoetric realization is homeomorphic to a sphere. Examples:
- The complex over {1,2,3} which contains the sets {1,2},{2,3},{3,1} and their subsets is a PL sphere, since its geometric realization is the perimeter of a triangle, which is homeomorphic to a circle (a 1-dimensional sphere).
- The complex that contains, in addition, the set {1,2,3} is not a PL sphere, since its geometric realization is a full triangle, which is homeomorphic to a disk (a 2-dimensional ball) and not to a sphere,
Is there a combinatorial definition of a PL sphere, that considers only the elements in the abstract simplicial complex, and does not require to go through the geometric realization?
One idea that I had, based on this Wikipedia page, is to use a recursive definition:
- A PL 0-sphere is a complex containing two disjoint points, e.g.{{x},{y}} for some points x,y.
- A PL $(n+1)$-sphere is a complex in which the link of every vertex is a PL $n$-sphere.
For example, in the complex {{1,2},{2,3},{3,1}}, the link of 1 is {{2},{3}}, which is a 0-sphere; similarly, the link of 2 is {{1},{3}} and the link of 3 is {{1},{2}}, which are 0-spheres too.
However, by this definition, the complex over containing the sets {1,2},{2,3},{3,1},{4,5},{5,6},{6,4} would also be considered a PL 1-sphere, although it is in fact homeomorphic to two disjoint circles.
Is it possible to amend my definition and provide a correct combinatorial definition of a PL sphere?
Initially, I was hoping for an algorithmic definition. Unfortunately, such a definition does not exist in general, since the problem of recognizing a sphere of dimension 5 and higher is undecidable (see the simplicial complex recognition problem)
But, I found a combinatorial definition that is not algorithmic. In this paper (Karim A. Adiprasito and Ivan Izmestiev, 2015) it is proved that every PL sphere becomes polytopal after finitely many derived subdivisions. The converse is obviously true too. So a definition could be: "$C$ is a PL-sphere if and only if there exists a finite integer $k$ such that, after some $k$ derived subdivisions, $C$ becomes polytopal".
Note that polytopality can be decided by a finite algorithm. It does not contradict the undecidability of detecting a PL sphere, since there is no algorithm that can compute how many derived subdivisions are needed to convert a given PL sphere to a polytope.