# What is a subdivision of an abstract simplicial complex?

I am looking for the definition of the subdivision of a simplicial complex. When the complex is defined in a geometric way, then the definition is pretty simple : the complex σ(C) is a subdivision of C if each simplex of σ(C) is contained in a simplex in C, and if each simplex of C is the union of finitely many simplexes in σ(C) (definition found in Herlihy-Shavit 1999).

When we deal with the abstract definition of simplicial complexes (i.e. a subset-closed family of subsets), I fail to find a similar definition. I found definitions of specific subdivisions (e.g. standard chromatic subdivision of the n-simplex) but not a general formulation like the one I gave. A difficulty is the fact that an abstract simplex cannot be the union of its subdivision, since the subdivision contains new edges that do not appear in the first simplex (they are not in the same "space" because there is no convenient notion of "space" in the abstract definition).

Thanks a lot for your help !

• The barycentric subdivions are easily described. They have one vertex $v_S$ for ever simplex $S$ of the original complex. The simplices of the barycentric subdivision correspond to chains of simpleces of the original complex ordered by inclusion. Commented Apr 8, 2021 at 12:59
• Another way to say what Liviu just said: the barycentric subdivision of $\sigma$ is the order complex of $\sigma$ viewed as a poset. Commented Apr 8, 2021 at 13:00
• Thank you for your answers! I am looking for a more general definition of a subdivision, which could concern both barycentric and chromatic subdivision. Like the minimal constraint, or relation, for a complex to be called the subdivision of another one, constraint satisfied by various notions of subdivision. Does something like that can exist ? Commented Apr 8, 2021 at 13:18
• Well you can always talk about the geometric realization of an abstract simplicial complex, so in a sense any property you can define 'concretely' also applies to abstract simplicial complexes. At the beginning of math.mit.edu/~rstan/pubs/pubfiles/89.pdf Stanley talks about several notions of subdivision for simplicial complex (but none are quite as abstract/formal as what you can do with Barycentric subdivision). Commented Apr 8, 2021 at 13:26
• Ok, thank you again for your answers, and for the reference ! Commented Apr 8, 2021 at 14:06

A generalization of a barycentric subdivision is the notion of a stellar subdivision, where one performs a barycentric subdivison of a simplex $$s$$ and subdivides all simplices accordingly that contain $$s$$ as a face. The inverse of a stellar subdivision is called a stellar weld. Now let $$K$$ and $$L$$ be two abstract simplicial complexes. If the topological realizations $$|K|$$ and $$|L$$ admit subdivisions that are isomorphic as simplicial complexes, then the Alexander-Newman theorem says that $$K$$ and $$L$$ are stellar equivalent, i.e. they are related by a finite sequence of stellar subdivisions and stellar welds. See Chapter 64.5 of my lecture notes for details and references https://www.uni-regensburg.de/Fakultaeten/nat_Fak_I/friedl/papers/1at-uptodate.pdf