5
$\begingroup$

I come from combinatorics and my notions of algebraic topology are very limited. I have a purely combinatorial definition of a certain set of "cells" and I want to know if what I have is "enough" to say that it is a CW-complex or if I need to prove more.

  • My cells of dimension 0 are just a discrete finite set.
  • My cells of dimension 1 are edges between those 0-cells (so basically, I have a graph)
  • My cells of dimension $n$ are given by certain subsets (not all) of 0-cells (the dimension here is a certain combinatorial property of this subset)

A face $f_1$ is included in face $f_2$ if the subset of 0-cells corresponding to $f_1$ is included in the subset of $f_2$

With the following properties:

A face of dimension $n$ is always a a certain subset of faces of dimension $n-1$ (so my poset of face inclusion is graded)

If $f_1$ is a face of dimension $n_1$ and $f_2$ a face of dimension $n_2$, if I intersect $f_1$ and $f_2$ (as sets of 0-cells), I get a face $f_3$ of dimension $n_3 \leq min(n_1,n_2)$ such that $n_3 = n_1$ iif $f_1 \subseteq f_2$ (and $n_3 = n_2$ iif $f_2 \subseteq f_1$).

In particular, if $f_1$ and $f_2$ are distinct and of similar dimension, their intersection will always be of dimension strictly smaller.

I am quite convinced that this "complex" is a CW-complex and I'm actually in the process of proving that it is a polytopal complex (each cell can be realized as a polytope).

My question is more of nomenclature. These properties are quite strong already, do they give a CW-complex? If not, what is missing? (And what is this thing called?)

Edit: I am adding another hypothesis which I believe is important. The 1-skeleton (so the graph on 0-cells) is actually a lattice and a face is always an interval of said lattice (but not every interval is a face)

The hypothesis are probably not enough as I really don't see how to define the "interior" and prove it's homeomorphic to a ball. In the case I am looking at, it is actually the case as I am proving it is actually a polytopal complex. But this is actually stronger than being a CW-complex... So I'd be interested in seeing examples of such structures which "don't work" (combinatorial complex which are not CW-complexes, or where you can prove CW-complex but not polytopal...)

Improve this question
$\endgroup$
10
  • 2
    $\begingroup$ You definitely need to prove more. An $n$-dimensional face of a CW complex is not just an abstract set of $0$-cells, it is an actual $n$-dimensional cell, in particular its interior is homeomorphic to an actual open ball in $n$-dimensional Euclidean space. Also, the face relation between cells is not just an abstract inclusion of sets of $0$-cells, it is an actual topological relation between an $n$-dimensional cell and a lower dimensional cell. $\endgroup$
    – Lee Mosher
    Commented Jun 27, 2019 at 15:05
  • $\begingroup$ That's what I thought. I am having trouble understanding how this topological properties translates in terms of combinatorics. I will probably end up proving that this is a polytopal complex, which solves the problem. But I also wonder if the combinatorial object I describe has a name. I call it a "combinatorial complex" because any other way to refer to it... $\endgroup$
    – Viviane
    Commented Jun 27, 2019 at 15:34
  • $\begingroup$ is it a simplicial complex, perhaps? (these are a particular type of CW-complexes, much easier to work with) $\endgroup$
    – Dima Pasechnik
    Commented Jun 27, 2019 at 20:02
  • $\begingroup$ No it's not, this I am sure. It has the same combinatorial properties though expect it's not made of triangle... $\endgroup$
    – Viviane
    Commented Jun 27, 2019 at 21:45
  • $\begingroup$ topologically it is often OK to triangulate your faces, cells, etc, just as triangulating faces of a planar graph does not break planarity (the latter would allow one to pass from general polytopes in $\mathbb{R}^3$ to simplicial ones) $\endgroup$
    – Dima Pasechnik
    Commented Jun 27, 2019 at 23:54

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .