Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Mr. or Mrs. Anonymous Coward created cell complex but didn’t have much to say. Maybe somebody feels like helping the Coward.
(Is such activity failed spam or failed contribution?)
I need to wake up early so can not help this time. But in any case there is more than one kind of cell complexes and categories of such in topology. Even for CW kind one can consider just cellular maps and all possible maps as morphisms, leading to different categories.
I am going to add the general definition: a cell complex is a (transfinite) composite of pushouts of generating cofibrations in a cofibrantly generated model category.
But not tonight.
Actually, does anyone know what a cell complex is? That is, is there some combinatorial notion (in the sense that simplicial complexes are combinatorial) where the geometric realization of each basic cell is the convex hull of finitely many points in Euclidean space (just as the geometric realization of a simplex is an affine simplex), and the geometric realization of a cell complex has the property that the intersection of two cells is a cell?
It seems to me this is rather nontrivial, and I’d be very interested if there were such a notion. Notice that I do not mean a CW complex; what I’m looking for belongs to “discrete mathematics”.
I seem to have killed a conversation I wanted to start! Perhaps I could try again.
The rough intuition of the notion of “cell complex” I am looking for would involve sets $C_n$ (whose elements are called $n$-cells) for each $n \geq 0$, and boundary maps $d_n: C_n \to P(C_{n-1})$, satisfying some axioms (which at first pass may appear a little clunky):
There are only finitely many and more than zero $(n-1)$-cells in the boundary of an $n$-cell. Each 1-cell has exactly two boundary 0-cells.
Let $C = \bigcup_n C_n$ and consider the smallest transitive relation $\lt$ on $C$ such that $c' \lt c$ if $c' \in d_n(c)$. (We say $c'$ is in the boundary of $c$). Then the restriction of the boundary maps to the down-set determined by a cell $c$ defines an exact chain complex in the category of $\mathbb{Z}_2$-vector spaces (when the boundary maps are interpreted as matrices with values in $\mathbb{Z}_2$).
Each cell is uniquely determined by the 0-cells on its boundary. That is, the map $d: C_n \to P(C_0)$ defined by relational composition of boundary maps is injective for $n \gt 0$.
In the notation of 3., if $c$ and $c'$ are cells and $d(c) \cap d(c')$ has at least two elements, then there exists a cell $c''$ in the boundary of $c$ and $c'$ such that $d(c'') = d(c) \cap d(c')$. (This $c''$ is unique, by 3.)
These are just a few axioms off the top of my head; I am not sure they are sufficient. “Sufficient” would imply that there is a “realization function” from $C_0$ to a Euclidean space of sufficiently high dimension, so that each cell $c$ is geometrically realized as the convex hull of the (realizations of) 0-cells in its boundary.
In 2. is $x'$ really $c'$?
There was a notion of polyhedral complex that in a manifestation as polyhedral T-complex came up in the thesis of one David Jones (from Wales, currently farming in mid-Wales). Also I thought, but cannot find, that Ronnie refers to cone complexes, and idea that might help.
Tim, thanks! I corrected that orphan $x'$.
Did you mean 3. looks dodgy? Sorry, I’m not seeing the difficulty; it is only trying to reflect the idea that the vertices of a cell c, which are the extreme points of c as a convex polytope, uniquely determine c as the convex hull.
For 6: look at the source!!! It is the quirky behaviour with respect to numbered articles.
Are you intending cell complexes always to be embedded in some Euclidean space? Otherwise you have to have some abstract notion of convex hull.
It may pay looking at how combinatorists look at this sort of thing. Bjorner’s 60th birthday party gives some links. Bjorner’s own work may be relevant.
Tim: thanks. As you can see, I’m struggling just taking baby steps here. I do want a more sophisticated conceptual description of how these things are to be geometrically realized, but for now my task is just to try to sketch out the intended intuition so that I can get some good responses.
Obviously I do want to know what the combinatorists have done, and not try to reinvent the wheel. Which of those links should I look at, and which of Björner’s papers should I look up?
1 to 10 of 10