# Cell complex

The class cell complexes is a combinatorial model of certain topological spaces. It can be seen as a generalization of the concept of maps, although it as a model of polytopes it is very weak. It does not require dyadicity, connectivity or even for the result to have finite rank.

## Definition

Informally a cell complex is a space which built up inductively, by attaching closed n -discs, ${\displaystyle D_{n}}$, called "cells", along their boundary (n-1) -spheres.

More formally the cell complex is the union of a sequence of spaces ${\displaystyle X_{k}}$ such that, ${\displaystyle X_{-1}=\emptyset }$ and ${\displaystyle X_{n}}$ is built by attaching closed n -discs to ${\displaystyle X_{n-1}}$ such that their boundaries map onto ${\displaystyle X_{n-1}}$.

## Examples

### Monogon

The monogon, although non-dyadic, can be interpreted as a cell complex. Starting with the empty set, ${\displaystyle X_{0}}$ consists of an single point (closed 0-discs are homeomorphic to single points) and ${\displaystyle X_{1}}$ consists of a single edge glued to ${\displaystyle X_{0}}$ at both ends. Optionally a ${\displaystyle X_{2}}$ can be constructed by adding a closed disk with ${\displaystyle X_{1}}$ as its boundary. This additional disc represents the top element of the monogon. Usually this element is omitted since we are more concerned with analyzing the surface of a polytope as a cell complex.

### The cube

The cube can be built as a cell complex. Starting from the empty set, ${\displaystyle X_{0}}$ consists of 8 disjoint points, these will be vertices of the cube. ${\displaystyle X_{1}}$ consists of the vertices in ${\displaystyle X_{0}}$ connected by 12 line segments to form the skeleton of the cube. Then ${\displaystyle X_{2}}$ is formed by attaching 6 closed discs to the skeleton of the cube. As with the monogon it is possible to add another cell representing the top element, but it is usually omitted.

### Doubly-wound monogon

In the monogon example it was possible to add a 2D disc to represent the top element of the monogon. However there is more than one way to do this. In the doubly wound case, the boundary of the top element can be a double cover of ${\displaystyle X_{1}}$. This has a distinct topology from the usual singly-wound case, but has the same incidence structure.

More generally any polygon can be wound any natural number of times.

## Comparison with abstract polytopes

Three is a natural way to describe both abstract polytopes and cell complexes with incidence geometries, thus inviting comparison. Each cell in the cell complex can be considered an element, with its type being its rank. Elements are incident with each other if they intersect, that is a cell is incident with all the lower rank cells that form its boundary, and all the higher rank cells that it lies on the boundary of. This gives an incidence structure, however it does not fully characterize the cell complex. For example the ordinary monogon and the doubly-wound monogon have the same incidence structure, but they are different. The incidence alone does not capture the fact that one is doubly wound.

In this sense cell complexes describe a very broad class of polytope-like objects. Unlike abstract polytopes, cell complexes can be non-dyadic, they can be disconnected, they can have ranks with no elements, and they can be infinite in rank. However there are still some abstract polytopes which are not cell complexes, that is there are incidence geometries which do not describe the incidence structure, but nonetheless are abstract polytopes. This is because the cells of a cell complex must be closed n -discs, where the elements of abstract polytopes are more flexible. For example the Petrial tesseract has cells which are homeomorphic to a torus, and thus not homeomorphic to a sphere, thus its incidence structure is not the incidence structure of any cell complex.