Cell Complex

A Cell Complex is a partitioning of a geometrical space into a collection of subspaces called cells, interconnected in a specific way. Cell complexes play a vital role in algebraic topology, simplifying topological spaces for analysis, and have applications in computational geometry and topological data analysis.

# Definition

Cell complex $X$ of dimension $n$ is a type of topological space constructed by a series of inclusions:

$$X_0 \subset X_1 \subset \cdots \subset X_{k} \subset X_{k+1} \subset \cdots X_n$$

Each $X_k$ is referred to as the k-skeleton of a complex. $X_k$ consists of all cells of dimension $k$ and lower. $X_{k+1}$ is obtained from $X_{k}$ by attaching $k$-dimensional cells to $(k+1)$-dimensional cells. These maps are referred to as *attaching maps*. Mathematically, the k-skeleton $X^k$ is defined iteratively:

$$X^k = X^{k-1} \cup \bigcup_{\alpha} \sigma_\alpha^k$$

where $\sigma_\alpha^k$ are k-cells. Each $k$-dimensional cell $\sigma^k$ is homeomorphic to a $k$-dimensional disk, (or equivalently, to $k$-dimensional polygon). That is:

• $\sigma^0$ is homeomorphic to a point.
• $\sigma^1$ is homeomorphic to a line segment.
• $\sigma^2$ is homeomorphic to a disk.
• $\sigma^3$ is homeomorphic to a ball.

and so on.

# Motivation

The understanding of topological spaces is simplified by the use of cell complexes, where higher-dimensional geometries can be studied using simpler, lower-dimensional structures. Cell complexes serve as a bridge between geometry and combinatorics, and they are integral in the study of homotopy, homology, and cohomology.

Cell complexes are integral in computational topology and can be used to approximate smooth manifolds, providing computational advantages. They also play a role in persistent homology, a key method in topological data analysis.

• Homology Groups: The homology groups of a cell complex can be computed using cellular homology, leading to efficient algorithms.
• Betti Numbers: These are significant invariants in algebraic topology and can be computed using cell complexes.

# Types

There are specific kinds of cell complexes that are particularly important:

• CW Complexes: specific type of cell complex studied in algebraic topology with more flexibility in attaching cells.
• Simplicial Complexes: All cells are simplices, and a complex itself attains a combinatorial structure, a subtype of Regular Cell Complexes.
• Delta Complexes: Simpler than general CW complexes but more flexible than simplicial complexes.