📂Topological Data Analysis

Definition of a Delta-Complex

Definition ¹

Definition of a simplex:

1. The $n$-simplex $\Delta^{n}$ is called the convex hull of affinely independent $v_{0}, v_{1} , \cdots , v_{n} \in \mathbb{R}^{n+1}$, and the vectors $v_{k}$ are called Vertices. Formally, it is as follows. $$ \Delta^{n} := \left\{ \sum_{k} t_{k} v_{k} : v_{k} \in \mathbb{R}^{n+1} , t_{k} \ge 0 , \sum_{k} t_{k} = 1 \right\} $$
2. The $n-1$-simplexes $\Delta^{n-1}$ created by removing a vertex from $\Delta^{n}$ are called the Faces of $\Delta^{n}$. The Boundary of $\Delta^{n}$ is the union of all its faces and is denoted by $\partial \Delta^{n}$.
3. The interior $\left( \Delta^{n} \right)^{\circ} := \Delta^{n} \setminus \partial \Delta^{n}$ of a simplex is called an Open Simplex.

A $\Delta$-Complex Structure on a topological space $X$ is a set of mappings $\sigma_{\alpha} : \Delta^{n} \to X$ depending on the index $\alpha$ with $n := n(\alpha)$ that satisfy the following three conditions:

• (i): The restriction function $\sigma_{\alpha} | \left( \Delta^{n} \right)^{\circ}$ in the open simplex $\left( \Delta^{n} \right)^{\circ}$ of $\sigma_{\alpha}$ is injective, and each point of $X$ is contained in exactly one image of $\sigma_{\alpha} | \left( \Delta^{n} \right)^{\circ}$.
• (ii): The restriction function on a face of $\Delta^{n}$ in $\sigma_{\alpha}$ is one of $\sigma_{\beta} : \Delta^{n-1} \to X$.
• (iii) Continuity: All $\sigma_{\gamma}$ must be continuous functions. In other words, $A \subset X$ being an open set in $X$ means that $\sigma_{\gamma}^{-1} (A)$ are open sets in the domain $\Delta^{n}$ of every $\sigma_{\alpha}$.

Explanation

Precautions

It’s important to understand that what is defined here is not exactly a complex but rather a Complex Structure, and that it’s merely a “set of mappings”. Having this set alone, without algebra or topology, won’t allow you to do much. You cannot even consider the intersection $\sigma_{1} \cap \sigma_{2}$ mentioned in the definition of a complex. However, condition (ii) plays its role instead, so conceptually calling it a complex is acceptable, but when specifics are required, it’s important to be able to discuss them.

Algebraic Topology

We will explore and create simplicial homology groups with these mappings as if they were a kind of character in free groups. By then, you might not even remember simplexes $\Delta^{n}$ or spaces $X$ at all, but that’s exactly why you need to study them properly once.

Example: Torus

Reading the text alone makes it quite difficult to understand, which is normal. Let’s look at the simplest example, a torus $X = T$.

Construction

In fact, to construct a torus, it’s sufficient to have just a square $S^{1} \times S^{1}$ rather than needing a simplex’s complex, or a simplicial complex. But, to engage in meaningful algebraic exploration with a $\Delta$-Complex Structure, you need the following $6$ mappings.

This is a top-down projection diagram of a torus. $\sigma_{a}$, $\sigma_{b}$, $\sigma_{v}$ are mappings that play a kind of ‘skeletal’ role in the naive method of making a torus. $\sigma_{b}$ rolls the square into a cylinder, and $\sigma_{a}$ joins the ends of that cylinder to make a donut. At this time, the vertices of the square have to converge exactly to one point, and $\sigma_{v}$ performs this role.

This is a side-view projection diagram of a torus. $\sigma_{U}$, $\sigma_{L}$ are mappings for the ‘surface’ that fills in between the frames. Once again, it’s emphasized that $\sigma_{c}$ is not necessarily needed just for the torus, but becomes relevant when viewing a square as the union of two triangles, acting as the mapping responsible for its boundary.

Comparison with Definition

According to the definition, the $\Delta$-Complex Structure of the torus $T^{2}$ is none other than the set of mappings $$ \left\{ \sigma_{U}, \sigma_{L}, \sigma_{a}, \sigma_{b}, \sigma_{c}, \sigma_{v} \right\} $$ Considering up to the $2$-simplicial complex, we only need to think about $n$ up to $n = 0,1,2$.

• The condition of (iii) continuity can be understood intuitively.
• $n = n(\alpha)$ about $\alpha = L, U$ is $2$. These maps send all points of $U^{\circ}, L^{\circ}$, which can be called the face of square $S^{1} \times S^{1}$, to $X$ without missing any.
• The face of $U,L$ is none other than the line segments $a$, $b$, $c$, and their corresponding $n = (\beta)$ is $1$. These map the endpoints of the line segments enclosing $U^{\circ}, L^{\circ}$ to $X$ excluding the endpoints. Condition (ii) is satisfied in this manner.
• Lastly, when $n = 0$, the point $v$ that is a $0$-simplex is the face of $a$, $b$, $c$, and $\sigma_{v}$ sends it to the last remaining point of $X$. Following the discussion, each point of $X$ belongs to precisely one of the six mappings’ images, thus satisfying condition (i).