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Topology

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1. Algebraic objects

2. Filtration

3. Appendix

1. Algebraic objects

⑴ Overview: A structure in which homology is used to compare topology, and once homology is used, the comparison problem becomes linear algebra

⑵ set / group / ring / field: A structure obtained by equipping a set $X$ with operations

① Example 1. point set $A$

② Example 2. $\mathbb{Z}/2\mathbb{Z}=\lbrace 0,1 \rbrace$. Introduced to simplify calculations

⑶ Complex = multiple algebraic objects + maps between them + conditions satisfied by the maps

① Example 1. $A=\lbrace a,b,c,d\rbrace$, $\triangle abc$, $\triangle bcd$: general Delaunay complex $K= \operatorname{Del}(A)=\lbrace\varnothing,a,b,c,d,ab,ac,bc,bd,cd,abc,bcd\rbrace$

Figure 1. Voronoi tessellation required to construct a Delaunay complex

○ If the Voronoi cells of two points $a$ and $b$ touch, then $ab\in\operatorname{Del}(A)$

○ If the Voronoi cells of three points meet at a common point, then $abc\in\operatorname{Del}(A)$

○ That is, if there exists a circle passing through $a$, $b$, and $c$ and there are no other points inside that circle, it becomes a Delaunay triangle

② Example 2. Chromatic Delaunay complex $\operatorname{Del}(\chi)$: When $\chi:A\to\sigma$ (color) is given, $\operatorname{Del}(\chi)$ is itself an extended complex that preserves the color-wise Delaunay structures while simultaneously incorporating mixed interactions within a single simplicial complex.

○ First, define the colors firstly and perform a Voronoi tessellation for each color

○ Second, Take all Voronoi cells of every color together and construct the nerve of the entire family.

○ For example, for Blue $B_i$ and Orange $O_k$, if $\operatorname{dom}(B_i, A_B) \cap \operatorname{dom}(O_k, A_O) \neq \emptyset,$ then $\lbrace B_i, O_k\rbrace \in \operatorname{Del}(\chi).$

○ Chromatic lifting is introduced to preserve the color-wise Delaunay structures while also accounting for mixed interactions.

○ Geometrically, the Delaunay triangulation remains unchanged regardless of the lifting height, as long as the height is nonzero.

○ For example, when an orange ball is positioned between two blue balls, it is difficult to establish a direct connection between the blue balls without chromatic lifting.

Figure 2. Construction of chromatic alpha complex using chromatic lifting

③ Example 3. $\operatorname{Alf}_r(\chi) = \lbrace\nu \in \operatorname{Del}(\chi) \mid \operatorname{Rad}(\nu) \le r\rbrace$

○ $\operatorname{Rad}^2(\nu) = \min_{x \in P} \max_{v \in \nu} \lVert x-v \rVert^2$

○ See the (chromatic) alpha complex below for a detailed explanation.

⑷ Vector space: A linear algebra space formed by collecting the simplices in a simplicial complex $K$ by dimension. Introduced for easier computation.

① A simplex is the simplest geometric object that generalizes a point, line segment, triangle, and tetrahedron to arbitrary dimensions.

② $C_p(K)$ = A vector space whose basis consists of the $p$-dimensional simplices of $K$

③ $C_0(K)$: Vertices represented as a linear algebra space (e.g., $\operatorname{span}\lbrace a,b,c \rbrace$)

④ $C_1(K)$: Edges represented as a linear algebra space (e.g., $\operatorname{span}\lbrace ab,bc,ac \rbrace$)

⑤ $C_2(K)$: Triangles, etc. represented as a linear algebra space (e.g., $\operatorname{span}\lbrace abc \rbrace$)

⑥ $C_3(K)$: Tetrahedra, etc. represented as a linear algebra space

⑦ Example 1. When $C_1(K)=\operatorname{span}_{\mathbb{Z}/2\mathbb{Z}}\lbrace ab,bc,ca \rbrace$, $ab+bc$ can be represented as $(1,1,0)$

⑸ Chain complex: For each boundary map $\partial_2,\partial_1$

① $\partial_p : C_p(K) \rightarrow C_{p-1}(K)$

② $Z_p=\ker\partial_p$: $p$-cycle. Related to that cycle in a geometric sense.

③ $B_p=\operatorname{im}\partial_{p+1}$: $p$-boundary. Related to that boundary in a geometric sense.

④ Always, $\operatorname{im}\partial_{p+1}\subseteq\ker\partial_p$ holds.

⑤ Example 1. An edge $e = ab$

Table 1. Examples of cycle and boundary

⑹ Homology

① Overview

○ The general definition of the homology of a chain complex is as follows

○ Quotient: $\forall x,y \in \ker \partial_p$, if $x-y \in \operatorname{im}\partial_{p+1}$, then $x$ and $y$ are equivalent in $\ker \partial_p / \operatorname{im}\partial_{p+1}$, denoted as $x \sim y$.

○ Quotient has many common points with set difference.

○ Homology is not a statistic that assigns a single overall direction or pattern to the entire dataset. Rather, it records multiple independent topological features in the space simultaneously, much like multiple basis elements or generators in a vector space. Therefore, it is not contradictory for one type of homological feature to appear in one region while a different type appears in another.

② Examples

○ Example 1. If $Z=\operatorname{span}\lbrace a,b \rbrace$ and $B=\operatorname{span}\lbrace ab\rbrace $, then $a,b\in Z$ and $a-b\in B$. Therefore, $a\sim b$ in $Z/B$.

○ Example 2. An edge $e = ab$

○ Example 3. An edge $e = ac$, where $b$ is the midpoint of $e$.

○ Example 4. When a triangle $abc$ is filled

○ Example 5. If the interior of the triangle $abc$ is absent and there are only three edges

○ Example 6. 4 dots, none of which are connected

○ Example 7. A graph with four vertices and four edges forming a cycle, with no triangles

○ Let’s review the definition of quotient.

○ Here, $[a]$ is the equivalent class that contains $a$ as a representative. In fact, $b, c, d, a+b+c, a+b+d, a+c+d, b+c+d$ can all be a representative of this equivalent class.

○ Example 8. 4 points, fully connected using Delaunay triangulation

③ Betti number: The number of generated holes forms a topological group, and the Betti number is a topological invariant

Table 2. Concept of Betti numbers

Table 3. Examples of Betti numbers

④ Gauss–Bonnet Theorem: Integrating curvature over the entire surface yields a topological invariant (the Euler characteristic), establishing a connection between geometry and topology.

2. Filtration

⑴ In probability theory, filtration is a family of increasing sigma algebras such as $\mathcal{F}_0\subseteq\mathcal{F}_1\subseteq\mathcal{F}_2\subseteq\cdots$.

⑵ In topology, filtration means $K_0\subseteq K_1\subseteq K_2\subseteq\cdots$

⑶ Persistent homology: For two time points $s\le t$, the following is called the $(s,t)$-persistent homology group.

① Limitation of persistent homology: Relies on qualitative analysis and cannot assign directionality

○ Solution 1. graph Laplacian (Kirchhoff 1847)

○ Solution 2. combinatorial Laplacian (B. Eckmann 1944)

○ Solution 3. persistent (combinatorial) Laplacian (Wang, Nguyen, Wei, 2019; Meng et al. 2021; Memoli et al. 2022; Liu and Wu 2023)

○ Solution 4. evolutionary homology (Zixuan Cang, Munch, & Wei, J. Appl. Comput. Topology, 2020)

○ Solution 5. persistent path Laplacian (Rui Wang, & Wei, Foundation of Data Science, 2023)

○ Solution 6. persistent hyperdigraph Laplacian (Dong Chen, Liu, Wu, & Wei, Foundation of Data Science, 2024)

○ Solution 7. persistent Mayer topology (Li Shen, Jian Liu, & Wei, Foundation of Data Science, 2024)

○ Solution 8. persistent Mayer Dirac (Suwayyid & Wei, J. Physics: Complexity, 2025)

○ Solution 9. persistent interaction topology (Jian Liu, Chen, & Wei, FoDS 2025)

○ Solution 10. persistent directed flag Laplacian (PDFL) (Ben Jones & Wei, Foundation of Data Science, 2025)

○ Directed edge (digraph, directed graph): A directed edge is defined as an ordered pair of vertices $u$ and $v$ and is denoted by $(u,v)$ or $u \to v$. A graph with directed edges is called a directed graph or digraph

○ face: Simply a filled simplex of some dimension

○ simplified complex: It is called a flag when every clique in the graph is a face

○ Directed flag: When every directed clique in a simplified complex is a face

○ Persistent directed flag complex: A directed simplicial complex in which every directed clique in a graph forms a simplex, these simplices persist through a filtration, and persistence captures the birth and death of homological features during that process

⑷ Persistent module: For $K_i=f^{-1}((-\infty,r_i])$

① In the paper, homology is applied to the filtration $K_0\subset K_1\subset\cdots$ to construct a persistent module $H_p(K_0)\to H_p(K_1)\to\cdots$

② During this process, the birth and death of features are recorded

⑸ Application 1. alpha complex filtration

① Definition: Gradually increase the radius $r$ from each point to see whether rings or connected structures appear

○ We are considering geometry $\to$ filtration $\to$ homology, and “geometry $\to$ filtration” is related to minimax.

② $\operatorname{Alf}_{r}(\chi)=\lbrace \nu\in\operatorname{Del}(\chi)\mid\operatorname{Rad}(\nu)\le r \rbrace$

○ $\operatorname{Rad}^{2}(\nu)=\min_{z\in P}\max_{v\in\nu}\lVert z-v\rVert^{2}$

○ Interpretation: Rad is the minimal size for which the ball contains a simplex.

○ $z\in P\iff$ there exists an empty stack passing through $\nu$ with $z$ as the common center

○ If $S_j$ contains no points of color $j$, the sphere is said to be empty with respect to that color

○ If the sphere is empty with respect to every color, it is called an empty stack

○ A red point lying inside a blue sphere does not violate the empty stack condition

○ This allows us to obtain the optimal center shared by the spheres of all colors and the radius of the sphere corresponding to each color from that center

○ This center does not need to be a biological center

③ Alpha complex filtration

○ If $r_1<r_2<r_3$, then $\operatorname{Alf}{r_1}(\chi)\subseteq\operatorname{Alf}{r_2}(\chi)\subseteq\operatorname{Alf}_{r_3}(\chi)\subseteq\operatorname{Del}(\chi)$, so this is a filtration.

Figure 3. Alpha complex filtration

○ The dimensional direction is the chain complex, and the scale direction is the filtration

④ Generalized discrete Morse function

○ Definition: Under certain conditions, alpha complex filtration has a $1$-norm relationship with the persistence diagram. Proved by KKT.

○ An extremizer reveals symmetry in the data and hidden structural support / active constraints in the function

Let real numbers $x_1,x_2,\cdots,x_6$ satisfy

Let $M$ be the maximum value of

Find the value of $M$.

The points satisfying the given constraints can be viewed as the intersection of a hypersphere and a hyperplane, which forms a hypercircle. Since the hypercircle is continuous and differentiable at every point, the maximum value $M$ is also a local maximum. Such a local maximum occurs at a point exhibiting a high degree of symmetry, where all remaining degrees of freedom disappear. Consequently, the six given variables do not all take distinct values, but instead split into two groups taking two distinct values. This can be analyzed using the KKT (Karush–Kuhn–Tucker) conditions.

○ Since the alpha complex is defined by minimax, it is an extremizer, and as in the example above, rather than simplices entering the filtration arbitrarily, simplices with the same radius are grouped into intervals $[\nu_{\min},\nu_{\max}]$, resulting in a high degree of symmetry.

○ Looking at Figure 2, for $r$ near $r_1$, $\operatorname{Alf}$_𝑟$(A)=\operatorname{Alf}_{r_1}(A)$

○ Such an interval is called a Morse interval.

○ This allows quantities such as the birth and persistence of a specific feature to be represented by the $1$-norm $r$

○ The Morse interval remains valid when extended to the chromatic alpha complex because its validity in alpha complex is tied to the independence of each vertex.

○ Case 1. generic: $\mathbf{d\ge m-q}$ (see the definition of chromatic alpha complex below)

○ If red consists of three points $R_1,R_2,R_3$, then $\lVert z-R_2\rVert=\lVert z-R_1\rVert$ and $\lVert z-R_3\rVert=\lVert z-R_1\rVert$ must hold, so there are two constraints

○ Generalizing this, when there are $m$ points and $q$ colors, there are a total of $m-q$ constraints

○ The common center $z\in\mathbb{R}^d$ has $d$ degrees of freedom, so $d\ge m-q$

○ generic: A state in which the points do not accidentally have overly special symmetries or algebraic relationships

○ Example 1. $q=2$, $d=2$: The maximum possible number of generic boundary points is $d+q=4$ (e.g. $2R+2B$)

○Example 2. $q=3$, $d=2$: The maximum possible number of generic boundary points is $d+q=5$ (e.g. $2R+2B+1G$)

○ If $d>m-q$, the remaining degrees of freedom are eliminated during the optimization process

○ Case 2. In a non-generic configuration with excessively strong symmetry, there are multiple dual solutions, so $\nu_{\min}$ cannot be uniquely determined, and the generalized discrete Morse interval structure breaks down

○ non-generic geometry $\to$ non-unique dual coefficient $\to$ non-unique $\nu_{\min}$ $\to$ the radius level set is not an interval

○ When applied to real data, Case 2 will probably occur very frequently, so the dual / representation unidentifiability problem is likely to continue arising

○ In other words, which local configuration supported the topology becomes unidentifiable $\to$ the discrete- Morse interval breaks down

Figure 4. Example in which uniqueness of the solution breaks down in a non-generic setting

○ Example 1. In the above homology (Example 6 ~ Example 8), we can find the following alpha complex filtration:

Table 4. Example of alpha complex filtration

○ Example 2. In the filtration of connected components, the components progressively merge into larger aggregates, as illustrated below.

Figure 5. Filtration of connected components

○ Example 3. In the filtration of loops, an acute triangle can give rise to a loop with positive persistence, whereas an obtuse triangle cannot.

Figure 6. Filtration of loops

○ Example 4. Among the following three, two represent the same homology class. However, the persistence diagram considers the feature that appears first (□), followed by the next one (△).

Figure 7. Selectivity in the persistence diagram of an alpha complex in the presence of linear dependence.

⑹ Application 2. Chromatic alpha complex filtration

① Definition: Until now, we have examined the homology of a single color; now we need to examine how blue and red are spatially mixed

Figure 8. Goal of the chromatic alpha complex

In $a+b$, $a$ is the number of colors required to create the cycle, and $b$ is the number of additional colors required to fill and kill that cycle

Case A
BBBBBBB
B     B
B     B
BBBBBBB

Case B
BBBBBBB
BRRRRRB
BRRRRRB
BBBBBBB

○ Case A : $H_1(\mathrm{blue})\ne 0$, $H_1(\mathrm{blue}+\mathrm{red})\ne 0$

○ Case B: $H_1(\mathrm{blue})\ne 0$, $H_1(\mathrm{blue}+\mathrm{red})=0$

② Induced map

○ If there is an inclusion $L\hookrightarrow K$ between spaces, a linear map $\kappa(L)\to H_p(K)$ is automatically induced

○ Here, $L$ is the blue complex and $K$ is the blue+red complex

○ image: The case where a blue ring remains a hole even after red is added. That is, $[z]\in H_1(L)$, $\kappa([z])\ne 0\iff\kappa([z])\in\operatorname{im}\kappa$

○ kernel: A structure that was a hole in blue but is filled because of another color. That is, $[z]\ne 0\in H_1(L)$, $\kappa([z])=0\in H_1(K)\iff[z]\in\ker\kappa$

○ coker: Something that lies in $H_1(K)$ but does not come from $H_1(L)$. That is, $\operatorname{coker}\kappa=H_p(K)/\operatorname{im}\kappa$

○ 6-pack: Understood as information theory from the perspective of group theory.

Table 5. 6-pack

○ Application: The 6-pack reveals the topological contribution of each color that is not distinguishable from Betti numbers alone.

Figure 9. The concept of coker

③ Rank-nullity theorem

○ $\dim H_p(L)=\dim\ker\kappa_p+\dim\operatorname{im}\kappa_p$

○ $\dim H_p(K)=\dim\operatorname{im}\kappa_p+\dim\operatorname{coker}\kappa_p$

○ Exact sequence

④ Relative homology

○ $C_p(K,L)=C_p(K)/C_p(L)$: If $L=\text{blue}$ and $K=\text{blue}+\text{red}$, then

○ blue–blue edge lies entirely in $L$, so it disappears in the quotient.

○ blue–blue–blue triangle disappears.

○ blue–red edge is not contained in $L$, so it remains.

○ red–red edge remains.

○ blue–red–red triangle remains.

○ blue–blue–red triangle remains.

○ $\bar{\partial}$_𝑝: $C_{p}(K,L) \longrightarrow C_{p-1}(K,L)$

○ $H_{p}(K,L) = \ker \bar{\partial}$_𝑝 / ${\operatorname{im} \bar{\partial}_{p+1}}$

○ Example 1. Viewed in the full complex $K$, this is a 2B2R 4-cycle, $B_1-B_2-R_4-R_5-B_1.$ In relative homology, the blue-only edge $B_1-B_2$ is quotiented out and becomes 0. Therefore, only the mixed/orange path $B_1-R_5-R_4-B_2$ remains as a representative. However, since both endpoints $B_1$ and $B_2$ lie in $L$, this path is closed relative to $L$ and therefore represents a relative cycle.

○ $\dim H_{p}(K,L) =$ $\dim \operatorname{coker}$_𝑝 $+ \dim \ker_{p-1}$

○ Exact sequence: $\cdots \to H$_𝑝$(L)$ ⟶κ_𝑝 $H$_𝑝$(K)$ ⟶𝑗_𝑝 $H$_𝑝$(K,L)$ ⟶δ_𝑝 $H$_𝑝−1$(L)$ ⟶κ_𝑝−1 $H$_𝑝−1$(K) \to \cdots$

○ $\delta$_𝑝$: H$_𝑝$(K,L) \to H$_𝑝−1$(L)$

○ $\ker \delta$_𝑝$ \cong \operatorname{coker} \kappa$_𝑝

○ $\operatorname{im} \delta$_𝑝$ = \ker \kappa$_𝑝−1

○ $\dim H$_𝑝$(K,L) = \dim \ker \delta$_𝑝$ + \dim \operatorname{im} \delta$_𝑝$ = \dim \operatorname{coker} \kappa$_𝑝$ + \dim \ker \kappa$_𝑝−1

○ Example 1. When a 1-dimensional hole in the blue region is filled by the red region, its homology class is captured by the kernel of the inclusion-induced map $\kappa_1 : H_1(L) \to H_1(K)$. In relative homology, however, the red filling disk (together with its blue boundary) represents a 2-dimensional relative cycle. Thus, a class in $\ker \kappa_1$ corresponds to a class in $H_2(K,L)$, effectively shifting the homological dimension up by one. Therefore, $\dim H_2(K,L)$ = $\dim \ker \kappa_1$ + $\dim \operatorname{coker}\kappa_2$ = 1+0 = 1.

Figure 10. Ordinary $H_1$ and relative $H_1$

3. Appendix

⑴ Can a sphere be turned inside out?

⑵ Stone–Weierstrass theorem

① The intersection of topology and analysis, where analysis is developed using continuous functions on topological spaces

Input: 2025.09.06 12:53

Modified: 2026.09.14 19:36