Simplicial Complexes, Part 2

Concrete Examples of Abstract Simplicial Complexes

Pre-requisites: previous blog post on abstract simplicial complexes.

In the next blog post, I will go over the main properties of abstract simplicial complexes.Howeever, before diving into the somewhat abstract properties of such very abstract constructs, I would like to go over a few examples of abstract simplicial complexes.Not all of these examplesare useful in algebraic topology, but they are all useful for getting to knowabstract simplicial complexes. Recall from last time that an abstract simplicialcomplex is a pair $(X, \Delta)$ where $X$ is a set and $\Delta$ is a collection offinite, non-empty subsets of $X$ called the "simplices" of the complex. Furthermore, everynon-empty subset of a simplex must also be a simplex.

Example 1: Let $X$ be any set. The smallest abstract simplicial complex structure on $X$ has no simplices.

In other words, $\Delta = \varnothing$.

Example 2: Let $X$ be any set. A slightly more interesting take on the previous example would be an abstract simplicial complex consisting of all of the singleton sets of $X$. For example, if $X = \{a, b, c\}$, then the simplices would be $\{a\}$, $\{b\}$, and $\{c\}$. While this abstract simplicial complex has simplices (unless $X$ is empty), it does not have any internal structure. That is, the simplices don't "interact" with each other because they are all disjoint.

Example 3: Let $X$ be any set. On the opposite extreme, we give $X$ all possible simplices. More specifically, we let every finite, non-empty subset of $X$ be a simplex. In symbols, $\Delta = \mathcal{P}(X) \setminus \varnothing$. This abstract simplicial complex is more interesting because simplices do intersect, but its description is still very simple. Because of this, it will be an important example for us going forward.

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The previous examples were set-theoretic, and although useful examples, did't use any extra structure on $X$. The next few examples are abstract simplicial complexes built out of metric spaces*, and can be used to investigate the properties of metric spaces. In particular, they have applications in topological data analysis. I won't go into the details of topological data analysis because I'm not familiar with the field, but I recommend checking it out for anyone who is interested in either algebraic topology or data science.

* For those unfamiliar with metric spaces, a metric space is a set $X$ equipt with a distance function $d: X \times X \rightarrow \mathbb{R}$ satisfying a few rules. The point of metric spaces is to generalize the idea of "distance" to as large a class as possible. If you are not familiar or not comfortable with metric spaces, you can assume that all metric spaces are actually subsets of Euclidean space (i.e. $\mathbb{R}^n$ for some $n$), with the normal definition of distance given by the Pythagorean Theorem. In these notes, I will use the notation $d(x, y)$ to denote the distance from $x$ to $y$.

Example 4: Let $X$ be a metric space, and choose $r > 0$. Define $\Delta_{VR,r}(X)$ to be the abstract simplicial complex on $X$ whose simplices are finite, nonempty subsets with diamater is at most $r$. That is, a simplex in $\Delta_{VR,r}(X)$ is a set $\{x_1, x_2, \dots, x_m\}$ for some $m \geq 1$ such that $d(x_i, x_j) \leq r$ for all $i, j$. This is called the Vietoris-Rips complex. The constant $r$ is the scale of the complex. The features that show up in the Vietoris-Rips complex at different scales correspond to geometric features of the metric space that are visible at certain scales. In a later blog post, I will go into how abstract simplicial complexes can be used to see the holes in a space; however, the Vietoris-Rips complex will not contain information about that hole if the hole much smaller than the scale. In a sense, the Vietoris-Rips complex only sees the metric space at scale $r$.

Figure 1(A) shows 8 points on the plane. Call these points $X$. $X$ is a metric space under the normal Euclidean distance between points. Figures 1(B) and 1(C) show the Vietoris-Rips complex built from this metric space at different scales. In Figures 1(B) and 1(C), the radius of the gray circles is the scale of the complex. These figures are drawn using the style outlined in the last blog post, although with slightly different colors.

Example 5: A relative of the Vietoris-Rips complex is the Čech complex. In this abstract simplicial complex, we again let $X$ be a metric space and choose $r > 0$. Let $\Delta_{C,r}(X)$ be the finite subsets $\{x_1, x_2, \dots x_m\} \subset X$ such that for some $y \in X$, $d(x_i, y) \lt r$ for all $i$. Sometimes this is stated as: $\{x_1, x_2, \dots x_m\}$ is a simplex if the intersection of the closed balls of radius $r$ centered at its vertices is non-empty. While more complicated than the Vietoris-Rips complex, the Čech complex has some theoretical properties that make it nicer for data analysis.

Figure 2(A) shows the same collection of points as Figure 1. These 8 points are the space $X$. Figures 2(B) and 2(C) show the Čech complex built from this metric space at different scales. The cyan triangle in Figure 2(C) represents a simplex of size 4.

By far, the most important use of abstract simplicial complexes in math is in algebraic topology, and algebraic topology is concerned with the study of shape. That will be the focus of future posts. As such most important example in this post for us will be Example 6: the abstract simplicial complex obtained from a triangulation* of a shape.

* Triangulation will be a central topic when we apply abstract simplicial complexes to topology, so I won't go into the full, rigorous definition here. Basically, a topological space (think: "shape") can often be decomposed as a union of points, lines, triangles, tetrahedra, and higher-dimensional analogues. Confusingly, these points, lines, triangles, and higher-dimensional analogues are also called simplices, but here I will always refer to them as topological simplices. Such a decomposition is called a triangulation of the space because the space is being cut into simplices, and simplices are kind of like triangles in different dimensions. In particular, a triangulation of a surface is literally a special way of cutting that surface into triangles.

Example 6: Let $K$ be a triangulated topological space. This means that we can write $K$ as a union $K = \bigcup_i A_i$ where each $A_i$ is a topological simplex. For reference, Figure 3 shows a triangulation of the sphere (although the triangles are flattened out, giving it the shape of an octahedron). We define the base set for the corresponding abstract simplicial complex to be the set of vertices, or zero-dimensional geometric simplices, used in the triangualtion of $K$. This set is written as $K^{(0)}$. So, the abstract simplicial complex for the triangulation in Figure 3 has a base set with 6 elements: $K^{(0)} = \{a, b, c, d, e, f\}$. The simplices of the abstract simplicial complex are in one-to-one correspondence with the geometric simplices of the triangulation. If $A$ is a topological simplex of the triangulation with vertices $x_1, x_2, \dots x_m$, then the corresponding (abstract simplicial complex) simplex is $\{x_1, x_2, \dots, x_m\}$.

We can now apply this constructino to the abstract simplicial complex obtained from the triangulation in Figure 3. The base set is:

The simplices are:

Remember that the dimension a simplex is one less than its size. So, this abstract simplicial complex has 8 0-dimensional simplices, 10 1-dimensional simplices, and 8 2-dimensional simplices.

Comparing the drawing of a triangulation in Figure 3 to the drawings of abstract simplicial complexes in Figures 1 and 2 (and in the last blog post), there is a clear similarity. In fact, if we made a drawing of the abstract simplicial complex associated with Figure 3, it would look exactly like Figure 3, but with different colors. This observation is even more general: there is a direct equivalence between triangulated topological spaces and abstract simplicial complexes that will be extremely important for us in the future. This equivalence allows us to use abstract simplicial complexes to investigate shapes.

Although useful for talking about shapes of objects and shapes of data, abstract simplicial complexes show up all over the place! The next few examples have nothing to do with shapes, and I know of no applications, but they may illustrate how abstract simplicial complexes can be found anywhere.

Example 7: Let $V$ be a vector space. We can construct an abstract simplicial complex whose base space is $V$ and whose base space and whose simplices are the non-empty linearly independent subsets of $V$.

Example 8: Let $V$ be a vector space. We can construct an abstract simplicial complex whose base space is $V$ and whose base space and whose simplices are the non-empty affine independent* subsets of $V$.

* A subset $\{x_1, x_2, x_3 \dots x_m\}$ of a vector space $V$ is affine independent if and only if $\{x_2 - x_1, x_3 - x_1 \dots x_m - x_1\}$ is linearly independent.

Example 9: Let $X$ be a room of people. Some people in $X$ are friends, and some are not. We can construct an abstract simplicial complex whose base set is $X$ and whose simplices are the non-empty friend groups, that is, subsets of $X$ in which every pair of people are friends. Alternatively, we can let the simplices be the non-empty groups of mutual strangers to obtain a different abstract simplicial complex.

Exercise: Verify that all of the above examples are abstract simplicial complexes. Don't worry about Example 6 if you are not familiar with triangulations.

Exercise+: For those who are familiar with triangulations, formulate an equivalence between abstract simplicial complexes and triangulated topological spaces.