How is the Cech complex a subcomplex of the Vietoris Rips complex?

Question

Wikipedia states: The Čech complex is a subcomplex of the Vietoris–Rips complex.

And they reference Ghrist which essentially makes the definitions:

Given a finite point cloud $X$ and $\epsilon >0$, we construct the Čech complex $C_\epsilon(X)$ as follows: Take the elements of $X$ as the vertex set of $C_\epsilon(X)$. Then, for each $\sigma\subset X$, let $\sigma\in C_\epsilon(X)$ if the set of $\epsilon$-balls centered at points of $\sigma$ has a nonempty intersection.

Similarly,

The Vietoris Rips Complex of scale $\epsilon$ on $X$ is an abstract simplicial complex whose simplices are all finite collections of points in $X$ of pairwise distance $\leq\epsilon$.

Ghrist also makes the claim that the Cech complex is a subcomplex of the Vietoris Rips complex $VR_\epsilon$, and sometimes a proper subcomplex (p.30), but does not state any scale for the Cech complex in that statement. I then take it as implicit that this means that $C_\epsilon(X)\leq VR_\epsilon(X)$, i.e. for the same $\epsilon$. But, I do not see how this can be:

Just simply take $X$ to be two points $x_1,x_2$ at distance $1$ from eachother. Then clearly $C_{0.6}(X)$ is two vertices and an edge, but $VR_{0.6}(X)$ is just two vertices.

Question: Either it seems it would be the other way round ($V\leq C$), or we have that $C_\epsilon\leq VR_{2\epsilon}$ or something. What is the relation, and/or what am I misunderstanding?

And, if it is, for example, $C_\epsilon\leq VR_{2\epsilon}$, what motivates a statement of the form 'The Čech complex is a subcomplex of the Vietoris–Rips complex' without any reference to scale?

Answer 1

As you have shown, in general $C_\epsilon(Q)\leq VR_\epsilon(Q)$ is false. But in fact both

1. $C_\epsilon(Q)\leq VR_{2\epsilon}(Q)$

2. $VR_\epsilon(Q)\leq C_\epsilon(Q)$

are true. I am not sure whether 2. is true for an arbitrary metric space $(Q, d)$, but here we have $Q \subset \mathbb R^n$ and $d$ is the usual Euclidean metric. Thus, given $n$ points $q_i \in Q$ such that $\lvert q_i - q_j \rvert < \epsilon$ for all $i,j$, we define the barycenter $b = \sum_{i=1}^n \frac{1}{n}q_i$ of the $q_i$ and easily verify that $b \in \bigcap _{i=1}^n B_\epsilon(q_i)$: We have $$\lvert b - q_j \rvert = \lvert\sum_{i=1}^n \frac{1}{n}q_i - \sum_{i=1}^n \frac{1}{n}q_j \rvert = \lvert\sum_{i=1}^n \frac{1}{n}(q_i - q_j) \rvert = \lvert \frac{1}{n} \sum_{i=1}^n (q_i - q_j) \rvert \le \frac{1}{n} \sum_{i=1}^n \lvert q_i - q_j \rvert\\ < \frac{1}{n} n \epsilon = \epsilon .$$

The statement

The Čech complex is a subcomplex of the Vietoris–Rips complex $VR_\epsilon(Q)$

suggests that $C_\epsilon(Q)\leq VR_{\epsilon}(Q)$, but it needs a more flexible interpretation. This is 1.

The problem with Ghrist's exposition is that is does not become clear what the purpose of the Vietoris–Rips / Čech complex really is.

I think you need to know Čech (co)homology if you want to understand that point. Unfortunately Ghrist does not introduce this properly. He only defines the Čech homology groups of open covers of a space $X$. The Čech (co)homology groups of a space $X$ are defined as follows:

The set $\mathscr C(X)$ of all open covers of $X$ is partially ordered by refinement; this makes $\mathscr C(X)$ an inverse system. The Čech homology groups (cohomology groups) of $X$ are defined as the inverse limit (direct limit) of the Čech homology groups (cohomology groups) of the open covers of $X$.

A similar construction gives us the Vietoris (co)homology groups of $X$.

For the limit processes we do not need the full set $\mathscr C(X)$, any cofinal subsystem will do. If $X$ is compact, the coverings by balls of a fixed diameter form such a cofinal subsystem. Now the above arguments concerning the Čech and Vietoris-Rips complexes show that the Čech and the Vietoris (co)homology groups agree.

This was a very rough explanation, but I hope it gives at least an idea what is going on here.