4 Simplicial Complexes on Finite Sets of Points
“To many, mathematics is a collection of theorems. For me, mathematics is a collection of examples; a theorem is a statement about a collection of examples and the purpose of proving theorems is to classify and explain the examples.” – John B Conway
Given a set of points S in a metric space, we describe ways to create simplicial complexes with the vertex set as S.
4.1 Cech and Rips Complex
Let (X,d) be a metric space and S be a finite subset of X (think \mathbb{R}^d). For a real number r > 0 we define the Cech complex \check{C}ech (S, r) to be the abstract simplicial complex defined as below:
\sigma \subset S is a simplex iff \displaystyle{\bigcap_{x \in \sigma} \bar{B}(x,r) \ne \emptyset}
Let (X,d) be a metric space and S be a finite subset of X (think \mathbb{R}^d). For a real number r > 0 we define the Rips complex Rips (S, r) to be the abstract simplicial complex defined as below:
\sigma \subset S is a simplex iff \displaystyle{ \text{Diam} (\sigma) \le 2r}
Exercise 4.1
- Describe the Cech and Rips complexes for the set of points S = \{(0, 0), (2, 0), (1, \sqrt{3})\} \subset \mathbb{R}^2, for scales r \in [0, \infty)
- Prove that \check{C}ech (S, r) \subset Rips (S, r) \subset \check{C}ech (S, 2r)
- Explain why \check{C}ech (S, r) is homotopy equivalent to the set \displaystyle{\bigcup_{s \in S} \bar{B}(s,r)}
4.2 Alpha Complex
Let S be a finite subset (of points in general position - no d+2 points lie on a sphere \mathbb{S}^{d-1}) in \mathbb{R}^d. For a real number r > 0 we define the Alpha complex Alpha (S, r) to be the abstract simplicial complex defined as below (here V_x is the Voronoi region corresponding to the point x \in S):
\sigma \subset S is a simplex iff \displaystyle{\bigcap_{x \in \sigma} \Big(\bar{B}(x,r) \cap V_x \Big) \ne \emptyset}