Intermediate between Vietoris–Rips complex and Čech Complex

Question

The Vietoris–Rips complex is an abstract simplicial complex that can be defined from any metric space M and distance $\delta$ by forming a simplex for every finite set of points that has diameter at most $\delta$. That is, if the distance between each pair of points in a set $S$ is at most $\delta$, then $S$ is included as a simplex.

The Čech complex (or nerve) is defined by having a simplex for every finite subset of balls with nonempty intersection.

It seems that the two definitions above represent two extremes, since the Vietoris–Rips complex only considers pairwise distances, while the Čech complex considers all possible combinations of intersections.

I am looking for any "intermediate" between these two complexes, or any other similar constructions that can construct a simplicial complex from a set of points.

Answer 1

I think you need to look back at the history of these ideas and you will see another image of what is happening. A good place to start is a paper by Dowker:

C. H. Dowker, Homology Groups of Relations, Annals of Maths, 56, (1952), 84 – 95.

He showed how any relation, $R$, between two sets, say $X$ and $Y$, gave two simplicial complexes. In one the simplices are finite sets of points in $X$ all related to a single $y$, and in the other the roles of $X$ and $Y$ are reversed.

The classical use of the names Čech and Vietoris relate to the situation of an open covering of a space $X$ and one takes $Y$ to be a set of indices for the open sets in the covering. The homology of the two simplicial sets are the classical Čech and Vietoris homology of the open covering. (Of course, classically having once taken the homology one considers the limit over the directed set of open coverings to get an invariant that depends on the space alone and not the open coverings.)

Dowker's paper proves that in general the two simplicial complexes have the same homology, for the simple reason that their geometric realisations are homotopy equivalent. (I think the proof that he gives is very pretty and I suspect will help you see some of the 'geometry' and combinatorics of what is going on! He puts a total order on the elements of $X$ and then defines two maps from the double barycentric subdivision of one complex to itself, one factoring through the other complex, the other being a simplicial approximation to the identity map. These two maps are shown to be 'contiguous'. A more detailed summary of the proof is given on the nLab page ( go to

https://ncatlab.org/nlab

and search for Dowker's theorem.)

Although Dowker's proof is quite elementary, various more complex proofs are known in the literature. Some of these use a 'bicomplex' where (back in the general setting of $R\subset X\times Y$), one looks at bi-indexed prisms, where $(x_0,\dotsc,x_m;y_0,\dotsc, y_n)$ is a $(m,n)$-prism if all the $x_i\mathrel Ry_j$, so in the open covering case all the $x$s are in the intersection of the $y$s. (This proof can be found in a paper

H. Abels and S. Holz, Higher generation by subgroups, J. Alg, 160, (1993), 311 – 341.

They look at a group $G$ and cover it by the cosets of a family of subgroups of $G$, but the idea is the same.)

The Wikipedia page on the Vietoris–Rips complex to which you link restricts to a very special case of the general construction. (I feel it misuses the term Čech complex as a result, but that does not matter.) The Vietoris – Čech construction has been used extensively in topological (strong) shape theory, see

D. A. Edwards and H. M. Hastings, 1976, Čech and Steenrod homotopy theories with applications to geometric topology, volume 542 of Lecture Notes in Maths, Springer–Verlag.

Dowker's theorem has also been used in combinatorics (and even in sociology); see this question: What are the applications of Dowker's theorem?

(although my answer is not more detailed than this one!).

For one application of the complexes, (but not of Dowker's result) see : Perspectives on A-homotopy theory and its applications, by Barcelo and Laubenbacher, and related papers.)

I hope that somewhere in this maze of results there will be an indication as to the possible generalisations that you seek. Without knowing the situation that you are looking at I cannot tell.

Answer 2

Other constructions that build a simplicial complex from a set of points in a metric space include alpha complexes, witness complexes, and lazy witness complexes.

If metric space $M$ is geodesic, then the Vietoris-Rips complex (with simplices the finite subsets of diameter at most $\delta$) has the same 1-skeleton as the Cech complex (the nerve of balls of radius $\frac{\delta}{2}$). Since the Vietoris-Rips complex is a flag or a clique complex, one could equivalently define (for $M$ geodesic) the Vietoris-Rips complex to be the maximal simplicial complex with 1-skeleton equal to the 1-skeleton of the Cech complex. This inspires one possible "intermediate" family of complexes between these two, given in the paragraph below.

Given $1\le k\le\infty$, let $\mbox{Cech}_k$ denote the maximal simplicial complex with k-skeleton equal to the k-skeleton of the Cech complex. Note that when $k=1$ and $M$ is geodesic, $\mbox{Cech}_1$ is the same as the Vietoris-Rips complex. (When $M$ is an arbitrary metric space, we still have $\mbox{Cech}_1\subseteq \mbox{Vietoris-Rips}$.) When $k=\infty$, we should interpret $\mbox{Cech}_\infty$ as the standard Cech complex. Furthermore, for any fixed ball radius, we have the following sequence of inclusions, $$ \mbox{Cech} =\mbox{Cech}_\infty \subseteq \ldots \subseteq \mbox{Cech}_{k+1} \subseteq \mbox{Cech}_k \subseteq \mbox{Cech}_{k-1}\subseteq \ldots \subseteq \mbox{Cech}_1 \subseteq \mbox{Vietoris-Rips},$$ where the last inclusion is an equality if metric space $M$ is geodesic.