# What are the applications of Dowker's theorem?

Let $R \subset X \times Y$ be any relation between sets $X$ and $Y$. CH Dowker constructed two simplicial complexes $K$ and $L$ associated to $R$:

1. a simplex in $K$ consists of finitely many elements $x \in X$ such that there exists a single $y \in Y$ with $(x,y) \in R$, and
2. a simplex in $L$ consists of finitely many elements $y \in Y$ such that there exists a single $x \in X$ with $(x,y) \in R$.

Clearly, these are simplicial complexes. The main theorem of Dowker is the construction of a natural isomorphism of homology groups of $K$ and $L$. There is even a proof outline on nlab. In fact there is a homotopy equivalence between the geometric realizations although that requires an ordering on the simplices and is therefore not natural. This may be found in the paper

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

The goal of this paper was to prove equivalence of Cech, Vietoris and Alexander (co)homology theories. My question is

What other applications (if any) have been found for this theorem?

In particular, the fact that the relation itself creates the simplicial complexes seems to make this theorem have very limited uses beyond the consequences proved already by Dowker: I can't seem to find much on the internet at least. I wonder if the experts have some idea...

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## 1 Answer

There is a considerable literature on `applications' of Dowker's result to sociology. This is sometimes doubtful in its depth! The development was started by R. Atkin.

As an example look at

http://www.ehu.es/ccwintco/uploads/1/11/Blanca-Cases-Q-analysis.pdf

As an offshoot of this there is fairly recent work in discrete maths (see work by Hélène Barcelo). I will not try to describe this other than saying it looks at an idea of the connectivity of a relation.

Back in the world of algebraic topology, it provides a way of proving that the pro-object in the homotopy category of simplicial sets, that is given by the Cech complex construction is in fact homotopy coherent. This provides a way of linking strong shape theory to the original form of shape theory. (It is not hard to prove this coherence directly although I only know one proof that has been written down in a thesis of one of my ex-students.)

(Edit: I forgot another example. You start with a group, $G$, and a family of subgroups, and ask to what extent invariants of the family give you invariants of the big group. This was the subject of a paper by Abels and Holz (Higher generation by subgroups , J. Alg, 160, (1993), 311– 341.) The family generates a covering of $G$ by its cosets. The two complexes given by that covering allow proof to be shortened and also in certain circumstances for links to Volodin's alegrbaic K-theory to be given.)

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