The purpose of this entry is simply to state and prove a result of Dowker that was mentioned at Vietoris complex. Here we will look at a version of Dowker’s original proof. In later sections we will see other ways of proving it. We will also discuss some of the interpretations and applications of the result and ask about possible generalisations.
To keep the discussion fairly self contained, certain ideas will be repeated from that, and possibly other, entries.
Let be sets and a relation between and , so . We write for .
(In addition to those given in Vietoris complex.)
If is a simplicial complex, its structure is specified by a collection of non-empty finite subsets of its set of vertices namely those sets of vertices declared to be simplices. This collection of simplices is supposed to be downward closed, i.e., if is a simplex and with , then is a simplex. For our purposes here, set to be the set of vertices of and , the set of simplices of with if is a vertex of the simplex .
Returning to the general situation, we define two simplicial complexes associated to , as follows:
Clearly the two constructions are in some sense dual to each other.
These two simplicial complexes were denoted and at Vietoris complex.
We next need some classical subdivision ideas.
Combinatorially, if is a simplicial complex with vertex set , then one associates to the partially ordered set of its simplices. Explicitly we write for the set of simplices of and for the partially ordered set with being the obvious inclusion. The barycentric subdivision, , of has as its set of vertices and a finite set of vertices of (i.e. simplices of ) is a simplex of if it can be totally ordered by inclusion. (Thus is the simplicial complex given by taking the nerve of the poset, .)
It is important to note that there is in general no natural simplicial map from to . If however is ordered in such a way that the vertices of any simplex in are totally ordered (for instance by picking a total order on ), then one can easily specify a map
\varphi : K' \rightarrow K
This preserves simplices, but reverses order so if then .
If one changes the order, then the resulting map is contiguous:
Let be two simplicial maps between simplicial complexes. They are contiguous if for any simplex of , forms a simplex in .
Contiguity gives a constructive form of homotopy applicable to simplicial maps.
If is a simplicial map, then it induces after subdivision. As there is no way of knowing / picking compatible orders on and in advance, we get that on constructing
\varphi_K : K' \rightarrow K
\varphi_L : L' \rightarrow L
that and will be contiguous to each other but rarely equal.
Returning to and , we order the elements of and . Then suppose is a vertex of , so , a simplex of and there is an element with . Set for one such .
If is a -simplex of , assume is its least vertex (in the inclusion ordering)
\varphi_L (y^\prime_0) \in y'_0 \subset y^\prime for each y_i \in \sigma
hence and the elements form a simplex in , so is a simplicial map. It, of course, depends on the ordering used and on the choice of , but any other choice for gives a contiguous map.
Reversing the roles of and in the above we get a simplicial map
\bar\psi : K_R' \rightarrow L_R.
Applying barycentric subdivisions again gives
\bar\psi' : K_R'' \rightarrow L_R'
and composing with gives a map
\psi \bar\psi' : K_R'' \rightarrow K_R.
Of course, there is also a map
\varphi_K \varphi'_K : K_R''\rightarrow K_R.
The two maps and are contiguous.
Before proving this, note that contiguity implies homotopy and that is homotopic to the identity map on after realisation, i.e., this proves the following modulo the proof of the proposition:
The homotopy depends on the ordering of the vertices and so is not natural.
Let be a simplex of and as usual assume is its least vertex, then for all
x_0'' \subset x_i''.
We have that is clearly order reversing so . Let , then for each , . Since , we have .
For each vertex of , hence as for each , so for each , however we therefore have
\varphi_k \varphi'_K (\sigma '') \cup \psi \bar \psi (\sigma''') = \bigcup \varphi_k \varphi'_K (x_i'')\cup \psi \bar \psi; x_i ''
forms a simplex in , i.e. and are contiguous.