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Dowker's theorem

Dowker's Theorem

Note

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.

The two nerves of a relation: Dowker’s construction

Let X,Y be sets and R a relation between X and Y, so RX×Y. We write xRy for (x,y)R.

Example

(In addition to those given in Vietoris complex.)

If K 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 σ is a simplex and τσ with τ, then τ is a simplex. For our purposes here, set X=V K to be the set of vertices of K and Y=S K, the set of simplices of K with xRy if x is a vertex of the simplex y.

Returning to the general situation, we define two simplicial complexes associated to R, as follows:

  1. K=K R:

    • the set of vertices is the set X;
    • a p-simplex of K is a set {x 0,,x p}X such that there is some yY with x iRy for i=0,1,,p.
  2. L=L R:

    • the set of vertices is the set, Y;
    • p-simplex of K is a set {y 0,,y p}Y such that there is some xX with xRy j for j=0,1,,p.

Clearly the two constructions are in some sense dual to each other.

Note

These two simplicial complexes were denoted V(R) and C(R) at Vietoris complex.

We next need some classical subdivision ideas.

Barycentric subdivisions

Combinatorially, if K is a simplicial complex with vertex set V K, then one associates to K the partially ordered set of its simplices. Explicitly we write S K for the set of simplices of K and (S K,) for the partially ordered set with being the obvious inclusion. The barycentric subdivision, K, of K has S K as its set of vertices and a finite set of vertices of K (i.e. simplices of K) is a simplex of K if it can be totally ordered by inclusion. (Thus K is the simplicial complex given by taking the nerve of the poset, (S K,).)

Note

It is important to note that there is in general no natural simplicial map from K to K. If however V K is ordered in such a way that the vertices of any simplex in K are totally ordered (for instance by picking a total order on V K), then one can easily specify a map

φ:KK\varphi : K' \rightarrow K

by:

  • if σ={x 0,,x p} is a vertex of K (so σS K), let φσ be the least vertex of σ in the given fixed order.

This preserves simplices, but reverses order so if σ 1σ 2 then φ(σ 1)φ(σ 2).

If one changes the order, then the resulting map is contiguous:

Contiguity

Let φ,~ψ:KL be two simplicial maps between simplicial complexes. They are contiguous if for any simplex σ of K, φ(σ)ψ(σ) forms a simplex in L.

Contiguity gives a constructive form of homotopy applicable to simplicial maps.

If ψ:KL is a simplicial map, then it induces ψ:KL after subdivision. As there is no way of knowing / picking compatible orders on V K and V L in advance, we get that on constructing

φ K:KK\varphi_K : K' \rightarrow K

and

φ L:LL\varphi_L : L' \rightarrow L

that φ Lψ and ψφ K will be contiguous to each other but rarely equal.

Returning to K R and L R, we order the elements of X and Y. Then suppose y is a vertex of L R, so y={y 0,,y p}, a simplex of L R and there is an element xX with xRy i,i=0,1,,p. Set ψy=x for one such x.

If σ={y 0,,y q} is a q-simplex of L R, assume y 0 is its least vertex (in the inclusion ordering)

φ L(y 0 )y 0y foreachy iσ\varphi_L (y^\prime_0) \in y'_0 \subset y^\prime for each y_i \in \sigma

hence ψy iRφ L(y 0) and the elements ψy 0,,ψy q form a simplex in K R, so ψ:L RK R is a simplicial map. It, of course, depends on the ordering used and on the choice of x, but any other choice x¯ for ψy gives a contiguous map.

Reversing the roles of X and Y in the above we get a simplicial map

ψ¯:K RL R.\bar\psi : K_R' \rightarrow L_R.

Applying barycentric subdivisions again gives

ψ¯:K RL R\bar\psi' : K_R'' \rightarrow L_R'

and composing with ψ:L RK R gives a map

ψψ¯:K RK R.\psi \bar\psi' : K_R'' \rightarrow K_R.

Of course, there is also a map

φ Kφ K:K RK R.\varphi_K \varphi'_K : K_R''\rightarrow K_R.
Proposition (Dowker, 1952, p. 88)

The two maps φ Kφ K and ψψ¯ are contiguous.

Before proving this, note that contiguity implies homotopy and that φφ is homotopic to the identity map on K R after realisation, i.e., this proves the following modulo the proof of the proposition:

Dowker’s Theorem

K RL R.

The homotopy depends on the ordering of the vertices and so is not natural.

Proof of Proposition

Let σ={x 0,x 1,,x q} be a simplex of K R and as usual assume x 0 is its least vertex, then for all i>0

x 0x i.x_0'' \subset x_i''.

We have that φ K is clearly order reversing so φ Kx iφ Kx 0. Let y=φ¯φ Kx 0, then for each xφ Kx 0, xRy. Since φ Kφ Kx iφ Kx iφ Kx 0, we have φ Kφ Kx iRy.

For each vertex x of x i,ψ¯xψ¯x i, hence as φ Kx 0x 0x i,y=ψ¯φ Kxx 0ψ¯x i for each x i, so for each x i,ψψ¯x iRy, however we therefore have

φ kφ K(σ)ψψ¯(σ)=φ kφ K(x i)ψψ¯;x i\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 K R, i.e. φ Kφ K and ψψ¯ are contiguous.

References

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

Revised on February 18, 2013 19:57:45 by Toby Bartels (64.89.53.62)