nLab
Vietoris complex

Idea

In the 1920s homology and cohomology were known for simplicial complexes and there were attempts to extend the definitions to first of all compact metric spaces and then more general spaces. Leopold Vietoris? (1927) came up with a construction and then shortly after Alexandrov and Čech gave a different one involving the nerve that now bears Čech's name. The input of the two methods is the same, we have a space $X$ and an open cover $𝒰$ of $X$.

It was noted that all the calculations of Vietoris homology gave the same answer as Čech homology. In 1952, C. H. Dowker showed why. His result is a beautiful mix of abstraction and concrete explicit calculation (and is not that well known unfortunately).

First some abstraction:

• We take the set $X$ and the open cover $𝒰$, together with the relation $x\in U$, i.e. an element $x$ will be considered to be related to the set $U$ in the cover $𝒰$, if it is an element of it!
• We replace this by an abstract setting of a set $X$, another set $Y$ and a relation $R\subseteq X\times Y$, from $X$ to $Y$.

Thinks: does what follows have a good generalisation to spans?

Definition:

The Vietoris complex of a relation $R$, as above, is the simplicial complex specified by

• Vertex set: the set of vertices is the set $X$;
• Simplices : a $p$-simplex of $K$ is a set $\{x_0,\dots ,x_p\}\subseteq X$ such that there is some $y\in Y$ with $x_i\mathrm{Ry}$ for $i=0,1,\dots ,p$

The Vietoris complex of $R$ will be denoted $V(R)$.

The special, and original, case where $R$ comes from an open cover $𝒰$ of $X$ will be denoted $V(X,𝒰)$ or simply $V(𝒰)$ and will be called the Vietoris complex of the open covering $𝒰$.

Remarks

In fact each relation will give two simplicial complexes, one as above, the other obtained by reversing the roles of $X$ and $Y$ in the above. In other words, in the usual way define the opposite relation by ’$R^{\mathrm{op}}\subseteq Y\times X$ is the relation in which ${\mathrm{yR}}^{\mathrm{op}}x$ if and only if $\mathrm{xRy}$’.

We define $C(R)=V(R^{\mathrm{op}})$ and call it the Čech nerve or Čech complex of the relation $R$.

If we look at this in the case of the relation coming from a space $X$ and an open covers $𝒰$ you get exactly the nerve of $𝒰$ as discussed in Čech methods.

There are natural questions that arise here. One is how to turn these simplicial complexes into simplicial sets, to which one reply is take a total order (or at least pick a partial order that any simplex is ordered in that partial order; another reply would be to take each $n$-simplex $(n+1)!$-times, one for each possible order.

Tim: I am not sure what happens to the homotopy types. It may be they both give the same. Mostly people take the total order way out, but that feels a bit evil!

Another obvious question is to compare $V(R)$ and $C(R)$: are they closely related? That is discussed more in Dowker's Theorem. It looks very much a situation for a combinatorial duality result.

A final question (and one I can answer!) is: are there useful applications of this construction other than in (co)homology. The answer is most decidedly ‘yes’.

(More to be added later.)

Gavin Wraith’s puzzle

Gavin Wraith posted a puzzle on the $n$Café whose solution uses the fact that the simplicial complexes $V(R)$ and $C(R)$ are homotopy equivalent:

• A Puzzle From Gavin Wraith

Perhaps this fact is Dowker's Theorem?