# Čech homology

Recall the definition of the nerve of an open cover from Čech methods.

## Preliminary Definitions

Given a (compact) space $X$ and a finite open cover, $\alpha $, of $X$, we can form a simplicial set, $C(X,\alpha )$, called the *nerve of the cover* whose $n$-simplices are $(n+1)$-strings of open sets from $\alpha $, i.e. $\u27e8{U}_{0},\dots ,{U}_{n}\u27e9$, each ${U}_{i}\in \alpha $, satisfying $\cap {U}_{i}\ne \varnothing $.

If $\beta $ is another cover such that for each $V\in \beta $, there is a $U\in \alpha $ with $V\subseteq U$, then the assignment $V\to U$ in this case defines a map

$$C(X,\alpha )\to C(X,\beta )$$`C(X,\alpha)\to C(X,\beta)`

dependent on the choice of $U$ for each $V$, but independent ‘up to homotopy’. This gives an inverse system of simplicial sets and homotopy classes of maps.

(i) The classical definition would require that the sets ${U}_{i}$, involved in the simplex were distinct, and that the ordering did not matter. That then gives a simplicial complex rather than a simplicial set. The simplicial complex definition yields something that is smaller but for some calculations is a lot less easy to work with.

(ii) The proof that the homotopy class of the ‘binding’ map from $C(X,\alpha )$to $C(X,\beta )$ is independent of the choices made is well known. It shows that any two choices yield ‘contiguous maps’ in as much as the images of a simplex under the two maps, given by the two choices, form two faces of a higher dimensional simplex. This allows an *explicit* homotopy of simplicial maps to be given.

Taking the simplicial homology groups of the $C(X,\alpha )$, gives, for each $n$, an inverse system of Abelian groups, which we will denote ${H}_{n}(X,\alpha )$.

## Definition

The ${n}^{\mathrm{th}}$ Čech homology group of the space, $X$, is defined to be

$${\stackrel{\u02c7}{H}}_{n}(X)=\mathrm{lim}{H}_{n}(X,\alpha )$$`\check{H}_n(X) = lim H_n(X,\alpha)`

where the limit is taken over all open covers of $X$.

It is to be noted that these groups do not constitute a homology theory in the sense of the Eilenberg-Steenrod axioms as the *exactness axiom* fails in general. There is a “corrected” theory known under the name strong homology.

## Literature and links

See also Čech methods.