nLab
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. $\langle U_0, \ldots, U_n\rangle$, each $U_i \in \alpha$, satisfying $\cap U_i \neq \emptyset$.

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

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

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.

Remarks:

(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 $n$th Čech homology group of the space $X$ is defined to be

$$\check{H}_n(X) = lim H_n(X,\alpha),$$

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

An expository account is available in Section 2 of

• David H. Fremlin, Singular homology for amateurs, PDF.

See also Čech methods.