nLab Čech homology

Idea

Given a topological space $X$, its Čech homology is the limit of simplicial homology-groups of the Čech nerves of its open covers, under refinement of open covers.

Compare to the singular homology of $X$, which is the simplicial homology of its singular simplicial complex. For well behaved topological spaces the two notions agree and are jointly known as computing the ordinary homology of $X$.

Definition

Given a topological space $X$ with an open cover,

$$\mathcal{U} \;=\; \Big\{ U_i \xhookrightarrow{open} X \Big\}_{i \in I}$$

we write

(1)$$C(X,\mathcal{U}) \;\coloneqq\; \pi_0 \big( U^{\times^\bullet_X} \big) \;\in\; sSet$$

for the simplicial set which is its Čech+nerve: whose $n$-simplices are the inhabited $(n + 1)$-fold intersections of the open subsets $U_i$ in $\mathcal{U}$.

Remark

For sufficiently well-behaved topological spaces $X$ (paracompact spaces) and good open covers, the Čech nerve (1) is simplicially homotopy equivalent to the singular simplicial complex of $X$ – this is the statement of the nerve theorem.

However, in general $C(X,\mathcal{U})$ may differ from the weak homotopy type of $X$ (even in the limit below) in which case Čech homology may differ from the singular homology of $X$. (See for instance the discussion at well group.)

If $\mathcal{U}$ is a refinement open cover, i.e. such that for each $U' \in \mathcal{U}$, there is a $U \in \mathcal{U}$ with $U' \subseteq U$, then these inclusions induce a morphism of simplicial sets

$$C(X,\mathcal{U}) \longrightarrow C(X,\mathcal{U}')$$

This yields an inverse system of simplicial sets.

Definition

The $n$th Čech homology group of the space $X$ is the limit

$$\check{H}_n(X) \;\coloneqq\; \underset{ \underset{\mathcal{U}}{\longleftarrow} }{lim} H_n(X,\,\mathcal{U})$$

over the inverse system of open covers $\alpha$, of the simplicial homology groups of the Čech nerve $C(X,\alpha)$:

$$H_n(X,\,\mathcal{U}) \;\coloneqq\; H_n\big(C(X,\,\mathcal{U})\big) \,.$$

Remark

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.

Related concepts

• Čech methods, nerve theorem

• singular homology, cellular homology

• ordinary homology

• étale homotopy type

• well group

References

Exposition:

• David H. Fremlin, Section 2 of: Singular homology for amateurs (2016) $[$pdf, pdf$]$

On closed covers in Čech homology:

• Edwin E. Floyd, Closed coverings in Čech homology theory, Trans. Amer. Math. Soc. 84 (1957) 319-337 [doi:10.1090/S0002-9947-1957-0087100-2, pdf]