nLab Čech homology

ech homology

Čech homology

Idea

Given a topological space XX, 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 XX, 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 XX.

Definition

Given a topological space XX with an open cover,

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

we write

(1)C(X,𝒰)π 0(U × X )sSet 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 nn-simplices are the inhabited (n+1)(n + 1)-fold intersections of the open subsets U iU_i in 𝒰\mathcal{U}.

Remark

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

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

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

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

This yields an inverse system of simplicial sets.

Definition

The nnth Čech homology group of the space XX is the limit

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

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

H n(X,𝒰)H n(C(X,𝒰)). 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.

References

Exposition:

On closed covers in Čech homology:

Last revised on June 29, 2022 at 15:37:14. See the history of this page for a list of all contributions to it.