nLab
Čech homology

Čech homology

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

Preliminary Definitions

Given a (compact) space XX and a finite open cover, α\alpha, of XX, we can form a simplicial set, C(X,α)C(X,\alpha), called the nerve of the cover whose nn-simplices are (n+1)(n + 1)-strings of open sets from α\alpha, i.e. U 0,,U n\langle U_0, \ldots, U_n\rangle, each U iαU_i \in \alpha, satisfying U i\cap U_i \neq \emptyset.

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

C(X,α)C(X,β)C(X,\alpha)\to C(X,\beta)

dependent on the choice of UU for each VV, 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 iU_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,α)C(X,\alpha)to C(X,β)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,α)C(X,\alpha), gives, for each nn, an inverse system of Abelian groups, which we will denote H n(X,α)H_n(X,\alpha).

Definition

The n thn^{th} Čech homology group of the space, XX, is defined to be

Hˇ n(X)=limH n(X,α)\check{H}_n(X) = lim H_n(X,\alpha)

where the limit is taken over all open covers of XX.

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.

See also Čech methods.

Revised on May 31, 2012 15:22:13 by Raeder? (80.203.119.14)