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Concept
Čech cohomology
Summary
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.
Let X be a topological space, and let be an open cover of X. Let denote the nerve of the covering. The idea of Čech cohomology is that, for an open cover consisting of sufficiently small open sets, the resulting simplicial complex should be a good combinatorial model for the space X. For such a cover, the Čech cohomology of X is defined to be the simplicial cohomology of the nerve. This idea can be formalized by the notion of a good cover. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of X, ordered by refinement. This is the approach adopted below.
Let X be a topological space, and let be a presheaf of abelian groups on X. Let be an open cover of X.
A q-simplex σ of is an ordered collection of q+1 sets chosen from , such that the intersection of all these sets is non-empty. This intersection is called the support of σ and is denoted |σ|.
Now let be such a q-simplex. The j-th partial boundary of σ is defined to be the (q−1)-simplex obtained by removing the j-th set from σ, that is:
The boundary of σ is defined as the alternating sum of the partial boundaries:
viewed as an element of the free abelian group spanned by the simplices of .
A q-cochain of with coefficients in is a map which associates with each q-simplex σ an element of , and we denote the set of all q-cochains of with coefficients in by . is an abelian group by pointwise addition.
The cochain groups can be made into a cochain complex by defining the coboundary operator by:
where is the restriction morphism from to (Notice that ∂jσ ⊆ σ, but σ ⊆ ∂jσ.)
A calculation shows that
The coboundary operator is analogous to the exterior derivative of De Rham cohomology, so it sometimes called
the differential of the cochain complex.
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