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CECH COHOMOLOGY

Čech cohomology is a particular type of cohomology in mathematics. It is named for the mathematician Eduard Čech.

Contents

Construction

Let X be a topological space with open cover U={Uα}α∈I, where I is a countable ordered set. We will simplify notation by writing intersections as Uα∩ Uβ = Uαβ, and so on for higher intersections. For every intersection Uα0···αn there are n + 1 inclusions defined as follows:

∂i : Uα0···αn→ Uα0···α(i − 1)α(i + 1)···αn

that is, ∂i skips the ith open set. We now define the cochain groups for this cohomology.

Let F be a presheaf. The 0-cochains on U with values in F are functions assigning an element of F(Uα) to every open set Uα. Using the properties of products, we may write

C0(U, F) = ∏α∈I F(Uα).

Then we define the 1-cochains to be elements of

C1(U, F) = ∏αβ F(Uαβ)

and so on, so that

Ci(U, F) = ∏α0···αi F(Uα0···αi ).

Next, the differential. Let δ : Ci(U, F)→Ci + 1(U, F) be the alternating difference of the F(∂i):

δ = F(∂0) − F(∂1) + ··· + (−1)i + 1F(∂i ).

One shows that δ² = 0, as required. By taking the cohomology of the chain complex formed we have the Čech cohomology of U with values in the presheaf F. We write

H_{\delta}C^*(\bold{U}, F) = H^*(\bold{U}, F).

Now, if V is a cover which refines U then there is a well-defined map in cohomology H∗(U, F)→H∗(V, F) and as such {H∗(U, F)}U is a direct system of groups. We call the direct limit of this system the Čech cohomology of X with values in F.

Let A be an abelian group and let FA be the constant presheaf which assigns to each nonempty open subset U of a topological space X the group A and to each inclusion of nonempty subsets V ⊂ U the identity homomorphism. Define F_A(\emptyset)=0. Then the Čech cohomology with A coefficients \check{H}^*(X;A) is defined to be the Čech cohomology of this presheaf.

A variant of Čech cohomology, called numerable Čech cohomology, is defined as above, except that all open covers considered are required to be numerable: that is, there is a partition of unity {ρi} such that each support {x | ρi(x) > 0} is contained in some element of the cover. If X is paracompact and Hausdorff, then numerable Čech cohomology agrees with the usual Čech cohomology.

Relation to other cohomology theories

If X is homotopy equivalent to a CW complex, then \check{H}^*(X;A) is naturally isomorphic to the singular cohomology H∗(X;A). If X is a differentiable manifold, then \check{H}^*(X;\mathbf{R}) is also naturally isomorphic to the de Rham cohomology. For less well-behaved spaces, Čech cohomology differs from singular cohomology. For example if X is the closed topologist's sine curve, then \check{H}^0(X;\mathbf{Z})=\mathbf{Z}, whereas H^0(X;\mathbf{Z})=\mathbf{Z}\oplus\mathbf{Z}.

If X is a differentiable manifold, the cover U of X is a "good cover" (i.e. all the sets Uα are contractible to a point, and all finite intersections of sets in U are either empty or contractible to a point), and F(U) = R for all open sets U in X (with all morphisms resU,V: F(U) → F(V), for V ⊂ U, being the identity map on R), then H∗(U, F) is isomorphic to the de Rham cohomology.

See also

References

  • R. Bott & L. Tu, Differential Forms in Algebraic Topology, Springer Graduate Texts in Mathematics 82, 1982.
  • Hatcher, Allen, Algebraic topology, Cambridge University Press (2002), ISBN 0521795400. For further discussion of Moore spaces, see Chapter 2, Example 2.40. A free electronic version of this book is available on the author's homepage.