Čech cohomology

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

1 Construction
2 Relation to other cohomology theories
3 See also
4 References

[edit] Construction

Let $X$ be a topological space, and let $\mathcal F$ be a presheaf of abelian groups on $X$ . For concreteness, one can take $\mathcal F$ to be the constant sheaf on $X$ with values in a fixed abelian group $A$ .

[edit] Cochains

Let $\mathcal U = \lbrace U_{\alpha} \rbrace_{\alpha \in I}$ be an open cover of $X$ where $I$ is an ordered set. We define the cochain groups on $\mathcal U$ with values in $\mathcal F$ as follows. The 0-cochains are functions assigning an element of $\mathcal F (U_{\alpha})$ to every open set $U α$ . Using the properties of products, we may write

$C^0(\mathcal U, \mathcal F) = \prod_{\alpha}\mathcal F(U_\alpha).$

The 1-cochains are defined to be elements of

$C^1(\mathcal U, \mathcal F) = \prod_{\alpha < \beta}\mathcal F(U_{\alpha}\cap U_{\beta})$

and so on, so that

$C^q(\mathcal U, \mathcal F) = \prod_{\alpha_0 < \alpha_1 < \cdots < \alpha_q}\mathcal F(U_{\alpha_0}\cap U_{\alpha_1}\cap\cdots\cap U_{\alpha_q}).$

The set of all $q$ -cochains forms an abelian group.

[edit] Differential

The cochain groups can be made into a cochain complex by defining a differential (or coboundary) operator

$\delta_q : C^q(\mathcal U, \mathcal F) \to C^{q+1}(\mathcal U, \mathcal F).$

We will simplify notation by writing intersections as $U_{\alpha} \cap U_{\beta} = U_{\alpha \beta}$ , and so on for higher intersections. For every intersection $U_{\alpha_{0} \cdots \alpha_{q}}$ there are $q + 1$ inclusions defined as follows:

$\partial_i : U_{\alpha_0\cdots\alpha_q} \to U_{\alpha_{0} \cdots \check{\alpha}_{i} \cdots \alpha_{q}}.$

That is, $\partial_{i}$ skips the $i$ th open set. Applying $\mathcal F$ to $\partial_{i}$ we get $q + 1$ restriction homomorphisms. The differential $δ$ is defined as the alternating sum of the $\mathcal F( \partial_{i})$ :

$\delta = \mathcal F(\partial_0) - \mathcal F(\partial_1) + \cdots + (-1)^{q+1}\mathcal F(\partial_{q+1}).$

Explicitly, for $\omega \in C^{q}(\mathcal U,\mathcal F)$ we have

$(\delta\omega)_{\alpha_{0} \cdots \alpha_{q+1}} = \sum_{i=0}^{q+1}(-1)^i\,\omega_{\alpha_0\cdots\check{\alpha}_i\cdots\alpha_q}.$

One shows that $δ 2 = 0$ , as required, so that $(C^{q}(\mathcal U, \mathcal F), \delta)$ does indeed form a cochain complex.

[edit] Cohomology

The Čech cohomology of U with values in the presheaf F is defined as the the cohomology of the above cochain complex. That is the qth Čech cohomology is given by

$H^q(\mathcal U,\mathcal F) = \ker\delta_q / \operatorname{im}\,\delta_{q-1}.$

The Čech cohomology of X is defined by considering refinement of open covers. If V is a refinement of U then there is a well-defined map in cohomology

$H^{\ast}(\mathcal U,\mathcal F) \to H^{\ast}(\mathcal V,\mathcal F).$

The open covers of X form a directed set under refinement, so the above map leads to a direct system of abelian groups. The Čech cohomology of X with values in F is defined as the direct limit of this system:

$H(X,\mathcal F) = \varinjlim_{\mathcal U} H(\mathcal U,\mathcal F).$

The Čech cohomology of X with coefficients in a fixed abelian group A, denoted H(X; A), is defined as H(X, F_A) where F_A is the constant sheaf on X determined by A.

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 {ρ_i} such that each support ${x | ρ i (x) > 0}$ is contained in some element of the cover. If X is paracompact and Hausdorff, then numerable Čech cohomology agrees with the usual Čech cohomology.

[edit] Relation to other cohomology theories

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

If X is a differentiable manifold, the cover U of X is a "good cover" (i.e. all the sets U_α are contractible to a point, and all finite intersections of sets in U are either empty or contractible to a point), then H^∗(U, R) is isomorphic to the de Rham cohomology.

[edit] See also

Sheaf cohomology

[edit] References

Bott, Raoul; Loring Tu (1982). Differential Forms in Algebraic Topology. New York: Springer. ISBN 0-387-90613-4.
Hatcher, Allen (2002). Algebraic Topology. Cambridge University Press. ISBN 0-521-79540-0. For further discussion of Moore spaces, see Chapter 2, Example 2.40.