Čech cohomology

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

Contents

[edit] Construction

Let X be a topological space, and let \mathcal{F} be a presheaf of abelian groups on X. Let \mathcal{U} be an open cover of X.

[edit] Simplex

A q-simplex \sigma := (,)_{i=0}^q U_i =: (U_0, \ldots, U_q) of \mathcal{U} is an ordered collection of q + 1 sets, such that \forall_{i=0}^q U_i \in \mathcal{U}, whose intersection |\sigma| := \bigcap_{i=0}^q U_i, called the support of σ, is non-empty. It has q+1 partial boundaries, each formed by removing one of the sets comprising the simplex. If 0 \le j \le q, then \partial_j \sigma := (,)_{i=0, i \ne j}^q U_i is the jth partial boundary of σ. The boundary of σ is \partial \sigma := \sum_{j=0}^q (-1)^j \partial_j \sigma, the alternating sum of the partial boundaries.

[edit] Cochain

A q-cochain of \mathcal{U} with coefficients in \mathcal{F} is a map which associates to each q-simplex σ an element of \mathcal{F}(|\sigma|) and we denote the set of all q-cochains of \mathcal{U} with coefficients in \mathcal{F} by C^q(\mathcal U, \mathcal F). C^q(\mathcal U, \mathcal F) is an abelian group by pointwise addition.

[edit] Differential

The cochain groups can be made into a cochain complex (C^{\textbf{.}}(\mathcal U, \mathcal F), \delta) by defining a coboundary operator (also called codifferential)

\delta_q : C^q(\mathcal U, \mathcal F) \to C^{q+1}(\mathcal{U}, \mathcal{F}) : \omega \mapsto \delta \omega, \quad (\delta \omega)(\sigma) := \sum_{j=0}^{q+1} (-1)^j \mathrm{res}^{|\partial_j \sigma|}_{|\sigma|} \omega (\partial_j \sigma),

(where \mathrm{res}^{|\partial_j \sigma|}_{|\sigma|} is restriction from | σ | to |\partial_j \sigma|), and showing that δ2 = 0.

[edit] Cocycle

A q-cochain is called a q-cocycle if it is in the kernel of δ and Z^q(\mathcal{U}, \mathcal{F}) := \ker \left( \delta_q : C^q(\mathcal U, \mathcal F) \to C^{q+1}(\mathcal{U}, \mathcal{F}) \right) is the set of all q-cocycles.

Thus a cochain f is a cocycle if for all q-simplices σ the cocycle condition \sum_{j=0}^n (-1)^j \mathrm{res}^{|\partial_j \sigma|}_{|\sigma|} f (\partial_j \sigma) = 0 holds. In particular, a 1-cochain f is a 1-cocycle if

\forall_{\{A, B, C\} \subset \mathcal{U}}\ U:=A \cap B \cap C,\ f(B \cap C)|_U - f(A \cap C)|_U + f(A \cap B)|_U = 0.

[edit] Coboundary

A q-cochain is called a q-coboundary if it is in the image of δ and B^q(\mathcal{U}, \mathcal{F}) := \mathrm{im} \left( \delta_{q-1} : C^{q-1}(\mathcal{U}, \mathcal{F}) \to C^{q}(\mathcal{U}, \mathcal{F}) \right) is the set of all q-coboundaries.

For example, a 1-cochain f is a 1-coboundary if there exists a 0-cochain h such that \forall_{\{A, B\} \subset \mathcal{U}}, U:=A \cap B, f(U) = (\delta h)(U) = h(A)|_U - h(B)|_U.

[edit] Cohomology

The Čech cohomology of \mathcal{U} with values in \mathcal{F} is defined to be the cohomology of the cochain complex (C^{\textbf{.}}(\mathcal{U}, \mathcal{F}), \delta). Thus the qth Čech cohomology is given by

H^q(\mathcal{U}, \mathcal{F}) := H^q((C^{\textbf{.}}(\mathcal U, \mathcal F), \delta)) = Z^q(\mathcal{U}, \mathcal{F}) / B^q(\mathcal{U}, \mathcal{F}).

The Čech cohomology of X is defined by considering refinements of open covers. If \mathcal{V} is a refinement of \mathcal{U} then there is a map in cohomology H^*(\mathcal U,\mathcal F) \to H^*(\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 H(X,\mathcal F) := \varinjlim_{\mathcal U} H(\mathcal U,\mathcal F) of this system.

The Čech cohomology of X with coefficients in a fixed abelian group A, denoted H(X; A), is defined as H(X, FA) where FA 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 {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.

[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 and the cover \mathcal{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 \mathcal{U} are either empty or contractible to a point), then \check{H}^{*}(\mathcal U;\mathbb{R}) is isomorphic to the de Rham cohomology.

[edit] See also

[edit] References

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