Čech cohomology

From Wikipedia, the free encyclopedia

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 $X$ . 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 $δ 2 = 0$ .

[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, 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 and the cover $\mathcal{U}$ 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 $\mathcal{U}$ 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

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.
Wells, Raymond (1980). Differential Analysis on Complex Manifolds. Springer-Verlag. ISBN 0-387-90419-0. ISBN 3-540-90419-0. Chapter 2 Appendix A

Categories: Algebraic topology | Homology theory

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Čech cohomology

From Wikipedia, the free encyclopedia

Contents

[edit] Construction

[edit] Simplex

[edit] Cochain

[edit] Differential

[edit] Cocycle

[edit] Coboundary

[edit] Cohomology

[edit] Relation to other cohomology theories

[edit] See also

[edit] References

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