Čech cohomology
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Čech cohomology is a particular type of cohomology in mathematics. It is named for the mathematician Eduard Čech.
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[edit] Construction
Let X be a topological space, and let be a presheaf of abelian groups on X. Let be an open cover of X.
[edit] Simplex
A q-simplex of is an ordered collection of q + 1 sets, such that , whose intersection , 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 , then is the jth partial boundary of σ. The boundary of σ is , the alternating sum of the partial boundaries.
[edit] Cochain
A q-cochain of with coefficients in is a map which associates to 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.
[edit] Differential
The cochain groups can be made into a cochain complex by defining a coboundary operator (also called codifferential)
- ,
(where is restriction from | σ | to ), and showing that δ2 = 0.
[edit] Cocycle
A q-cochain is called a q-cocycle if it is in the kernel of δ and is the set of all q-cocycles.
Thus a cochain f is a cocycle if for all q-simplices σ the cocycle condition holds. In particular, a 1-cochain f is a 1-cocycle if
- .
[edit] Coboundary
A q-cochain is called a q-coboundary if it is in the image of δ and 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 .
[edit] Cohomology
The Čech cohomology of with values in is defined to be the cohomology of the cochain complex . Thus the qth Čech cohomology is given by
- .
The Čech cohomology of X is defined by considering refinements of open covers. If is a refinement of then there is a map in cohomology 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.
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 is naturally isomorphic to the singular cohomology H * (X;A). If X is a differentiable manifold, then 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 whereas
If X is a differentiable manifold and the cover of X is a "good cover" (i.e. all the sets Uα are contractible to a point, and all finite intersections of sets in are either empty or contractible to a point), then is isomorphic to the de Rham cohomology.
[edit] See also
[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