- Čech cohomology
-
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.
Contents
Motivation
Let X be a topological space, and let
be an open cover of X. Define a simplicial complex
, called the nerve of the covering, as follows:
- There is one vertex for each element of
.
- There is one edge for each pair
such that
.
- In general, there is one k-simplex for each k+1-element subset
of
for which
.
Geometrically, the nerve
is essentially a "dual complex" (in the sense of a dual graph, or Poincaré duality) for the covering
.
The idea of Čech cohomology is that, if we choose a cover
consisting of sufficiently small, connected open sets, the resulting simplicial complex
should be a good combinatorial model for the space X. For such a cover, the Čech cohomology of X is defined to be the simplicial cohomology of the nerve.
This idea can be formalized by the notion of a good cover, for which every open set and every finite intersection of open sets is contractible. However, a more general approach is to take the direct limit of the cohomology groups of the nerve over the system of all possible open covers of X, ordered by refinement. This is the approach adopted below.
Construction
Let X be a topological space, and let
be a presheaf of abelian groups on X. Let
be an open cover of X.
Simplex
A q-simplex σ of
is an ordered collection of q + 1 sets chosen from
, such that the intersection of all these sets is non-empty. This intersection is called the support of σ and is denoted | σ | .
Now let
be such a q-simplex. The j-th partial boundary of σ is defined to be the q-1-simplex obtained by removing the j-th set from σ, that is:
The boundary of σ is defined as the alternating sum of the partial boundaries:
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.
Differential
The cochain groups can be made into a cochain complex
by defining a coboundary operator (also called codifferential)
,
(where
is the restriction morphism from
to
), and showing that δ2 = 0.
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 (q-1)-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
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
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
, is defined as
where
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.
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
. 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.
If X is compact Hausdorff, then Čech cohomology (with coefficients in a discrete group) is isomorphic to Alexander-Spanier cohomology.
See also
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. http://www.math.cornell.edu/~hatcher/AT/ATpage.html. 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
- Cohomology theories
- Homology theory
- There is one vertex for each element of
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