Rotation matrix

2014-01-30T20:12:41Z

192.35.35.34: /* In two dimensions */

[[File:Penrose-dreieck.svg|thumb|A [[Penrose triangle]] depicts a nontrivial element of the first cohomology of an [[annulus (mathematics)|annulus]] with values in the group of distances from the observer<ref>{{Citation |first=Roger |last=Penrose | |authorlink=Roger Penrose |date=1992 |title=On the Cohomology of Impossible Figures |journal=[[Leonardo (journal)|Leonardo]] |volume=25 |issue=3/4 |pages=245–247 |doi=10.2307/1575844}}. Reprinted from {{Citation |first=Roger |last=Penrose | |authorlink=Roger Penrose |date=1991 |title=On the Cohomology of Impossible Figures / La Cohomologie des Figures Impossibles |journal=Structural Topology |volume=17 |pages=11–16 |url=http://www.iri.upc.edu/people/ros/StructuralTopology/ST17/st17.html |accessdate=January 16, 2014}}</ref>]]

In [[mathematics]], specifically [[algebraic topology]], '''Čech cohomology''' is a [[cohomology]] theory based on the intersection properties of [[open set|open]] [[cover (topology)|covers]] of a [[topological space]]. It is named for the mathematician [[Eduard Čech]].

==Motivation==
Let ''X'' be a topological space, and let <math>\mathcal{U}</math> be an open cover of ''X''. Define a [[simplicial complex]] <math>N(\mathcal{U})</math>, called the [[nerve of a covering|nerve]] of the covering, as follows:
* There is one vertex for each element of <math>\mathcal{U}</math>.
* There is one edge for each pair <math>U_1,U_2\in\mathcal{U}</math> such that <math>U_1 \cap U_2 \ne \emptyset</math>.
* In general, there is one ''k''-simplex for each ''k+1''-element subset <math>\{U_0,\ldots,U_k\}\,\!</math> of <math>\mathcal{U}</math> for which <math>U_0\cap\cdots\cap U_k\ne\emptyset\,\!</math>.
Geometrically, the nerve <math>N(\mathcal{U})</math> is essentially a "dual complex" (in the sense of a [[dual graph]], or [[Poincaré duality]]) for the covering <math>\mathcal{U}</math>.

The idea of Čech cohomology is that, if we choose a "nice" cover <math>\mathcal{U}</math> consisting of sufficiently small open sets, the resulting simplicial complex <math>N(\mathcal{U})</math> 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 homology|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 [[Open cover#Refinement|refinement]]. This is the approach adopted below.

==Construction==

Let <math>X</math> be a [[topological space]], and let <math>\mathcal{F}</math> be a [[presheaf]] of [[abelian group]]s on <math>X</math>. Let <math>\mathcal{U}</math> be an [[open cover]] of <math>X</math>.

===Simplex===
A ''q''-'''simplex''' <math>\sigma</math> of <math>\mathcal{U}</math> is an ordered collection of <math>q+1</math> sets chosen from <math>\mathcal{U}</math>, such that the intersection of all these sets is non-empty. This intersection is called the ''support'' of <math>\sigma</math> and is denoted <math>|\sigma|</math>.

Now let <math>\sigma = (U_i)_{i \in \{ 0 , \ldots , q \}}</math> be such a ''q''-simplex. The ''j-th partial boundary'' of <math>\sigma</math> is defined to be the ''q-1''-simplex obtained by removing the ''j''-th set from <math>\sigma</math>, that is:
:<math>\partial_j \sigma := (U_i)_{i \in \{ 0 , \ldots , q \} \setminus \{j\}}.</math>
The ''boundary'' of <math>\sigma</math> is defined as the alternating sum of the partial boundaries:
:<math>\partial \sigma := \sum_{j=0}^q (-1)^{j+1} \partial_j \sigma.</math>

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

===Differential===
The cochain groups can be made into a [[cochain complex]] <math>(C^{\textbf{.}}(\mathcal U, \mathcal F), \delta)</math> by defining the '''coboundary operator'''
<math>\delta_q : C^q(\mathcal U, \mathcal F) \to C^{q+1}(\mathcal{U},
\mathcal{F}) </math> by

<math> \quad (\delta_q \omega)(\sigma) := \sum_{j=0}^{q+1} (-1)^j \mathrm{res}^{|\partial_j \sigma|}_{|\sigma|} \omega (\partial_j \sigma)</math>,

where <math>\mathrm{res}^{|\partial_j \sigma|}_{|\sigma|}</math> is the [[Sheaf (mathematics)|restriction morphism]] {{H:title|Notice that ∂ⱼσ ⊆ σ, but |σ| ⊆ |∂ⱼσ| |from}} <math>\mathcal F(|\partial_j \sigma|)</math> to <math>\mathcal F(|\sigma|).</math>

A calculation shows that <math>\delta_{q+1} \circ \delta_q = 0 </math>.

The coboundary operator is also sometimes called
the [[codifferential]].

====Cocycle====
A ''q''-cochain is called a ''q''-cocycle if it is in the kernel of δ, hence <math>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)</math> 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 <math>\sum_{j=0}^{q-1} (-1)^j \mathrm{res}^{|\partial_j \sigma|}_{|\sigma|} f (\partial_j \sigma) = 0</math> holds. In particular, a 1-cochain ''f'' is a 1-cocycle if
:<math>\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.</math>

====Coboundary====
A ''q''-cochain is called a ''q''-coboundary if it is in the image of ''δ'' and <math>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)</math> 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 <math>\forall_{\{A, B\} \subset \mathcal{U}}, U:=A \cap B, f(U) = (\delta h)(U) = h(A)|_U - h(B)|_U.</math>

===Cohomology===

The '''Čech cohomology''' of <math>\mathcal{U}</math> with values in <math>\mathcal{F}</math> is defined to be the cohomology of the cochain complex <math>(C^{\textbf{.}}(\mathcal{U}, \mathcal{F}), \delta)</math>. Thus the ''q''th Čech cohomology is given by
:<math>\check{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})</math>.

The Čech cohomology of ''X'' is defined by considering [[Cover (topology)#Refinement|refinement]]s of open covers. If <math>\mathcal{V}</math> is a refinement of <math>\mathcal{U}</math> then there is a map in cohomology <math>\check{H}^*(\mathcal U,\mathcal F) \to \check{H}^*(\mathcal V,\mathcal F).</math>
The open covers of ''X'' form a [[directed set]] under refinement, so the above map leads to a [[direct system (mathematics)|direct system]] of abelian groups. The '''Čech cohomology''' of ''X'' with values in ''F'' is defined as the [[direct limit]] <math>\check{H}(X,\mathcal F) := \varinjlim_{\mathcal U} \check{H}(\mathcal U,\mathcal F)</math> of this system.

The Čech cohomology of ''X'' with coefficients in a fixed abelian group ''A'', denoted <math>\check{H}(X;A)</math>, is defined as <math>\check{H}(X,\mathcal{F}_A)</math> where <math>\mathcal{F}_A</math> 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 <math>\{x|\rho_i(x)>0\}</math> is contained in some element of the cover. If ''X'' is [[paracompact]] and [[Hausdorff space|Hausdorff]], then numerable Čech cohomology agrees with the usual Čech cohomology.

==Relation to other cohomology theories==

If <math>X</math> is [[homotopy equivalent]] to a [[CW complex]], then the Čech cohomology <math>\check{H}^{*}(X;A)</math> is [[naturally isomorphic]] to the [[singular homology|singular cohomology]] <math> H^*(X;A) \,</math>. If ''X'' is a [[differentiable manifold]], then <math>\check{H}^*(X;\mathbb{R})</math> 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 [[topologist's sine curve|closed topologist's sine curve]], then <math>\check{H}^1(X;\mathbb{Z})=\mathbb{Z},</math> whereas <math>H^1(X;\mathbb{Z})=0.</math>

If ''X'' is a differentiable manifold and the cover <math>\mathcal{U}</math> of ''X'' is a "good cover" (''i.e.'' all the sets ''U''α are [[Contractible space|contractible]] to a point, and all finite intersections of sets in <math>\mathcal{U}</math> are either empty or contractible to a point), then
<math>\check{H}^{*}(\mathcal U;\mathbb{R})</math> 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]].

==In algebraic geometry==
Čech cohomology can be defined more generally for objects in a [[site (mathematics)|site]] '''C''' endowed with a topology. This applies, for example, to the Zariski site or the etale site of a [[scheme (mathematics)|scheme]] ''X''. The Čech cohomology with values in some [[sheaf (mathematics)|sheaf]] ''F'' is defined as
:<math>\check H^n (X, F) := \varinjlim_{\mathcal U} \check H^n(\mathcal U, F).</math>
where the [[colimit]] runs over all coverings (with respect to the chosen topology) of ''X''. Here <math>\check H^n(\mathcal U, F)</math> is defined as above, except that the ''r''-fold intersections of open subsets inside the ambient topological space are replaced by the ''r''-fold [[fiber product]]
:<math>\mathcal U^{\times^r_X} := \mathcal U \times_X \dots \times_X \mathcal U.</math>

As in the classical situation of topological spaces, there is always a map
:<math>H^n(X, F) \rightarrow \check H^n(X, F)</math>
from [[sheaf cohomology]] to Čech cohomology. It is always an isomorphism in degrees ''n'' = 0 and 1, but may fail to be so in general. For the [[Zariski topology]] on a [[Noetherian topological space|Noetherian]] [[separated scheme]], Čech and sheaf cohomology agree for any [[quasi-coherent sheaf]]. For the [[etale topology]], the two cohomologies agree for any sheaf, provided that any finite set of points in the base scheme ''X'' are contained in some open affine subscheme. This is satisfied, for example, if ''X'' is [[quasi-projective variety|quasi-projective]] over an [[affine scheme]].<ref>{{Citation | last1=Milne | first1=James S. | title=Étale cohomology | url=http://books.google.com/books?isbn=978-0-691-08238-7 | publisher=[[Princeton University Press]] | series=Princeton Mathematical Series | isbn=978-0-691-08238-7 | id={{MR|559531}} | year=1980 | volume=33}}, section III.2</ref>

The possible difference between Cech cohomology and sheaf cohomology is a motivation for the use of [[hypercovering]]s: these are more general objects than the Cech [[nerve (category theory)|nerve]]
:<math>N_X \mathcal U : \dots \rightarrow \mathcal U \times_X \mathcal U \times_X \mathcal U \rightarrow \mathcal U \times_X \mathcal U \rightarrow \mathcal U.</math>
A hypercovering ''K''&lowast; of ''X'' is a [[simplicial object]] in '''C''', i.e., a collection of objects ''K''''n'' together with boundary and degeneracy maps. Applying a sheaf ''F'' to ''K''&lowast; yields a [[simplicial abelian group]] ''F''(''K''&lowast;) whose ''n''-th cohomology group is denoted ''H''''n''(''F''(''K''&lowast;)). (This group is the same as <math>\check H^n(\mathcal U, F)</math> in case ''K'' equals <math>N_X \mathcal U </math>.) Then, it can be shown that there is a canonical isomorphism
:<math>H^n (X, F) = \varinjlim_{K_*} H^n(F(K_*)),</math>
where the colimit now runs over all hypercoverings.<ref>{{Citation | last1=Artin | first1=Michael | author1-link=Michael Artin | last2=Mazur | first2=Barry | author2-link=Barry Mazur | title=Etale homotopy | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Lecture Notes in Mathematics, No. 100 | year=1969}}, Theorem 8.16</ref>

==References==
<references />

*{{cite book | last = Bott | first = Raoul | authorlink = Raoul Bott | coauthors = Loring Tu | title = Differential Forms in Algebraic Topology | year = 1982 | publisher = Springer | location = New York | isbn = 0-387-90613-4}}
*{{cite book | last = Hatcher | first = Allen | authorlink = Allen Hatcher | year = 2002 | title = Algebraic Topology | publisher = Cambridge University Press | isbn = 0-521-79540-0 | url = http://www.math.cornell.edu/~hatcher/AT/ATpage.html}}
*{{cite book | last = Wells | first = Raymond | authorlink = Raymond O'Neil Wells, Jr. | year = 1980 | title = Differential Analysis on Complex Manifolds | publisher = Springer-Verlag}} ISBN 0-387-90419-0. ISBN 3-540-90419-0. Chapter 2 Appendix A

{{DEFAULTSORT:Cech cohomology}}
[[Category:Algebraic topology]]
[[Category:Cohomology theories]]
[[Category:Homology theory]]

Creation and annihilation operators

2012-05-18T21:46:42Z

192.35.35.34: /* Derivation for quantum harmonic oscillator */ The signs in the expressions for the Hamiltonian using operators (aa^+) and (a^+a) were interchanged and are now correct. Refer to the Wikipedia page on "Quantization_of_the_electromagnetic_field".

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Rotation matrix

Creation and annihilation operators