Cohomology

Cohomology is a mathematical concept that is used in many areas, originally in algebraic topology . The word cohomology is used in two different meanings: on the one hand for the basic construction of the cohomology of any coquette complex , on the other hand for the application of this basic construction to specific coquette complexes, which one can e.g. B. from a manifold ( De-Rham cohomology ), a topological space ( singular cohomology ), a simplicial complex ( simplicial cohomology ) or a group ( group cohomology ) . A general construction method for generalized cohomology theories uses so-called spectra .

The concept was developed independently by Andrei Kolmogorow and James W. Alexander in the 1930s .

Cohomology of a coquette complex

Basic construction

Be a coquette complex . That means: ${\displaystyle (C^{\bullet },d^{\bullet })}$

1. for each there is an Abelian group (in general: an object of an Abelian category ) ${\displaystyle k\in \mathbb {Z} }$ ${\displaystyle C^{k}}$
2. for each there is a group homomorphism (in general: a morphism), called a differential or Korand operator ${\displaystyle k\in \mathbb {Z} }$ ${\displaystyle d^{k}\colon C^{k}\to C^{k+1}}$
3. each counts as a picture ${\displaystyle k\in \mathbb {Z} }$ ${\displaystyle d^{k}\circ d^{k-1}=0}$ ${\displaystyle C^{k-1}\to C^{k+1}}$

From this one can construct the following groups:

• ${\displaystyle Z^{k}=\ker(d^{k})}$. Elements of hot coccycles. ${\displaystyle Z^{k}}$ ${\displaystyle k}$
• ${\displaystyle B^{k}={\text{im}}(d^{k-1})}$. Elements of hot braids. Because of condition 3. is , every Korand is a cocycle. Two coccycles are called cohomologous if their difference is a Korand. To be cohomologous is an equivalence relation . ${\displaystyle B^{k}}$ ${\displaystyle k}$ ${\displaystyle B^{k}\subseteq Z^{k}}$
• ${\displaystyle H^{k}(C^{\bullet },d^{\bullet })=Z^{k}/B^{k}}$, called the -th cohomology group of . Their elements are equivalence classes of coccycles for the equivalence relation "cohomologous". Exactly then applies when is exact at the point . The cohomology group is therefore a measure of inaccuracy. ${\displaystyle k}$ ${\displaystyle (C^{\bullet },d^{\bullet })}$ ${\displaystyle H^{k}(C^{\bullet },d^{\bullet })=0}$ ${\displaystyle (C^{\bullet },d^{\bullet })}$ ${\displaystyle k}$

At this point, cohomology and homology are almost synonymous: For a coquette complex , with , a chain complex, and . ${\displaystyle (C^{\bullet },d^{\bullet })}$ ${\displaystyle ({\tilde {C}}_{\bullet },{\tilde {d}}_{\bullet })}$ ${\displaystyle {\tilde {C}}_{k}=C^{-k}}$ ${\displaystyle {\tilde {d}}_{k}=d^{-k}}$ ${\displaystyle H^{k}(C^{\bullet },d^{\bullet })=H_{-k}({\tilde {C}}_{\bullet },{\tilde {d}}_{\bullet })}$

Are and two coquette complexes and a chain map , d. H. applies to all , gives functorial homomorphisms . If two chain figures are homotopic , is . ${\displaystyle (C_{1}^{\bullet },d_{1}^{\bullet })}$ ${\displaystyle (C_{2}^{\bullet },d_{2}^{\bullet })}$ ${\displaystyle f^{\bullet }:C_{1}^{\bullet }\to C_{2}^{\bullet }}$ ${\displaystyle f^{k+1}\circ d_{1}^{k}=d_{2}^{k}\circ f^{k}}$ ${\displaystyle k}$ ${\displaystyle f_{*}^{k}:H^{k}(C_{1}^{\bullet },d_{1}^{\bullet })\to H^{k}(C_{2}^{\bullet },d_{2}^{\bullet })}$ ${\displaystyle f^{\bullet },g^{\bullet }:C_{1}^{\bullet }\to C_{2}^{\bullet }}$ ${\displaystyle f_{*}=g_{*}}$

The long exact sequence

Given a short exact sequence of coquette complexes:

${\displaystyle 0\to C_{1}^{\bullet }{\stackrel {f^{\bullet }}{\longrightarrow }}C_{2}^{\bullet }{\stackrel {g^{\bullet }}{\longrightarrow }}C_{3}^{\bullet }\to 0}$

( these are left out for the sake of clarity). This means and are chain pictures, and each is ${\displaystyle d_{i}^{\bullet }}$ ${\displaystyle f^{\bullet }}$ ${\displaystyle g^{\bullet }}$ ${\displaystyle k}$

${\displaystyle 0\to C_{1}^{k}{\stackrel {f^{k}}{\longrightarrow }}C_{2}^{k}{\stackrel {g^{k}}{\longrightarrow }}C_{3}^{k}\to 0}$

exactly. Then there are so called compound homomorphisms , so that the sequence ${\displaystyle \delta ^{k}:H^{k}(C_{3}^{\bullet })\to H^{k+1}(C_{1}^{\bullet })}$

${\displaystyle \dots {\stackrel {\delta ^{k-1}}{\longrightarrow }}H^{k}(C_{1}^{\bullet }){\stackrel {f_{*}^{k}}{\longrightarrow }}H^{k}(C_{2}^{\bullet }){\stackrel {g_{*}^{k}}{\longrightarrow }}H^{k}(C_{3}^{\bullet }){\stackrel {\delta ^{k}}{\longrightarrow }}H^{k+1}(C_{1}^{\bullet }){\stackrel {f_{*}^{k+1}}{\longrightarrow }}H^{k+1}(C_{2}^{\bullet }){\stackrel {g_{*}^{k+1}}{\longrightarrow }}H^{k+1}(C_{3}^{\bullet }){\stackrel {\delta ^{k+1}}{\longrightarrow }}\dots }$

is exact.

${\displaystyle \delta ^{k}}$can be constructed like this: Let (coccycles in ). Because is surjective, it has an archetype . It is so is for one . Now is , but because is injective, it follows that there is a -Cocycle, and one can set. (For a complete proof the proof of the well-definedness is missing, that is, that a Korand is if there is a Korand.) Arguments of this type are called diagram hunting . ${\displaystyle a\in Z_{3}^{k}}$ ${\displaystyle (C_{3}^{\bullet },d_{3}^{\bullet })}$ ${\displaystyle g^{k}}$ ${\displaystyle a}$ ${\displaystyle b\in C_{2}^{k}}$ ${\displaystyle g^{k+1}d_{2}^{k}b=d_{3}^{k}g^{k}b=d_{3}^{k}a=0}$ ${\displaystyle d_{2}^{k}b=f^{k+1}c}$ ${\displaystyle c\in C_{1}^{k+1}}$ ${\displaystyle f^{k+2}d_{1}^{k+1}c=d_{2}^{k+1}f^{k+1}c=d_{2}^{k+1}d_{2}^{k}b=0}$ ${\displaystyle f^{k+2}}$ ${\displaystyle d_{1}^{k+1}c=0}$ ${\displaystyle c}$ ${\displaystyle (k+1)}$ ${\displaystyle \delta ^{k}[a]=[c]}$ ${\displaystyle c}$ ${\displaystyle a}$

The snake lemma is a special case of this construction.

Derived categories

In many applications there is no clearly defined coquette complex whose cohomology one would like to form, but one must or at least can make choices that do not affect the end result, the cohomology. The derived category is a modification of the category of coquette complexes in which these different choices are already isomorphic, so that the final step, the formation of the cohomology, is no longer necessary to achieve uniqueness.

Cohomology theories

General

A typical cohomology theory takes the form of groups for , where is a space and in the simplest case is an Abelian group. Other common properties are: ${\displaystyle H^{k}(X,A)}$ ${\displaystyle k\geq 0}$ ${\displaystyle X}$ ${\displaystyle A}$

• ${\displaystyle H^{k}(X,A)}$is contravariant in and covariant in ${\displaystyle X}$ ${\displaystyle A}$
• There is a long exact cohomology sequence.
• There are products so that when there is a ring it becomes a graduated ring . ${\displaystyle H^{p}(X,A)\times H^{q}(X,B)\to H^{p+q}(X,A\otimes B)}$ ${\displaystyle \textstyle \bigoplus _{k\geq 0}H^{k}(X,A)}$ ${\displaystyle A}$

While many of the theories of cohomology are interrelated and often give similar results in cases where several theories are applicable, there is no all-encompassing definition.

A few more examples follow.

De Rham cohomology

→ Main article : De Rham cohomology

Be a smooth manifold . The De Rham cohomology of is the cohomology of the complex ${\displaystyle X}$ ${\displaystyle H_{\text{dR}}^{\bullet }(X)}$ ${\displaystyle X}$

${\displaystyle 0\to C^{\infty }(X){\stackrel {\text{d}}{\longrightarrow }}\Omega ^{1}(X){\stackrel {\text{d}}{\longrightarrow }}\Omega ^{2}(X){\stackrel {\text{d}}{\longrightarrow }}\dots }$

(supplemented to the left with zeros), where the global differential forms are of degree and the Cartan derivative . ${\displaystyle \Omega ^{k}(X)}$ ${\displaystyle k}$ ${\displaystyle {\text{d}}}$

If there is a smooth map between smooth manifolds, the retraction of differential forms interchanges with the Cartan derivative, i.e. defines a chain map that induces homomorphisms . ${\displaystyle f\colon X\to Y}$ ${\displaystyle f^{*}\colon \Omega ^{k}(Y)\to \Omega ^{k}(X)}$ ${\displaystyle f^{*}}$ ${\displaystyle H_{\text{dR}}^{k}(Y)\to H_{\text{dR}}^{k}(X)}$

The umbrella product of differential forms induces a product structure . ${\displaystyle H_{\text{dR}}^{*}(X)}$

Vector bundle with flat connection are an appropriate coefficient category for the De Rham cohomology.

Singular cohomology

→ Main article : Singular cohomology

Let be a topological space and an Abelian group. Continue to be the standard simplex. The side faces of a simplex are themselves again simplices, corresponding to the embedding , for . Now be the set of continuous mappings in a topological space . By linking with you get images . In the next step, let the free Abelian group be on the set , and be defined by for . It is , therefore is a chain complex, the singular chain complex of . Finally , if and , one obtains the singular coquette complex of , whose cohomology is the singular cohomology . ${\displaystyle X}$ ${\displaystyle A}$ ${\displaystyle \Delta ^{k}=\{x\in \mathbb {R} ^{k+1}:x_{0}+\dots +x_{k}=1,\ {\text{alle}}\ x_{i}\geq 0\}}$ ${\displaystyle k}$ ${\displaystyle \partial _{k}^{i}\colon \Delta ^{k-1}\to \Delta ^{k}}$ ${\displaystyle x\mapsto (x_{0},\dots ,x_{i-1},0,x_{i},\dots ,x_{k-1})}$ ${\displaystyle i=0,\dots ,k}$ ${\displaystyle X_{k}}$ ${\displaystyle \Delta ^{k}\to X}$ ${\displaystyle X}$ ${\displaystyle \partial _{k}^{i}}$ ${\displaystyle X_{k}\to X_{k-1}}$ ${\displaystyle C_{k}}$ ${\displaystyle X_{k}}$ ${\displaystyle \partial _{k}\colon C_{k}\to C_{k-1}}$ ${\displaystyle \textstyle \partial _{k}(\sigma )=\sum _{i=0}^{k}(-1)^{i}(\sigma \circ \partial _{k}^{i})}$ ${\displaystyle \sigma :\Delta ^{k}\to X}$ ${\displaystyle \partial _{k-1}\partial _{k}=0}$ ${\displaystyle (C_{\bullet },\partial _{\bullet })}$ ${\displaystyle X}$ ${\displaystyle C^{k}={\text{Hom}}(C_{k},A)}$ ${\displaystyle d^{k}\colon C^{k}\to C^{k+1}}$ ${\displaystyle d^{k}s=s\circ \partial _{k}}$ ${\displaystyle X}$ ${\displaystyle H^{k}(X,A)}$

${\displaystyle A}$is called the coefficient ring of the cohomology theory.

The cohomology with coefficients is referred to as integral cohomology . ${\displaystyle A=\mathbb {Z} }$

For a continuous mapping , a chain mapping is obtained , from this a chain mapping and thus a functional homomorphism . ${\displaystyle X\to Y}$ ${\displaystyle C_{\bullet }(X)\to C_{\bullet }(Y)}$ ${\displaystyle C^{\bullet }(Y,A)\to C^{\bullet }(X,A)}$ ${\displaystyle H^{k}(Y,A)\to H^{k}(X,A)}$

For a subspace is a subcomplex of , and with one obtains a short exact sequence of chain complexes, which by using of gives a short exact sequence of coquette complexes: ${\displaystyle Y\subseteq X}$ ${\displaystyle C_{\bullet }(Y)}$ ${\displaystyle C_{\bullet }(X)}$ ${\displaystyle C_{\bullet }(X,Y):=C_{\bullet }(X)/C_{\bullet }(Y)}$ ${\displaystyle {\text{Hom}}(?,A)}$

${\displaystyle 0\to C^{\bullet }(X,Y,A)\to C^{\bullet }(X,A)\to C^{\bullet }(Y,A)\to 0}$

According to the general construction, this results in a long, exact cohomology sequence:

${\displaystyle \dots \to H^{k}(X,Y,A)\to H^{k}(X,A)\to H^{k}(Y,A)\to H^{k+1}(X,Y,A)\to H^{k+1}(X,A)\to H^{k+1}(Y,A)\to \dots }$

The so-called universal coefficient theorem can be used to compare the cohomology groups and for different coefficient groups . ${\displaystyle H^{k}(X,A)}$ ${\displaystyle H^{k}(X,B)}$ ${\displaystyle A,B}$

Samuel Eilenberg and Norman Steenrod have given a list of simple properties that a cohomology theory for topological spaces should have, the Eilenberg-Steenrod axioms . There is essentially only one cohomology theory that satisfies the axioms, and singular cohomology is one.

Group cohomology

→ Main article : group cohomology

The group cohomology has two arguments: a group and one - module . In the coefficient argument , the cohomology is covariant and there is a long exact cohomology sequence. In the argument , the cohomology is contravariant in a suitable sense, e.g. B. if one chooses a fixed Abelian group with trivial operation as the coefficient. The relationship between the cohomology of a group and a factor group or a normal divisor is described by the Hochschild-Serre spectral sequence . ${\displaystyle H^{k}(G,A)}$ ${\displaystyle G}$ ${\displaystyle G}$ ${\displaystyle A}$ ${\displaystyle A}$ ${\displaystyle G}$

Cohomology ring

The direct sum becomes a graduated commutative ring with the cup product , the so-called cohomology ring of room X. ${\displaystyle \oplus _{i=0}^{\infty }H^{i}(X;R)}$

Non-Abelian cohomology

Various constructions which provide a cohomology for non-Abelian coefficients, but which are mostly limited to and , do not fit into the scheme of the basic construction given above , e.g. B. in group or sheaf homology. Jean Giraud worked out an interpretation of the nonabelian cohomology for with the help of tanning . ${\displaystyle H^{k}(X,G)}$ ${\displaystyle k=0}$ ${\displaystyle k=1}$ ${\displaystyle k=2}$

See also

• Spectrum (topology) for the construction of generalized cohomology theories

literature

• IM Gelfand , Y. Manin : Homological Algebra (= Encyclopaedia of Mathematical Sciences. Volume 138: Algebra. V). 1st edition, 2nd printing. Springer, Berlin et al. 1999, ISBN 3-540-65378-3 .
• Jean Giraud : Cohomologie non abélienne (= The basic teachings of the mathematical sciences in individual presentations with special consideration of the areas of application 179). Springer, Berlin et al. 1971, ISBN 3-540-05307-7 .
• Allen Hatcher: Algebraic Topology . Cambridge University Press, Cambridge et al. 2002, ISBN 0-521-79540-0 .
• Charles A. Weibel: An introduction to homological algebra (= Cambridge Studies in Advanced Mathematics 38). Cambridge University Press, Cambridge et al. 2003, ISBN 0-521-55987-1 .