Singular cohomology

The singular cohomology is a method from the mathematical branch of algebraic topology that any topological space a sequence of abelian groups assigns. To put it clearly, it counts the different-dimensional holes in a room. The singular cohomology is defined as the cohomology of the singular coquette complex . Just like singular homology , it is an invariant of the topological space on which it is based. In contrast to singular homology, however, it has the advantage that the sequence of its cohomology groups together with the cup product form a ring .

Singular coquette complex

Let be a topological space and an Abelian group . With is singular chain complex of designated. For every natural number define ${\displaystyle X}$ ${\displaystyle G}$ ${\displaystyle C_{p}(X)}$ ${\displaystyle X}$ ${\displaystyle p>0}$

${\displaystyle C^{p}(X;G):=\operatorname {Hom} (C_{p}(X),G)\,,}$

where the group of group homomorphisms is from to . The elements of are called singular coquettes with coefficients in or short coquettes. ${\displaystyle \operatorname {Hom} (C_{p}(X),G)}$ ${\displaystyle C_{p}(X)}$ ${\displaystyle G}$ ${\displaystyle C^{p}(X;G)}$ ${\displaystyle G}$ ${\displaystyle p}$

The boundary operator of the singular chain complex induces a boundary operator ${\displaystyle \partial _{p}\colon C_{p}(X)\to C_{p-1}(X)}$

${\displaystyle \delta _{p}\colon C^{p}(X;G)\to C^{p+1}(X;G)}$,

the Korand operator is called. It can be characterized by what follows. This gives the coquette complex ${\displaystyle \delta \phi (c)=\phi (\partial c)}$ ${\displaystyle \delta _{p+1}\delta _{p}=0}$

${\displaystyle \ldots \longrightarrow C^{p-1}(X;G){\stackrel {\delta _{p-1}}{\longrightarrow }}C^{p}(X;G){\stackrel {\delta _{p}}{\longrightarrow }}C^{p+1}(X;G){\stackrel {\delta _{p+1}}{\longrightarrow }}\ldots }$,

the singular coquette complex is called.

Singular cohomology

The singular cohomology is now the cohomology with respect to the singular coquette complex. A -coquette is called a cocycle if it is true and a Korand if there is one with . In the following, the group of coccycles and the group of coranders are designated with. Both groups are subgroups of . The singular cohomology with coefficients in is then defined as the quotient group ${\displaystyle p}$ ${\displaystyle \phi }$ ${\displaystyle \delta \phi =0}$ ${\displaystyle \psi \in C^{p-1}(X;G)}$ ${\displaystyle \delta \psi =\phi }$ ${\displaystyle Z^{p}(X;G)}$ ${\displaystyle B^{p}(X;G)}$ ${\displaystyle C^{p}(X;G)}$ ${\displaystyle H^{p}(X;G)}$ ${\displaystyle G}$

${\displaystyle H^{p}(X;G):=Z^{p}(X;G)/B^{p}(X;G)}$.

The following interpretation of the terms "Kozykel" and "Korand" results directly from the definitions. A coquette is a cocycle if and only if it disappears on the edges, i.e. it applies to all . A coquette is a Korand when it disappears on cycles, so for everyone with it . In particular, two cocycles represent the same cohomology class if they have the same values ​​on all cycles, i.e. for all with . ${\displaystyle \phi :C_{p}(X)\rightarrow G}$ ${\displaystyle \phi }$ ${\displaystyle \phi (\partial c)=0}$ ${\displaystyle c\in C_{p+1}(X)}$ ${\displaystyle \phi (z)=0}$ ${\displaystyle z\in C_{p}(X)}$ ${\displaystyle \partial z=0}$ ${\displaystyle \phi ,\phi ^{\prime }:C_{p}(X)\rightarrow \mathbb {R} }$ ${\displaystyle \phi (z)=\phi ^{\prime }(z)}$ ${\displaystyle z\in C_{p}(X)}$ ${\displaystyle \partial z=0}$

The elements of are called cohomology classes (with coefficients in ). ${\displaystyle H^{*}(X;G)}$ ${\displaystyle G}$

properties

Contravariant functor

The singular cohomology is a contravariant functor from the category of topological spaces to the category of Abelian groups. So the functor has the following two properties. Let and be two continuous mappings between topological spaces. With and induced Kohomologiehomomorphismen be called. Then applies ${\displaystyle f\colon X\to Y}$ ${\displaystyle g\colon Y\to Z}$ ${\displaystyle f^{*}}$ ${\displaystyle g^{*}}$

${\displaystyle (g\circ f)^{*}=f^{*}\circ g^{*}}$.

In addition, the cohomology homomorphism induced by the identical mapping is again the identical mapping.

Long exact sequence

For a topological subspace , the singular complex is a subcomplex of , and with one gets a short exact sequence of chain complexes, which by using of gives a short exact sequence of coquette complexes: ${\displaystyle Y\subseteq X}$ ${\displaystyle C_{\bullet }(Y)}$ ${\displaystyle C_{\bullet }(X)}$ ${\displaystyle C_{\bullet }(X,Y):=C_{\bullet }(X)/C_{\bullet }(Y)}$ ${\displaystyle \operatorname {Hom} (\cdot ,G)}$

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

This results in the long, exact cohomology sequence using methods of homological algebra

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

The groups are called relative singular cohomology groups. ${\displaystyle H^{k}(X,Y,G)}$

Topological invariant

The singular cohomology groups are topological invariants of the underlying space. So let and be two topological spaces and a homeomorphism , then for all and for every Abelian group the cohomology groups and are isomorphic . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle p}$ ${\displaystyle G}$ ${\displaystyle H^{p}(X;G)}$ ${\displaystyle H^{p}(Y;G)}$

Homotopy invariance

Homotopic mappings induce the same mappings . Homotopy equivalences induce isomorphisms . ${\displaystyle f,g\colon X\to Y}$ ${\displaystyle f^{*},g^{*}\colon H^{*}(Y,G)\to H^{*}(X,G)}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle f^{*}\colon H^{*}(Y,G)\to H^{*}(X,G)}$

Mayer-Vietoris sequence

Let be a (not disjoint) decomposition with ${\displaystyle X=A\cup B}$

${\displaystyle X=int(A)\cup int(B)}$.

Then there is an exact sequence

${\displaystyle \cdots \rightarrow H^{n}(X;G)\rightarrow H^{n}(A;G)\oplus H^{n}(B;G)\rightarrow H^{n}(A\cap B;G)\rightarrow H^{n+1}(X;G)\rightarrow \cdots }$.

De Rham cohomology and simplicial cohomology

If is a differentiable manifold , then is isomorphic to De Rham cohomology . If is homeomorphic to the geometric realization of a simplicial complex , then is isomorphic to the simplicial cohomology . ${\displaystyle X}$ ${\displaystyle H^{*}(X,\mathbb {R} )}$ ${\displaystyle H_{dR}^{*}(X)}$ ${\displaystyle X}$ ${\displaystyle \mid K\mid }$ ${\displaystyle K}$ ${\displaystyle H^{*}(X,G)}$ ${\displaystyle H^{*}(K,G)}$

Cup product

→ Main article : Cup product

In contrast to singular homology, it is possible with singular cohomology classes to define an associative, graduated commutative and distributive product on them. This is called the cup product and induces a ring structure on the cohomology groups .

See also

• Universal coefficient theorem

Individual evidence

1. ^ ^{A b} John M. Lee: Introduction to Topological Manifolds (= Graduate Texts in Mathematics 202). Springer-Verlag, New York NY et al. 2000, ISBN 0-387-98759-2 , p. 329.
2. ^ ^{A b} John M. Lee: Introduction to Topological Manifolds (= Graduate Texts in Mathematics 202). Springer-Verlag, New York NY et al. 2000, ISBN 0-387-98759-2 , p. 330.