Cohomology

Cohomology is a mathematical concept that comes in many areas used originally in algebraic topology. The word cohomology is used here in two different meanings: on the one hand, for the basic structure of the cohomology of any Kokettenkomplexes, on the other hand, for the application of this basic construction to concrete Kokettenkomplexe that one example from a manifold ( De Rham cohomology ), a topological space ( singular cohomology ), a simplicial ( simplicial cohomology ) or a group ( Gruppenkohomologie ) receives. A general construction method for generalized cohomology theories used so-called spectra.

• 2.1 General
• 2.2 De Rham cohomology
• 2.3 Singular cohomology
• 2.4 Gruppenkohomologie

Cohomology of a Kokettenkomplexes

Basic construction

Be a Kokettenkomplex. This means:

From this one can construct the following groups:

• . Elements of hot - Kozykeln.
• . Elements of hot - Koränder. Because of condition 3 is every coboundary is a cocycle so. Two hot kohomolog cocycle if their difference is a coboundary. To be Kohomolog, is an equivalence relation.
• Called the -th cohomology group of. Its elements are equivalence classes of Kozykeln for the equivalence relation " kohomolog ". If and only if it is at the exact location. The cohomology group is thus a measure of non- exactness.

At this point, cohomology and homology are still virtually synonymous: For a Kokettenkomplex is a chain complex, and.

Are Kokettenkomplexe and two and a chain map, ie applies to everyone, we obtain functorial homomorphisms. Are two pictures chain homotopic is.

The long exact sequence

Be a short exact sequence of Kokettenkomplexen added:

( which are omitted for clarity ). This means and are chain images and for each

Exactly. Then there are the so-called Verbindungshomomorphismen so that the sequence

Is exact.

Can be constructed: Let ( cocycle in ). Since is surjective, has a preimage. It is, so is for one. Now, however, because is injective, it follows, then, is a - cocycle, and you can set. (Still missing to a complete proof of the proof of well- definedness, ie that is a coboundary if a coboundary is. ) Arguments of this type are called diagram chase.

The Schlangenlemma is a special case of this construction.

Derived Categories

In many applications do not explicitly identified Kokettenkomplex is given, whose cohomology we want to make, but you have to, or at least can make choices, but do not affect the final result, the cohomology. The derived category is a modification of the category of Kokettenkomplexe in which these different options are already isomorphic, so that the last step, the formation of the cohomology is no longer necessary in order to achieve uniqueness.

Cohomology theories

General

Typical cohomology is in the form of groups, wherein a space, and in the simplest case an Abelian group. Other common features are:

• Is contravariant and covariant in in
• There is a long exact Kohomologiesequenz.
• There are products, so that to a graduated ring when even a ring.

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

Here are a few examples.

De Rham cohomology

Let be a smooth manifold. The de Rham cohomology of the cohomology of the complex

( supplemented with zeros to the left ) wherein the global differential forms of degree and Cartan derivative is.

Is a smooth map between smooth manifolds, reversed the withdrawal of differential forms with the Cartan derivative, ie defines a chain map, the induced homomorphisms.

The wedge product of differential forms induces a product structure.

Vector bundle with flat connection are a suitable coefficient category for the de Rham cohomology.

Singular cohomology

Be a topological space and an abelian group. Be on the Standard - Simplex. The side faces of a simplex are themselves simplices, corresponding to the embeddings for. Now let the set of continuous maps in a topological space. By chaining you get pictures. The next step is the free abelian group on the set, and defined by for. It is, therefore, a chain complex, the singular chain complex of. If, finally, and, we obtain the singular Kokettenkomplex of whose cohomology is the singular cohomology.

Is referred to as the coefficients of the cohomology ring.

As an integral cohomology of the cohomology is denoted by coefficients.

For a continuous map gives a chain map, from a chain map and thus a functorial homomorphism.

For a subspace is a subcomplex of, and one obtains a short exact sequence of chain complexes, which results through the application of a short exact sequence of Kokettenkomplexen:

This is obtained according to the general construction a long exact Kohomologiesequenz:

To compare the cohomology groups and for different groups of coefficients one can use the so-called universal Koeffiziententheorem.

Samuel Eilenberg and Norman Steenrod have specified a list of simple features that should have a cohomology theory for topological spaces, the Eilenberg - Steenrod axioms. There are essentially only a cohomology theory satisfying the axioms, and singular cohomology is one such.

Gruppenkohomologie

The Gruppenkohomologie has two arguments: a group and a module. The coefficient of the cohomology argument is covariant, and there is a long exact Kohomologiesequenz. In argument, the cohomology is contravariant in a suitable sense, for example, if a fixed abelian group is chosen as the coefficients with trivial operation. The relationship between the cohomology of a group and a factor group or a normal subgroup is described by the Hochschild -Serre spectral sequence.

Cohomology ring

The direct sum is the cup product to a graded commutative ring, called the cohomology ring of the space X.

Nonabelian cohomology

Not in the scheme of the above basic design to fit various designs that provide a non-abelian cohomology of coefficients, but most are limited and, for example, in groups or Garbenkohomologie. Jean Giraud has developed an interpretation of the nonabelian cohomology for using tanning.