Homology (mathematics)

Homology (Greek: όμος, homos = same, λόγος, logos = meaning ) is a mathematical object. It was developed in the field of algebraic topology. Later homologies were also considered as purely algebraic objects, which developed the branch of homological algebra. Homology is a sequence of mathematical objects, the homology groups.

In the area of ​​algebraic topology are the homologies or the homology groups invariants of a topological space, so they help to distinguish topological spaces.

Construction of homology groups

It is generally as follows: A mathematical object is a chain complex is initially allocated, including information about. A chain complex is a sequence of modules connected by homomorphisms, so that the sequential execution of any two of these figures, the zero picture is: for each n This means that the image of the ( n 1 ) th image always in the core of the nth image is included. Now we define the n-th homology group of X as the quotient module

A chain complex is exactly when the image of the (n 1) th picture is always the core of the n-th image; the homology groups of measure ie, "how inexact " the associated chain complex.

Examples

The first example comes from algebraic topology: simplicial homology of a simplicial complex. Here is the free module over the n-dimensional oriented simplices of. The pictures hot pictures edge and form the simplex with vertices

To the alternating sum of the " border areas "

From.

For modules over a body ( ie vector spaces ) describes the dimension of the n-th homology group of the number of n-dimensional holes.

With this example, one can define a simplicial homology for every topological space. The chain complex for is defined so that the free module over all continuous maps from the n-dimensional unit simplex is after. The homomorphisms arise from the simplicial boundary maps.

In homological algebra one uses homology to define derived functors. One considers there an additive functor and a module. The chain complex for is constructed as follows: is a free module and an epimorphism, is a free module, which is to have the property that there exists an epimorphism, therefore obtains a sequence of free modules and homomorphisms, and by the application of a chain complex. The n-th homology of this complex is dependent, as can be shown, only up and down. One writes and calls the n-th derived functor of.

Homology functors

The chain complexes form a category: A morphism - they say: a chain image - from the chain complex in the chain complex is a sequence of module homomorphisms such that for each n, the n-th homology group can be of a functor from the category of chain complexes in the category modules over interpret.

If the chain complex depends on functorial ( ie each morphism induces a chain map from the chain complex of the by), then the functors of the category, is one of the, in the category of modules.

One difference between homology and cohomology is that the chain complexes in the cohomology of contravariant dependent and therefore the homology groups ( which are then called cohomology groups and are referred to in this context ) are contravariant functors. Furthermore, one has a canonical ring structure usually on the graduate cohomology, there is not anything like it on the level of homology.

Properties

Is a chain complex, so that all ( but finitely many ) are zero and all others are finitely generated free modules, then you can the Euler characteristic

Define. One can show that the Euler characteristic can also be calculated on the homology level:

In algebraic topology provides the two ways, the invariant for the object from which the chain complex was generated to calculate.

Each short exact sequence

Of chain complexes gives a long exact sequence of homology groups

All pictures of this exact sequence are induced by the maps between the chain complexes, except the figures, the connecting homomorphisms are called and whose existence is proved by the Schlangenlemma.