# Homology theory

A homology ( ancient Greek ὁμός homos , "similar, equal", and λόγος logos , here: "relationship, analogy, proportion") is a mathematical object . It is a sequence of mathematical objects, the homology groups . The singular homology is one of the most important characteristics of a homology . Homologies were developed in the field of algebraic topology . Later they were also viewed as purely algebraic objects, from which the branch of homological algebra developed. The original motivation for defining homology groups was the observation that shapes can be distinguished by their holes (for example in the classification of surfaces ). However, since holes are "not there" it is not obvious how to define holes mathematically. Homology is a mathematical approach to formalize the existence of holes. Certain “very fine” holes are invisible to homology; here u can. U. the homotopy groups, which are more difficult to determine , can be used.

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

## Construction of homology groups

The general procedure is as follows: First a chain complex is assigned to a mathematical object , which contains information about . A chain complex is a sequence of modules over a solid ring , connected by homomorphisms , so that the sequential execution of two of these maps is the zero map : for each . This means that the image of the -th figure is always contained in the core of the -th figure. The -th homology group of is now defined as the quotient module${\ displaystyle X}$${\ displaystyle X}$ ${\ displaystyle A_ {0}, A_ {1}, \ dots}$ ${\ displaystyle d_ {n} \ colon A_ {n} \ to A_ {n-1}}$${\ displaystyle d_ {n} \ circ d_ {n + 1} = 0}$${\ displaystyle n}$${\ displaystyle (n + 1)}$${\ displaystyle n}$${\ displaystyle n}$${\ displaystyle X}$

${\ displaystyle \, H_ {n} (X) = \ mathrm {ker} (d_ {n}) / \ mathrm {im} (d_ {n + 1}).}$

A chain complex is called exact if the image of the -th image is always the core of the -th image; the homology groups of thus measure “how inexact” the assigned chain complex is. ${\ displaystyle (n + 1)}$${\ displaystyle n}$${\ displaystyle X}$${\ displaystyle X}$

## Examples

The first example comes from algebraic topology : the simplicial homology of a simplicial complex . Here is the free module over the n -dimensional oriented simplices of . The figures are called edge figures and form the simplex with the corners ${\ displaystyle X}$${\ displaystyle A_ {n}}$${\ displaystyle X}$${\ displaystyle d_ {n}}$

${\ displaystyle (a [0], a [1], \ dots, a [n])}$

on the alternating sum of the "edge areas"

${\ displaystyle \ sum _ {i = 0} ^ {n} (- 1) ^ {i} (a [0], \ dots, a [i-1], a [i + 1], \ dots, a [n])}$

from.

For modules over a body (i.e. vector spaces), the dimension of the n -th homology group of describes the number of n -dimensional holes of . ${\ displaystyle X}$${\ displaystyle X}$

With this example one can define a simplicial homology for every topological space . The chain complex for is defined in such a way that the free module over all continuous mappings is from the n -dimensional unit simplex after . The homomorphisms result from the simplicial edge maps. ${\ displaystyle X}$${\ displaystyle A_ {n}}$${\ displaystyle X}$${\ displaystyle d_ {n}}$

In homological algebra , homology is used to define derived functors . There one considers an additive functor and a module . The chain complex for is constructed as follows: Let be a free module and an epimorphism , let be a free module, which is supposed to have the property that an epimorphism exists, so you get a sequence of free modules and homomorphisms and by using a chain complex. The n th homology of this complex depends on how it can be shown only by and from. One writes and calls the nth derived functor of . ${\ displaystyle F}$${\ displaystyle X}$${\ displaystyle X}$${\ displaystyle F_ {1}}$${\ displaystyle p_ {1} \ colon F_ {1} \ to X}$${\ displaystyle F_ {2}}$${\ displaystyle p_ {2} \ colon F_ {2} \ to \ mathrm {ker} \, p_ {1}}$${\ displaystyle \ ldots}$${\ displaystyle F_ {n}}$${\ displaystyle p_ {n} \ colon F_ {n} \ to F_ {n-1}}$${\ displaystyle F}$${\ displaystyle H_ {n}}$${\ displaystyle F}$${\ displaystyle X}$${\ displaystyle H_ {n} =: D ^ {n} F (X)}$${\ displaystyle D ^ {n} F}$${\ displaystyle F}$

## Homology functors

The chain complexes form a category : A morphism - they say: a chain mapping - from the chain complex to the chain complex is a sequence of module homomorphisms , so that for every n . The n th homology group can be understood as a functor from the category of chain complexes to the category of modules above the underlying ring . ${\ displaystyle (A_ {n}, d_ {n} ^ {A})}$${\ displaystyle (B_ {n}, d_ {n} ^ {B})}$${\ displaystyle f_ {n} \ colon A_ {n} \ to B_ {n}}$${\ displaystyle f_ {n-1} \ circ d_ {n} ^ {A} = d_ {n} ^ {B} \ circ f_ {n}}$${\ displaystyle H_ {n}}$${\ displaystyle R}$

If the chain complex is functionally dependent (i.e. each morphism induces a chain mapping from the chain complex of to that of ) then the functors of the category to which it belongs are in the category of modules. ${\ displaystyle X}$${\ displaystyle X \ to Y}$${\ displaystyle X}$${\ displaystyle Y}$${\ displaystyle H_ {n}}$${\ displaystyle X}$

A difference between homology and cohomology is that the chain complexes in cohomology depend contravariantly on and therefore the homology groups (which are then called cohomology groups and are referred to in this context as) are contravariant functors. Furthermore, one usually has a canonical ring structure on the graduate cohomology group, there is nothing comparable on the level of homology. ${\ displaystyle X}$${\ displaystyle H ^ {n}}$

## properties

If a chain complex is such that all (except for a finite number) are zero and all others are finitely generated free modules, then one can use the Euler characteristic${\ displaystyle (A_ {n}, d_ {n})}$${\ displaystyle A_ {n}}$

${\ displaystyle \ chi = \ sum (-1) ^ {n} \, \ mathrm {rank} \, (A_ {n})}$

define. It can be shown that the Euler characteristic can also be expressed in terms of homology:

${\ displaystyle \ chi = \ sum (-1) ^ {n} \, \ mathrm {rank} (H_ {n})}$

In algebraic topology, this provides two ways of calculating the invariant for the object from which the chain complex was generated. ${\ displaystyle \ chi}$${\ displaystyle X}$

${\ displaystyle 0 \ rightarrow A \ rightarrow B \ rightarrow C \ rightarrow 0}$

of chain complexes provides a long exact sequence of homology groups

${\ displaystyle \ cdots \ rightarrow H_ {n} (A) \ rightarrow H_ {n} (B) \ rightarrow H_ {n} (C) \ rightarrow H_ {n-1} (A) \ rightarrow H_ {n-1 } (B) \ rightarrow H_ {n-1} (C) \ rightarrow H_ {n-2} (A) \ rightarrow \ cdots \,}$

All images in this exact sequence induced by the mappings between the chain complexes except the figures , the connecting homomorphisms are called and their existence with the snake lemma is proved. ${\ displaystyle H_ {n} (C) \ rightarrow H_ {n-1} (A)}$