|
|
|
|
Homology (mathematics)
|
| |
|
| |
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek ?µ?? homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. See homology theory for more background, or singular homology for a concrete version for topological spaces, or group cohomology for a concrete version for groups.
For a topological space, the homology groups are generally much easier to compute than the homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.
n an object such as a topological space ,
one first defines a chain complex
encoding information about .
A chain complex is a sequence of abelian groups or modules
connected by homomorphisms
,
which we call boundary operators.
That is,
,
where
denotes the trivial group and
for
.
We also require the composition of any two consecutive boundary operators to be zero.
That is, for all
,
,
i.e., the constant map to the group identity in
.
This means
.
Now since each
is abelian,
is a normal subgroup of
.
And we want to mod out by this subgroup,
i.e., consider everything in
equivalent and partition
using this equivalence relation.
We define the n-th homology group of X to be the factor group (or quotient module)
We also use the notation
and
,
so
Computing these two groups is usually rather difficult since they are very large groups.
On the other hand, we do have tools which make the task easier.
The simplicial homology groups of a simplicial complex are defined using the simplicial chain complex ,
with the free abelian group generated by the -simplices of .
Discussion
Ask a question about 'Homology (mathematics)' Start a new discussion about 'Homology (mathematics)' Answer questions from other users |
Encyclopedia
In mathematics (especially algebraic topology and abstract algebra), homology (in Greek ?µ?? homos "identical") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group. See homology theory for more background, or singular homology for a concrete version for topological spaces, or group cohomology for a concrete version for groups.
For a topological space, the homology groups are generally much easier to compute than the homotopy groups, and consequently one usually will have an easier time working with homology to aid in the classification of spaces.
Construction of homology groups
The procedure works as follows.
Given an object such as a topological space ,
one first defines a chain complex
encoding information about .
A chain complex is a sequence of abelian groups or modules
connected by homomorphisms
,
which we call boundary operators.
That is,
,
where
denotes the trivial group and
for
.
We also require the composition of any two consecutive boundary operators to be zero.
That is, for all
,
,
i.e., the constant map to the group identity in
.
This means
.
Now since each
is abelian,
is a normal subgroup of
.
And we want to mod out by this subgroup,
i.e., consider everything in
equivalent and partition
using this equivalence relation.
We define the n-th homology group of X to be the factor group (or quotient module)
We also use the notation
and
,
so
Computing these two groups is usually rather difficult since they are very large groups.
On the other hand, we do have tools which make the task easier.
The simplicial homology groups of a simplicial complex are defined using the simplicial chain complex ,
with the free abelian group generated by the -simplices of . The singular homology groups are defined for any topological space , and agree with the simplicial homology groups for a simplicial complex.
A chain complex is said to be exact if the image of the (n + 1)-th map is always equal to the kernel of the nth map. The homology groups of therefore measure "how far" the chain complex associated to is from being exact.
Cohomology groups are formally similar: one starts with a cochain complex, which is the same as a chain complex but whose arrows, now denoted point in the direction of increasing n rather than decreasing n; then the groups and follow from the same description and
, as before.
Examples
The motivating example comes from algebraic topology: the simplicial homology of a simplicial complex . Here is the free abelian group or module whose generators are the n-dimensional
oriented simplexes of . The mappings are called the boundary mappings and send the simplex with vertices
to the sum
(which is considered if ).
If we take the modules to be over a field, then the dimension
of the n-th homology of turns out to be the number of "holes" in at dimension n.
Using this example as a model, one can define a singular homology for any topological space . We define a chain complex for by taking to be the free abelian group (or free module) whose generators are all continuous maps from n-dimensional simplices into . The homomorphisms arise from the boundary maps of simplices.
In abstract algebra, one uses homology to define derived functors, for example the Tor functors. Here one starts with some covariant additive functor and some module . The chain complex for is defined as follows: first find a free module and a surjective homomorphism . Then one finds a free module and a surjective homomorphism . Continuing in this fashion, a sequence of free modules and homomorphisms can be defined. By applying the functor to this sequence, one obtains a chain complex; the homology of this complex depends only on and and is, by definition, the n-th derived functor of , applied to .
Homology functors
Chain complexes form a category: A morphism from the chain complex to the chain complex is a sequence of homomorphisms such that for all n. The n-th homology Hn can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups (or modules).
If the chain complex depends on the object X in a covariant manner (meaning that any morphism X ? Y induces a morphism from the chain complex of X to the chain complex of Y), then the Hn are covariant functors from the category that X belongs to into the category of abelian groups (or modules).
The only difference between homology and cohomology is that in cohomology the chain complexes depend in a contravariant manner on X, and that therefore the homology groups (which are called cohomology groups in this context and denoted by Hn) form contravariant functors from the category that X belongs to into the category of abelian groups or modules.
Properties
If is a chain complex such that all but finitely many are zero, and the others are finitely generated abelian groups (or finite dimensional vector spaces), then we can define the Euler characteristic
(using the rank in the case of abelian groups and the Hamel dimension in the case of vector spaces). It turns out that the Euler characteristic can also be computed on the level of homology:
and, especially in algebraic topology, this provides two ways to compute the important invariant for the object which gave rise to the chain complex.
Every short exact sequence
of chain complexes gives rise to a long exact sequence of homology groups
All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps The latter are called connecting homomorphisms and are provided by the snake lemma.
See also
|
| |
|
|
