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For a topological space, the homology groups are generally much easier to compute than the Homotopy Group s, 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 the object X, one first defines a '' of the ''n''+1-th map is contained in the Kernel of the ''n''-th, and we can define the n-th homology group of X to be the Factor Group (or factor module)

: H_n(X) = \mathrm{ker}(d_n) / \mathrm{im}(d_{n+1}).

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 ''n''th map. The homology groups of X therefore measure "how far" the chain complex associated to X is from being exact.


EXAMPLES


The motivating example comes from X. Here A_n is the Free Abelian Group or module whose generators are the ''n''-dimensional
oriented simplexes of X. The mappings are called the ''boundary mappings'' and send the simplex with vertices

: (a a[1 , \dots, a[n])

to the sum

: \sum_{i=0}^n (-1)^i(a \dots, a[i-1 , a \dots, a[n ).

If we take the modules to be over a field, then the dimension
of the ''n''-th homology of X turns out to be the number of "holes" in X at dimension ''n''.

Using this example as a model, one can define a simplicial homology for any Topological Space X. We define a chain complex for X by taking A_n to be the free abelian group (or free module) whose generators are all Continuous maps from ''n''-dimensional Simplices into X. The homomorphisms d_n arise from the boundary maps of simplices.

In homomorphism p_1 : F_1 ightarrow X . Then one finds a free module F_2 and a surjective homomorphism p_2 : F_2 ightarrow \mathrm{ker}(p_1) . Continuing in this fashion, a sequence of free modules F_n and homomorphisms p_n can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology H_n of this complex depends only on F and X and is, by definition, the ''n''-th derived functor of F, applied to X.


HOMOLOGY FUNCTORS


Chain complexes form a 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 ightarrow Y induces a morphism from the chain complex of X to the chain complex of Y), then the H_n are covariant Functor s 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 H_n) form ''contravariant'' functors from the category that X belongs to into the category of abelian groups or modules.


PROPERTIES


If (d_n : A_n ightarrow A_{n-1}) is a chain complex such that all but finitely many A_n are zero, and the others are finitely generated abelian groups (or finite dimensional vector spaces), then we can define the '' Euler Characteristic ''

: \chi = \sum (-1)^n \, \mathrm{rank}(A_n)

(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:

: \chi = \sum (-1)^n \, \mathrm{rank}(H_n)

and, especially in algebraic topology, this provides two ways to compute the important invariant \chi for the object X which gave rise to the chain complex.

Every Short Exact Sequence

: 0 ightarrow A ightarrow B ightarrow C ightarrow 0

of chain complexes gives rise to a Long Exact Sequence of homology groups

: \cdots ightarrow H_n(A) ightarrow H_n(B) ightarrow H_n(C) ightarrow H_{n-1}(A) ightarrow H_{n-1}(B) ightarrow H_{n-1}(C) ightarrow H_{n-2}(A) ightarrow \cdots \,

All maps in this long exact sequence are induced by the maps between the chain complexes, except for the maps H_n(C) ightarrow H_{n-1}(A) . These latter are called ''connecting homomorphisms'' and are provided by the Snake Lemma .


SEE ALSO