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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.
The procedure works as follows: Given the object X, one first defines a chain complex that encodes information about X. A chain complex is a sequence of abelian groups or modules A0, A1, A2... connected by homomorphisms dn : An -> An-1, such that the composition of any two consecutive maps is zero: dn o dn+1 = 0 for all n. This means that the image 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)
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.
The motivating example comes from algebraic topology: the simplicial homology of a simplicial complexIn set theory, a chain is a total order subset of a poset. See also Ascending Chain Condition and Descending Chain Condition. Algebraic topology In algebraic topology, a simplicial k chain is a formal linear combination of k simplices. Integration on chai X. Here An is the free abelian groupIn abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Unlike 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
to the sum of
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 An to be the free abelian group (or free module) whose generators are all continuous maps from n-dimensional simplicesTopology Geometry Geometry In geometry, a simplex is an n dimensional figure, being the convex hull of a set of n + 1) affinely independent points in some Euclidean space of dimension n or higher i. a set of points such that no m plane contains more than into X. The homomorphisms dn arise from the boundary maps of simplices.
In abstract algebra, one uses homology to define derived functorHomological algebra Category theory In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. It is not rs, for example the Tor functorThe Tor functors are the derived functors of the tensor product functor in mathematics. They were first defined in generality to express the Kunneth theorem and universal coefficient theorem in algebraic topology. Specifically, suppose R is a ring, and des. Here one starts with some covariant additive functor F and some module X. The chain complex for X is defined as follows: first find a free module F1 and a surjective homomorphism p1 : F1 -> X. Then one finds a free module F2 and a surjective homomorphism p2 : F2 -> ker(p1). Continuing in this fashion, a sequence of free modules Fn and homorphisms pn can be defined. By applying the functor F to this sequence, one obtains a chain complex; the homology Hn of this complex depends only on F and X and is, by definition, the n-th derived functor of F, applied to X.