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In homological algebra, 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. They tend to be written out like so:
A variant on the concept of chain complex is that of cochain complex. A cochain 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+1 o dn = 0 for all n:
The idea is basically the same.
Chain complexes are mainly used to define homology and cohomology.
Suppose we are given a topological space X.
Define Cn(X) for natural n to be the free abelian group formally generated by singular simplices in X, and define the boundary map
where the hat denotes the omission of a vertex. That is, the boundary of a singular simplex is alternating sum of restrictions to its faces. It can be shown ∂² = 0, so is a chain complex; the singular homology is the homology of this complex; that is,
The differential k-forms on any smooth manifold M form an abelian group (in fact an R- vector space) called Ωk(M) under additionAddition is one of the basic operations of arithmetic. In its simplest form, addition combines two numbers terms summands , the augend and addend into a single number, the sum . Adding more numbers corresponds to repeated addition. By extension, addition. The exterior derivativeIn mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. It is important in the theory of integration on manifolds, and is the differential use d = d k maps Ωk(M) → Ωk+1(M), and d 2 = 0 follows essentially from symmetry of second derivatives, so the vector spaces of k-forms along with the exterior derivative are a cochain complex:
The homology of this complex is the de Rham cohomology