# Cochain

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A homogeneous element of an Abelian cochain group (or, in the general case, a module), i.e. a graded Abelian group equipped with an endomorphism of degree such that . The endomorphism is called the coboundary mapping or the coboundary.

A cochain group usually arises as a group or , where is an arbitrary Abelian group, called the coefficient group, and is a chain group, i.e. a graded Abelian group equipped with an endomorphism of degree (the boundary mapping or boundary) with . In this situation the mapping on the group is defined as the adjoint of , where , .

Given a topological space , one defines the group of singular chains as the Abelian group of formal finite sums , where and the are arbitrary singular simplices in , i.e. continuous mappings of the standard simplex into . A singular cochain in with coefficients in is defined as a homogeneous element of the group .

Similarly, a simplicial -cochain of a simplicial complex in with coefficients in an Abelian group is defined as a homomorphism , where is the group of -chains of , i.e. the group of formal finite sums , where and the are -simplices in . In particular, a cochain in the sense of Aleksandrov–Čech in an arbitrary topological space is a cochain of the nerve of an open covering of .

If is a -complex (and denotes the -skeleton of ), then the Abelian group is called the group of -dimensional cellular cochains of the complex . The coboundary homomorphism is put equal to the connecting mappings of the triple .

In practice, the group is frequently provided with an additional multiplicative structure, i.e. it is a graded algebra. In these cases the coboundary mapping possesses the Leibniz property: , where the element is assumed to be homogeneous of degree . An example of such a graded cochain algebra is the algebra of differential forms on a smooth manifold, in which the exterior differential acts as coboundary.

#### References

 [1] P.J. Hilton, S. Wylie, "Homology theory. An introduction to algebraic topology" , Cambridge Univ. Press (1960) [2] S. MacLane, "Homology" , Springer (1963)