Let X be a topological spaceMathworldPlanetmath and let K be a simplicial approximation to X. An n-chain on X is a finite formal sum of oriented n-simplices in K. The group of such chains is denoted by Cn(X) and is called the nth chain group of X. In other words, Cn(X) is the free abelian groupMathworldPlanetmath generated by the oriented n-simplices in K.

We have defined chain groups for simplicial homologyMathworldPlanetmath. Their definition is similar in singular homologyMathworldPlanetmath and the homology of CW complexes. For example, if Y is a CW complex, then its nth chain group is the free abelian group on the cells of Yn, the n-skeleton of Y.

The formal boundary of an oriented n-simplex σ=(v0,,vn) is given by the alternating sum of the oriented n-simplices forming the topological boundary of σ, that is,


The boundary of a 0-simplex is 0.

Since n-simplices form a basis for the chain group Cn(X), this extends to give a group homomorphismMathworldPlanetmath n:Cn(X)Cn-1(X), called the boundary mapPlanetmathPlanetmath. An n-chain is closed if its boundary is 0 and exact if it is the boundary of some (n+1)-chain. Closed n-chains are also called cycles. Every exact n-chain is also closed. This implies that the sequence


is a complex of free abelian groups. This complex is usually called the chain complex of X corresponding to the simplicial complex K. Note that while the chain groups Cn(X) depend on the choice of simplicial approximation K, the resulting homology groups


do not.

Title n-chain
Canonical name Nchain
Date of creation 2013-03-22 13:46:20
Last modified on 2013-03-22 13:46:20
Owner mps (409)
Last modified by mps (409)
Numerical id 11
Author mps (409)
Entry type Definition
Classification msc 16E05
Synonym chain
Related topic LongExactSequenceOfHomologyGroups
Defines closed n-chain
Defines exact n-chain
Defines boundary map