$n$ -chain

Let $X$ be a topological space 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 $C_{n}(X)$ and is called the $n$ th chain group of $X$ . In other words, $C_{n}(X)$ is the free abelian group generated by the oriented $n$ -simplices in $K$ .

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

The formal boundary of an oriented $n$ -simplex $\sigma=(v_{0},\dots,v_{n})$ is given by the alternating sum of the oriented $n$ -simplices forming the topological boundary of $\sigma$ , that is,

\partial_{n}(\sigma)=\sum_{j=0}^{n}(-1)^{j}(v_{0},\dots,v_{j-1},v_{j+1},\dots,% v_{n}).

The boundary of a $0$ -simplex is $0$ .

Since $n$ -simplices form a basis for the chain group $C_{n}(X)$ , this extends to give a group homomorphism $\partial_{n}\colon C_{n}(X)\to C_{n-1}(X)$ , called the boundary map. 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

\xymatrix{\cdots\ar[r]&C_{n+1}(X)\ar[r]^{\partial_{n+1}}&C_{n}(X)\ar[r]^{% \partial_{n}}&C_{n-1}\ar[r]&\cdots}

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 $C_{n}(X)$ depend on the choice of simplicial approximation $K$ , the resulting homology groups

H_{n}(X)=\frac{\ker\partial_{n}}{\operatorname{im}\,\partial_{n+1}}

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

n-chain

$n$ -chain