# $n$-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 Nchain 2013-03-22 13:46:20 2013-03-22 13:46:20 mps (409) mps (409) 11 mps (409) Definition msc 16E05 chain LongExactSequenceOfHomologyGroups closed n-chain exact n-chain boundary map