$n$-chainNChain2003-07-18 19:17:392007-03-09 16:36:33Definitionclosed n-chainexact n-chainboundary mapchain complexcell% this is the default PlanetMath preamble. as your knowledge
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\DeclareMathOperator{\im}{im}Let $X$ be a topological space and let $K$ be a simplicial approximation to $X$. An \emph{$n$-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 \emph{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 \emph{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 \emph{boundary map}. An $n$-chain is \emph{closed} if its boundary is 0 and \emph{exact} if it is the boundary of some $(n+1)$-chain. Closed $n$-chains are also called \emph{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 \emph{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
\[
H_n(X) = \frac{\ker\partial_n}{\im\,\partial_{n+1}}
\]
do not.
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