$n$-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 $\sigma =(v_0,\mathrm{\dots },v_n)$ is given by the alternating sum of the oriented $n$-simplices forming the topological boundary of $\sigma$, that is,

$${\partial }_n(\sigma )=\sum _{j=0}^n{(-1)}^j(v_0,\mathrm{\dots },v_{j-1},v_{j+1},\mathrm{\dots },v_n).$$

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 ${\partial }_n:C_n(X)\to C_{n-1}(X)$, 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

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

$$H_n(X)=\frac{\ker {\partial }_n}{\mathrm{im}{\partial }_{n+1}}$$

do not.