# $n$-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

$$\text{xymatrix}\mathrm{\cdots}\text{ar}[r]\mathrm{\&}{C}_{n+1}(X)\text{ar}{[r]}^{{\partial}_{n+1}}\mathrm{\&}{C}_{n}(X)\text{ar}{[r]}^{{\partial}_{n}}\mathrm{\&}{C}_{n-1}\text{ar}[r]\mathrm{\&}\mathrm{\cdots}$$ |

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

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

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 |