# long exact sequence (of homology groups)

If $X$ is a topological space^{}, and $A$ and $B$ are subspaces^{} with $X\supset A\supset B$,
then there is a long exact sequence:

$$\begin{array}{ccccccccccc}\hfill \mathrm{\cdots}\hfill & \hfill \to \hfill & \hfill {H}_{n}(A,B)\hfill & \hfill \stackrel{{i}_{*}}{\to}\hfill & \hfill {H}_{n}(X,B)\hfill & \hfill \stackrel{{j}_{*}}{\to}\hfill & \hfill {H}_{n}(X,A)\hfill & \hfill \stackrel{{\partial}_{*}}{\to}\hfill & \hfill {H}_{n-1}(A,B)\hfill & \hfill \to \hfill & \end{array}$$ |

where ${i}_{*}$ is induced by the inclusion $i:(A,B)\hookrightarrow (X,B)$, ${j}_{*}$ by the inclusion $j:(X,B)\hookrightarrow (X,A)$, and $\partial $ is the following map: given $a\in {H}_{n}(X,A)$, choose a chain representing it. $\partial a$ is an $(n-1)$-chain of $A$, so it represents an element of ${H}_{n-1}(A,B)$. This is ${\partial}_{*}a$.

When $B$ is the empty set^{}, we get the long exact sequence of the pair $(X,A)$:

$$\begin{array}{ccccccccccc}\hfill \mathrm{\cdots}\hfill & \hfill \to \hfill & \hfill {H}_{n}(A)\hfill & \hfill \stackrel{{i}_{*}}{\to}\hfill & \hfill {H}_{n}(X)\hfill & \hfill \stackrel{{j}_{*}}{\to}\hfill & \hfill {H}_{n}(X,A)\hfill & \hfill \stackrel{{\partial}_{*}}{\to}\hfill & \hfill {H}_{n-1}(A)\hfill & \hfill \to \hfill & \end{array}$$ |

The existence of this long exact sequence follows from the short exact sequence^{}

$$\begin{array}{ccccccccc}\hfill 0\hfill & \hfill \to \hfill & \hfill {C}_{*}(A,B)\hfill & \hfill \stackrel{{i}_{\mathrm{\u266f}}}{\to}\hfill & \hfill {C}_{*}(X,B)\hfill & \hfill \stackrel{{j}_{\mathrm{\u266f}}}{\to}\hfill & \hfill {C}_{*}(X,A)\hfill & \hfill \to \hfill & \hfill 0\hfill \end{array}$$ |

where ${i}_{\mathrm{\u266f}}$ and ${j}_{\mathrm{\u266f}}$ are the maps on chains induced by $i$ and $j$,
by the Snake Lemma^{}.

Title | long exact sequence (of homology groups) |
---|---|

Canonical name | LongExactSequenceofHomologyGroups |

Date of creation | 2013-03-22 13:14:50 |

Last modified on | 2013-03-22 13:14:50 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 8 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 55N10 |

Related topic | NChain |

Related topic | ProofOfSnakeLemma |