# relative homology groups

If $X$ is a topological space^{}, and $A$ a subspace^{}, then the inclusion map^{} $A\hookrightarrow X$
makes ${C}_{n}(A)$ into a subgroup of ${C}_{n}(X)$. Since the boundary map on ${C}_{*}(X)$ restricts to
the boundary map on ${C}_{*}(A)$, we can take the quotient complex ${C}_{*}(X,A)$,

$$ |

The homology groups of this complex ${H}_{n}(X,A)$, are called the relative homology groups
of the pair $(X,A)$. Under relatively mild hypotheses, ${H}_{n}(X,A)={H}_{n}(X/A)$ where $X/A$ is
the set of equivalence classes^{} of the relation^{} $x\sim y$ if $x=y$ or if $x,y\in A$, given the quotient
topology (this is essentially $X$, with $A$ reduced to a single point). Relative homology groups are
important for a number of reasons, principally for computational ones, since they fit into long
exact sequences, which are powerful computational tools in homology^{}.

Title | relative homology groups |
---|---|

Canonical name | RelativeHomologyGroups |

Date of creation | 2013-03-22 13:14:47 |

Last modified on | 2013-03-22 13:14:47 |

Owner | bwebste (988) |

Last modified by | bwebste (988) |

Numerical id | 5 |

Author | bwebste (988) |

Entry type | Definition |

Classification | msc 55N10 |