relative homology groups
If is a topological space, and a subspace, then the inclusion map makes into a subgroup of . Since the boundary map on restricts to the boundary map on , we can take the quotient complex ,
The homology groups of this complex , are called the relative homology groups of the pair . Under relatively mild hypotheses, where is the set of equivalence classes of the relation if or if , given the quotient topology (this is essentially , with 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 |
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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 |