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 |
---|---|
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 |