relative homology groups

If X is a topological spaceMathworldPlanetmath, and A a subspaceMathworldPlanetmath, then the inclusion mapMathworldPlanetmath AX makes Cn(A) into a subgroup of Cn(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 Hn(X,A), are called the relative homology groups of the pair (X,A). Under relatively mild hypotheses, Hn(X,A)=Hn(X/A) where X/A is the set of equivalence classesMathworldPlanetmathPlanetmath of the relationMathworldPlanetmathPlanetmath xy if x=y or if x,yA, 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 homologyMathworldPlanetmath.

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