In algebraic topology, a branch of mathematics, the (singular) homology of a topological space relative to a subspace is a construction in singular homology, for pairs of spaces.
Intuitively, it helps determine what part of an absolute homology group comes from which subspace.
denotes the singular chains on the space X.
, we then have a complex By definition, the nth relative homology group of the pair of spaces
[1] The above short exact sequences specifying the relative chain groups give rise to a chain complex of short exact sequences.
An application of the snake lemma then yields a long exact sequence The connecting map
takes a relative cycle, representing a homology class in
is a point in X, is the n-th reduced homology group of X.
The excision theorem says that removing a sufficiently nice subset
leaves the relative homology groups
, then using the long exact sequence of pairs and the excision theorem, one can show that
is the same as the n-th reduced homology groups of the quotient space
Relative homology readily extends to the triple
by The exactness of the sequence implies that the Euler characteristic is additive, i.e., if
-th local homology group of a space
, denoted is defined to be the relative homology group
One easy example of local homology is calculating the local homology of the cone (topology) of a space at the origin of the cone.
Recall that the cone is defined as the quotient space where
Note the previous construction can be proven in algebraic geometry using the affine cone of a projective variety
Using the excision theorem there is an isomorphism of relative homology groups hence the local homology of a point reduces to the local homology of a point in a closed ball
Because of the homotopy equivalence and the fact the only non-trivial part of the long exact sequence of the pair
Just as in absolute homology, continuous maps between spaces induce homomorphisms between relative homology groups.
be the natural projections which take elements to their equivalence classes in the quotient groups.
Chain maps induce homomorphisms between homology groups, so
fulfilling the mild regularity condition that there exists a neighborhood of
We can immediately use this fact to compute the homology of a sphere.
Applying the exact sequence of relative homology gives the following:
Because the disk is contractible, we know its reduced homology groups vanish in all dimensions, so the above sequence collapses to the short exact sequence:
is the deformation retract of a suitable neighborhood of itself in
Another insightful geometric example is given by the relative homology of