Relative homology

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. The relative homology is useful and important in several ways. Intuitively, it helps determine what part of an absolute homology group comes from which subspace.
Definition.
Given a subspace formula_1, one may form the short exact sequence
where formula_3 denotes the singular chains on the space "X". The boundary map on formula_3 leaves formula_5 invariant and therefore descends to a boundary map formula_6 on the quotient. If we denote this quotient by formula_7, we then have a complex
By definition, the th relative homology group of the pair of spaces formula_9 is
One says that relative homology is given by the relative cycles, chains whose boundaries are chains on "A", modulo the relative boundaries (chains that are homologous to a chain on "A", i.e., chains that would be boundaries, modulo "A" again).
Properties.
The above short exact sequences specifying the relative chain groups gives rise to a chain complex of short exact sequences. An application of the snake lemma then yields a long exact sequence
The connecting map "formula_12" takes a relative cycle, representing a homology class in formula_13, to its boundary (which is a cycle in "A").
It follows that formula_14, where formula_15 is a point in "X", is the "n"-th reduced homology group of "X". In other words, formula_16 for all formula_17. When formula_18, formula_19 is the free module of one rank less than formula_20. The connected component containing formula_15 becomes trivial in relative homology.
The excision theorem says that removing a sufficiently nice subset formula_22 leaves the relative homology groups formula_13 unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that formula_13 is the same as the "n"-th reduced homology groups of the quotient space formula_25.
Relative homology readily extends to the triple formula_26 for formula_27.
One can define the Euler characteristic for a pair formula_28 by
The exactness of the sequence implies that the Euler characteristic is "additive", i.e., if formula_27, one has
Local homology.
The formula_32-th local homology group of a space formula_33 at a point formula_15, denoted
is defined to be the relative homology group formula_36. Informally, this is the "local" homology of formula_33 close to formula_15.
Local homology of the cone CX at the origin.
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 formula_40 has the subspace topology. Then, the origin formula_41 is the equivalence class of points formula_42. Using the intuition that the local homology group formula_43 of formula_44 at formula_15 captures the homology of formula_44 "near" the origin, we should expect this is the homology of formula_47 since formula_48 has a homotopy retract to formula_33. Computing the local cohomology can then be done using the long exact sequence in homology
Because the cone of a space is contractible, the middle homology groups are all zero, giving the isomorphism
since formula_48 is contractible to formula_33.
In algebraic geometry.
Note the previous construction can be proven in Algebraic geometry using the affine cone of a projective variety formula_33 using Local cohomology.
Local homology of a point on a smooth manifold.
Another computation for local homology can be computed on a point formula_55 of a manifold formula_56. Then, let formula_57 be a compact neighborhood of formula_55 isomorphic to a closed disk formula_59 and let formula_60. 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 formula_62. Because of the homotopy equivalence
and the fact
the only non-trivial part of the long exact sequence of the pair formula_65 is
hence the only non-zero local homology group is formula_67.
Functoriality.
Just as in absolute homology, continuous maps between spaces induce homomorphisms between relative homology groups. In fact, this map is exactly the induced map on homology groups, but it descends to the quotient.
Let formula_9 and formula_69 be pairs of spaces such that formula_1 and formula_71, and let formula_72 be a continuous map. Then there is an induced map formula_73 on the (absolute) chain groups. If formula_74, then formula_75. Let
formula_76
be the natural projections which take elements to their equivalence classes in the quotient groups. Then the map formula_77 is a group homomorphism. Since formula_78, this map descends to the quotient, inducing a well-defined map formula_79 such that the following diagram commutes:
Chain maps induce homomorphisms between homology groups, so formula_80 induces a map formula_81 on the relative homology groups.
Examples.
One important use of relative homology is the computation of the homology groups of quotient spaces formula_25. In the case that formula_83 is a subspace of formula_33 fulfilling the mild regularity condition that there exists a neighborhood of formula_83 that has formula_83 as a deformation retract, then the group formula_87 is isomorphic to formula_88. We can immediately use this fact to compute the homology of a sphere. We can realize formula_89 as the quotient of an n-disk by its boundary, i.e. formula_90. Applying the exact sequence of relative homology gives the following:<br> formula_91
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:
formula_92
Therefore, we get isomorphisms formula_93. We can now proceed by induction to show that formula_94. Now because formula_95 is the deformation retract of a suitable neighborhood of itself in formula_96, we get that formula_97.
Another insightful geometric example is given by the relative homology of formula_98 where formula_99. Then we can use the long exact sequence
Using exactness of the sequence we can see that formula_101 contains a loop formula_102 counterclockwise around the origin. Since the cokernel of formula_103 fits into the exact sequence
it must be isomorphic to formula_105. One generator for the cokernel is the formula_106-chain formula_107 since its boundary map is
Notes.
i.e., the boundary formula_109 maps formula_110 to formula_111