Relative homology

Summary

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

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Given a subspace  , one may form the short exact sequence

 

where   denotes the singular chains on the space X. The boundary map on   descendsa to   and therefore induces a boundary map   on the quotient. If we denote this quotient by  , we then have a complex

 

By definition, the nth relative homology group of the pair of spaces   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).[1]

Properties

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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  , to its boundary (which is a cycle in A).[2]

It follows that  , where   is a point in X, is the n-th reduced homology group of X. In other words,   for all  . When  ,   is the free module of one rank less than  . The connected component containing   becomes trivial in relative homology.

The excision theorem says that removing a sufficiently nice subset   leaves the relative homology groups   unchanged. If   has a neighbourhood   in   that deformation retracts to  , 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   for  .

One can define the Euler characteristic for a pair   by

 

The exactness of the sequence implies that the Euler characteristic is additive, i.e., if  , one has

 

Local homology

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The  -th local homology group of a space   at a point  , denoted

 

is defined to be the relative homology group  . Informally, this is the "local" homology of   close to  .

Local homology of the cone CX at the origin

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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   has the subspace topology. Then, the origin   is the equivalence class of points  . Using the intuition that the local homology group   of   at   captures the homology of   "near" the origin, we should expect this is the homology of   since   has a homotopy retract to  . Computing the local homology 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   is contractible to  .

In algebraic geometry

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Note the previous construction can be proven in algebraic geometry using the affine cone of a projective variety   using Local cohomology.

Local homology of a point on a smooth manifold

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Another computation for local homology can be computed on a point   of a manifold  . Then, let   be a compact neighborhood of   isomorphic to a closed disk   and let  . 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   is

 

hence the only non-zero local homology group is  .

Functoriality

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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   and   be pairs of spaces such that   and  , and let   be a continuous map. Then there is an induced map   on the (absolute) chain groups. If  , then  . Let

 

be the natural projections which take elements to their equivalence classes in the quotient groups. Then the map   is a group homomorphism. Since  , this map descends to the quotient, inducing a well-defined map   such that the following diagram commutes:[3]

 

Chain maps induce homomorphisms between homology groups, so   induces a map   on the relative homology groups.[2]

Examples

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One important use of relative homology is the computation of the homology groups of quotient spaces  . In the case that   is a subspace of   fulfilling the mild regularity condition that there exists a neighborhood of   that has   as a deformation retract, then the group   is isomorphic to  . We can immediately use this fact to compute the homology of a sphere. We can realize   as the quotient of an n-disk by its boundary, i.e.  . 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:

 

Therefore, we get isomorphisms  . We can now proceed by induction to show that  . Now because   is the deformation retract of a suitable neighborhood of itself in  , we get that  .

Another insightful geometric example is given by the relative homology of   where  . Then we can use the long exact sequence

 

Using exactness of the sequence we can see that   contains a loop   counterclockwise around the origin. Since the cokernel of   fits into the exact sequence

 

it must be isomorphic to  . One generator for the cokernel is the  -chain   since its boundary map is

 

See also

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Notes

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^ i.e., the boundary   maps   to  

References

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  • "Relative homology groups". PlanetMath.
  • Joseph J. Rotman, An Introduction to Algebraic Topology, Springer-Verlag, ISBN 0-387-96678-1
Specific
  1. ^ Hatcher, Allen (2002). Algebraic topology. Cambridge, UK: Cambridge University Press. ISBN 9780521795401. OCLC 45420394.
  2. ^ a b Hatcher, Allen (2002). Algebraic topology. Cambridge: Cambridge University Press. pp. 118–119. ISBN 9780521795401. OCLC 45420394.
  3. ^ Dummit, David S.; Foote, Richard M. (2004). Abstract algebra (3 ed.). Hoboken, NJ: Wiley. ISBN 9780471452348. OCLC 248917264.