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 $Asubset\; X$, one may form the short exact sequence

:

where denotes the singular chains on the space "X". The boundary map on leaves invariant and therefore descends to a boundary map on the quotient. The correponding homology is called relative homology:

:

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

:$cdots\; o\; H\_n(A)\; o\; H\_n(X)\; o\; H\_n\; (X,A)\; stackrel\{delta\}\{\; o\}\; H\_\{n-1\}(A)\; o\; cdots\; .$

The connecting map "δ" takes a relative cycle, representing a homology class in "H_n"("X", "A"), to its boundary (which is a chain in "A").

It follows that "H_n"("X", "x"₀), where "x"₀ is a point in "X", is the "n"-th reduced homology group of "X". In other words, "H_i"("X", "x"₀) = "H_i"("X") for all "i" > 0. When "i" = 0, "H"₀("X", "x"₀) is the free module of one rank less than "H"₀("X"). The connected component containing "x"₀ becomes trivial in relative homoloy.

The excision theorem says that removing a sufficiently nice subset "Z" ⊂ "A" leaves the relative homology groups "H_n"("X", "A") unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that "H_n"("X", "A") is the same as the "n"-th reduced homology groups of the quotient space "X"/"A".

The "n"-th local homology group of a space "X" at a point "x"₀ is defined to be "H_n"("X", "X" - "x"₀). Informally, this is the "local" homology of "X" close to "x"₀.

Relative homology readily extends to the triple ("X", "Y", "Z") for "Z" ⊂ "Y" ⊂ "X".

One can define the Euler characteristic for a pair "Y" ⊂ "X" by

:$chi\; (X,\; Y)\; =\; sum\; \_0\; ^n\; (-1)^j\; ;\; mbox\{rank\}\; ;\; H\_j\; (X,\; Y).$

The exactness of the sequence implies that the Euler characteristic is "additive", i.e. if "Z" ⊂ "Y" ⊂ "X", one has

:$chi\; (X,\; Z)\; =\; chi\; (X,\; Y)\; +\; chi\; (Y,\; Z).,$

Functoriality

The map can be considered to be a functor

:

where Top² is the category of pairs of topological spaces and Comp is the category of chain complexes.

References

*
* Joseph J. Rotman, "An Introduction to Algebraic Topology", Springer-Verlag, ISBN 0-387-96678-1

ee also

*Relative contact homology