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 δ takes a relative cycle, representing a homology class in Hn(X, A), to its boundary (which is a cycle in A).
It follows that Hn(X, x0), where x0 is a point in X, is the n-th reduced homology group of X. In other words, Hi(X, x0) = Hi(X) for all i > 0. When i = 0, H0(X, x0) is the free module of one rank less than H0(X). The connected component containing x0 becomes trivial in relative homology.
The excision theorem says that removing a sufficiently nice subset Z ⊂ A leaves the relative homology groups Hn(X, A) unchanged. Using the long exact sequence of pairs and the excision theorem, one can show that Hn(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 x0 is defined to be Hn(X, X - {x0}). Informally, this is the "local" homology of X close to x0.
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
The exactness of the sequence implies that the Euler characteristic is additive, i.e. if Z ⊂ Y ⊂ X, one has
Read more about this topic: Relative Homology
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—Ralph Waldo Emerson (18031882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (16321704)