Relative Homology - Properties

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 H_n(X, A), to its boundary (which is a cycle in A).

It follows that H_n(X, x₀), where x₀ is a point in X, is the n-th reduced homology group of X. In other words, H_i(X, x₀) = H_i(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 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 x₀ is defined to be H_n(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