The relative chain complex is , and its homology is the relative homology . A pair produces a long exact sequence for the triple.
A pair is good when is closed and is a deformation retract of some neighborhood in . Such a pair satisfies the hypotheses needed to compare relative homology with the reduced homology of a quotient.
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Relative homology is a concept in algebraic topology that extends the notion of homology groups to pairs of spaces. Specifically, if we have a topological space \( X \) and a subspace \( A \subseteq X \), the relative homology groups \( H_n(X, A) \) provide information about the structure of \( X \) relative to the subspace \( A \).