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.

Given a subspace ${\displaystyle A\subset X}$, one may form the short exact sequence

where ${\displaystyle C_{\bullet }(X)}$ denotes the singular chains on the space X. The boundary map on ${\displaystyle C_{\bullet }(X)}$ leaves ${\displaystyle C_{\bullet }(A)}$ invariant and therefore descends to a boundary map on the quotient. The corresponding homology is called relative homology:

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