Relative homology and long exact sequences

For a subspace $A\subseteq X$, singular simplices lying in A form a subcomplex of $C_*(X;R)$. Define relative chains as $C_*(X,A;R)=C_*(X;R)/C_*(A;R)$ and then take homology. A chain c represents a relative cycle precisely when its boundary lies in the A-subcomplex. It may have a nonzero absolute boundary. Relative boundaries are images of higher chains after passing to the quotient. These definitions are valid for arbitrary pairs of spaces with fixed coefficients.

Taking homology of a quotient complex is not generally the same as quotienting the two homology groups. A subspace can contain a cycle that becomes a boundary in the whole space, producing additional relative classes one degree higher. For suitable good pairs with A nonempty, such as nonempty CW subcomplex pairs, relative homology also agrees with reduced homology of the quotient space obtained by collapsing A. That identification needs its hypotheses; the relative chain construction itself does not require pretending that every arbitrary quotient has the same behavior.

For a relative cycle represented by c, its boundary is an A-chain and has zero boundary because $\partial^2=0$. Send its relative class to the class of that boundary in $H_{n-1}(A;R)$. Changing c by an A-chain changes its boundary by an A-boundary, and changing c by an absolute boundary changes it by zero. These checks make the connecting homomorphism well-defined. Its degree drop is essential: an n-dimensional relative filling detects an (n−1)-dimensional boundary.

The short exact sequence of chain complexes produces a natural long exact sequence. Exactness at each term says that the incoming image equals the outgoing kernel. It does not say that every map is injective or surjective, and it does not describe the maps solely from knowing adjacent groups. Two maps from the integers to the integers, multiplication by one and multiplication by five, have different cokernels even though their domain and codomain are identical. Keep the inclusion and connecting maps visible in every calculation.

An oriented disk relative to its boundary has an integer relative class whose connecting image is the oriented boundary sphere. Ordinary disk homology alone does not contain that top-dimensional class. For a nonempty connected disk and its nonempty boundary, the degree-zero part of the exact sequence also matters; replacing an ordinary H₀ term by zero without using a reduced sequence gives the wrong conclusion. State whether a calculation uses ordinary or reduced groups, and preserve that choice through the whole sequence.

Changing an orientation can change a connecting map's generator by a minus sign while leaving its kernel, image, and resulting isomorphism types unchanged. Such a sign choice is harmless only when applied consistently to all related formulas. In computations with several cells or several pieces, inconsistent orientations can destroy the chain identity and produce false ranks or torsion. Relative homology is therefore both a conceptual localization tool and a useful bookkeeping framework for translating geometric attachments into algebraic maps.