Relative Homology and the Long Exact Sequence of a Pair

Core Idea

Relative homology H_n(X, A) measures the homology of X "modulo" its subspace A — it detects holes in X that are not already present in A. The short exact sequence of chain complexes 0 -> C_*(A) -> C_*(X) -> C_*(X)/C_*(A) -> 0 gives rise to a long exact sequence ... -> H_n(A) -> H_n(X) -> H_n(X, A) -> H_{n-1}(A) -> ..., which is the principal tool for computing homology by decomposing spaces into simpler pieces. The connecting homomorphism in this sequence encodes how the topology of the boundary of A interacts with the topology of the whole space.

Explainer

Given a pair (X, A) where A is a subspace of X, the relative chain group C_n(X, A) is defined as the quotient C_n(X)/C_n(A). Since A is a subspace, C_n(A) (generated by singular simplices mapping into A) is naturally a subgroup of C_n(X). The boundary operator on C_n(X) descends to a well-defined boundary operator on C_n(X, A) — if a chain lies in C_n(A), its boundary lies in C_{n-1}(A) — and the resulting quotient chain complex has homology groups H_n(X, A) = ker(d_n)/im(d_{n+1}) computed in C_*(X, A). A relative n-cycle is represented by a chain in X whose boundary lies in A (not necessarily zero), and a relative n-boundary is a chain in X that differs from a chain in A by a boundary.

The long exact sequence of the pair (X, A) is the centerpiece of computational homology. The short exact sequence of chain complexes 0 -> C_*(A) -> C_*(X) -> C_*(X, A) -> 0 gives rise, by a general algebraic construction (the snake lemma), to a long exact sequence in homology: ... -> H_n(A) -i_*-> H_n(X) -j_*-> H_n(X, A) -partial-> H_{n-1}(A) -i_*-> H_{n-1}(X) -> ... Here i_* is induced by the inclusion A hookrightarrow X, j_* is induced by the quotient C_*(X) -> C_*(X, A), and the connecting homomorphism partial is the key new map. It takes a relative cycle (a chain in X whose boundary lies in A), extracts that boundary, and views it as a cycle in A.

The long exact sequence is the Swiss army knife of homology computation. By knowing two of the three terms (H_*(A), H_*(X), H_*(X,A)), one can often deduce the third. The standard application: if X = Dⁿ (contractible), then H_k(Dⁿ) = 0 for k > 0, and the sequence gives isomorphisms H_k(Dⁿ, S^n-1) = H_{k-1}(S^n-1) for k >= 2. Since H_k(Dⁿ, S^n-1) is isomorphic to the reduced homology of Dⁿ/S^n-1 = Sⁿ, this provides an inductive computation of H_*(Sⁿ). More generally, whenever a space X is built by attaching cells to a subspace A, the long exact sequence relates the new homology created by the attachment to the homology of A and the relative homology of the pair.

Relative homology has an important topological interpretation. For "good" pairs (X, A) — meaning A is a deformation retract of some neighborhood in X, which holds in all practical cases — the relative homology H_n(X, A) is isomorphic to the reduced homology of the quotient space X/A (where all of A is collapsed to a point). This is a consequence of the excision theorem and provides a geometric picture: relative homology measures what homology classes of X "survive" after we crush A to a point. This perspective is essential for cellular homology, where the relative homology of successive skeleta (Xⁿ, X^n-1) detects exactly the new cells attached at each stage.

The long exact sequence generalizes beyond pairs. For a triple A subset B subset X, there is a long exact sequence relating H_*(X, A), H_*(X, B), and H_*(B, A). The Mayer-Vietoris sequence, which relates the homology of a union to the homology of its pieces, can be derived from the long exact sequence of a pair. These tools, together with excision, form the complete computational toolkit for singular homology — allowing the systematic decomposition of complex spaces into simpler pieces whose homology can be determined.