Relative CohomologyEdit

Relative cohomology is a cornerstone of algebraic topology that analyzes how the cohomological structure of a space X interacts with a distinguished subspace A ⊆ X. It formalizes the idea of measuring features of X that become trivial when restricted to A, or equivalently, the obstruction to extending a cochain from A to all of X. This construction sits naturally alongside absolute cohomology and is organized by exact sequences that encode how the pieces fit together.

From a practical standpoint, relative cohomology is a flexible tool for understanding spaces with boundary-like data, inclusions, or decompositions. It plays a central role in many areas of geometry, topology, and their applications, and it has several concrete realizations, including singular, de Rham, and Čech variants. The theory also provides a bridge to powerful duality principles and computational techniques that reveal global structure from local information.

Historically, the relative theory emerged in the mid-20th century as part of the development of a robust axiomatic framework for (co)homology. It was developed and clarified by early contributors such as Leray and later integrated into the standard toolkit by Eilenberg and Steenrod and their successors. Today, the relative theory is taught as a routine device in the study of spaces with inclusions, and it complements the study of absolute invariants by incorporating boundary or attachment data.

Definitions and basic constructions

Relative cochains and cohomology

Let X be a topological space and A ⊆ X a subspace. There are two common, equivalent ways to form the relative cochain groups:

Quotient model: C^n(X, A) = C^n(X) / C^n(A), where C^n(Y) denotes the group of singular n-cochains on Y and C^n(A) is viewed inside C^n(X) by restriction.
Vanishing-on-A model: C^n(X, A) = {φ ∈ C^n(X) | φ|A = 0}. The differential on these groups is inherited from the ambient cochain complex.

Either model yields a cochain complex whose cohomology is the relative cohomology H^n(X, A). This theory reduces to the ordinary, absolute cohomology when A is empty, i.e., H^n(X, ∅) ≅ H^n(X).

For a more geometric perspective, one can view relative cochains as capturing cochains on X that are “compatible with” A, in the sense that their restriction to A vanishes (in the vanishing-on-A picture) or, equivalently, as classes modulo those that come from A (in the quotient picture).

Refer to relative cohomology for the formal setup and the standard proofs that these two viewpoints agree.

The long exact sequence of a pair

A fundamental feature of relative cohomology is that it participates in a long exact sequence associated to a pair (X, A). For every n, there is a natural exact sequence

... → H^n(X, A) → H^n(X) → H^n(A) → H^{n+1}(X, A) → H^{n+1}(X) → ...

The maps have concrete descriptions: the first map is induced by the inclusion X → X with the subspace A ignored, the second map is restriction to A, and the connecting homomorphism δ shifts degree and encodes the obstruction to lifting a class from A back to X.

This exact sequence provides a precise measure of how much the cohomology of X differs from that of A, and it underpins many computational strategies, such as deducing H^(X, A) from known information about H^(X) and H^*(A).

Functoriality and naturality

Given a continuous map f: (X, A) → (Y, B) with f(A) ⊆ B, there is an induced map f^*: H^n(Y, B) → H^n(X, A) that respects the long exact sequences of the pairs. This functoriality is essential for comparing spaces and for transferring computations along maps, and it is a standard feature of any cohomology theory that behaves well with inclusions.

Variants and models

Singular relative cohomology uses singular cochains, a framework that mirrors the classical singular homology and provides a very general and flexible model.
Relative de Rham cohomology applies to smooth manifolds and uses differential forms; the complex Ω^(X, A) of forms vanishing on A yields H^_dR(X, A), with natural comparisons to the singular theory via the de Rham theorem.
Čech relative cohomology provides another approach, often convenient in sheaf-theoretic or topological settings with good covers.

In all these variants, the core ideas—the quotient or vanishing-to-A viewpoint, the long exact sequence, and the functorial behavior—remain the guiding principles.

Examples and computations

Disk and boundary: Consider the pair (D^n, S^{n-1}) of the n-disk with its boundary. One has H^k(D^n, S^{n-1}) ≅ Z if k = n and 0 otherwise. This reflects the idea that the boundary S^{n-1} contributes the only nontrivial relative cohomology class in degree n.
Trivial subspace: For any space X, H^n(X, ∅) ≅ H^n(X). The relative theory recovers the absolute theory when the subspace carries no information.
Whole space as subspace: H^n(X, X) ≅ 0 for all n. Since every cochain vanishes on X, the relative groups collapse.
Use of excision: If A ⊆ U ⊆ X with the closure of U contained in the interior of A, excision allows one to replace (X, A) by (X \ U, A \ U) for the purpose of computing relative cohomology, simplifying many calculations, and linking local data to global invariants.
Manifolds with boundary and duality: For a compact oriented manifold M with boundary ∂M, the relative cohomology H^*(M, ∂M) pairs with relative homology and, via Poincaré duality (in the oriented case), yields powerful global invariants that reflect both the interior and boundary geometry.
Relative de Rham intuition: For smooth manifolds, one can study relative cohomology class representatives as differential forms that vanish on the boundary in the de Rham model, providing a concrete analytic viewpoint.

Connections and applications

Relationship to absolute invariants: The long exact sequence of a pair ties together H^(X), H^(A), and H^*(X, A), showing precisely how information is gained or lost when passing from A to X.
Duality principles: Relative cohomology is central to duality theorems such as Poincaré-Lefschetz duality, which relates the cohomology of a manifold with boundary to the homology of its boundary.
Computation techniques: The Mayer–Vietoris sequence, together with excision, provides efficient strategies for calculating relative cohomology by breaking a space into simpler pieces.
Connections to other theories: Relative cohomology participates in the categorical and derived viewpoints of modern topology, including mapping cone constructions and derived functors, while remaining accessible through more concrete chain-level descriptions.

Computation techniques and tools

Mayer–Vietoris sequence for relative cohomology: A tool for patching local data to obtain global information about H^*(X, A) from open covers and the restrictions of A to those cover pieces.
Excision and homotopy invariance: Critical properties that justify replacing parts of a space without changing the relative invariants, thereby simplifying calculations.
Relative long exact sequence as a computational backbone: The standard workflow uses known H^(X) and H^(A) to extract H^*(X, A) and to understand how inclusions affect cohomology.