CochainsEdit

Cochains are a fundamental construct in algebraic topology that encode how local geometric data on a space can be assembled into global algebraic information. They sit in a dual position to chains: while chains track oriented, combinatorial pieces of a space, cochains assign values to those pieces and interact with them through a coboundary operation. This pair of ideas—cochains and their coboundaries—provides a flexible framework for defining and manipulating cohomology theories, which are among the most robust tools for distinguishing spaces up to homeomorphism or homotopy equivalence.

The cochain perspective is attractive because it makes many constructions explicit and computationally accessible. There are several common models of cochains, each built to suit different problems. One can work with singular cochains, Čech cochains, de Rham cochains, or other variants, and all of these lead to the same essential invariants after passing to cohomology. For a space X and an abelian group G, cochains come in a graded sequence C^n(X; G) connected by the coboundary map, and they form a cochain complex. The formalism is versatile enough to support additional structures, such as products on cochains that induce ring structures on cohomology.

History and foundations

Cochains entered the mathematical toolkit in the mid-20th century as part of a broader effort to organize topological information algebraically. Early work by Eilenberg, Steenrod, and their collaborators established the axiomatic backbone for cohomology theories, while contemporaries developed concrete models that could be calculated in practice. The development of singular cochains provided a hands-on approach to compute invariants of spaces, while Čech and de Rham cochains offered alternatives better suited to open coverings or smooth manifolds, respectively. For an overview of how these perspectives relate, see Cohomology and Cech cohomology.

Formalism

Cochains and the coboundary operator

Let X be a topological space and G an abelian group. The group of n-cochains C^n(X; G) consists of all functions from the set of singular n-simplices of X to G. A singular n-simplex is a continuous map from the standard n-simplex Δ^n into X, so a cochain assigns an element of G to every such map.

The coboundary operator δ: C^n(X; G) → C^{n+1}(X; G) is defined by a standard alternating sum reflecting how the boundary of an (n+1)-simplex decomposes into its n-dimensional faces. Concretely, for φ ∈ C^n(X; G) and a singular (n+1)-simplex σ, δφ(σ) is computed by evaluating φ on the various face maps of σ and combining the results with alternating signs. A key property is δ∘δ = 0, which ensures the meaningful definition of cohomology.

These ideas can be expressed through a cochain complex (C^n(X; G), δ), where the successive application of δ maps cochains to higher-degree cochains and annihilates those that come from lower-degree cochains.

Cochain complexes and cohomology

The cohomology groups are defined as H^n(X; G) = ker δ / im δ, the quotient of n-cocycles (cochains with δφ = 0) by n-coboundaries (cochains that are themselves δ of some (n−1)-cochain). These groups measure the extent to which local data cannot be globally assembled without obstruction.

Different models of cochains yield the same cohomology up to natural isomorphism, so long as the spaces and maps are treated consistently. For this reason, researchers can choose the model most convenient for computation or the one best suited to the geometry at hand.

Variants of cochains

Singlar cochains: use all singular n-simplices to build C^n(X; G). This model is flexible and broadly applicable, especially for abstract topological spaces. See Singular cochain.
Simplicial or simplicial cochains: employ a fixed triangulation or simplicial structure on X, sometimes simplifying explicit calculations in combinatorial topology. See Simplicial cochain.
Čech cochains: rely on open coverings of X and encode data on intersections of covering sets; this model is particularly handy for sheaf-theoretic viewpoints and for spaces with a rich open covering structure. See Čech cohomology.
de Rham cochains: for smooth manifolds, one can use differential forms to model cochains, leading to de Rham cohomology. This differential-geometric approach connects topology with analysis. See de Rham cohomology.
Alexander–Spanier cochains: use a refinement based on neighborhoods and local data, providing another flexible framework for cohomology. See Alexander–Spanier cochain. Each model has its own technical advantages, and under broad conditions they yield canonically isomorphic cohomology theories.

Products and structures

Cochains admit additional algebraic operations that pass to cohomology. The most important is the cup product, a bilinear operation ⌣ on cochains that respects degrees and induces a graded-commutative ring structure on cohomology: H^*(X; G) becomes a graded ring with the cup product. This structure provides a powerful tool for distinguishing spaces and for encoding how local pieces of X interact. See Cup product.

Maps between spaces pull back cochains: a continuous map f: Y → X induces a pullback on cochains f^: C^n(X; G) → C^n(Y; G), which in turn induces a homomorphism f^: H^n(X; G) → H^n(Y; G). This functorial behavior makes cohomology a robust invariant under continuous deformations.

Examples and intuition

A point: for any abelian group G, H^0(point; G) ≅ G and H^n(point; G) = 0 for n > 0. This reflects the intuition that a single point has no interesting higher-dimensional cohomology.
The circle S^1: with coefficients in G, H^0(S^1; G) ≅ G and H^1(S^1; G) ≅ G, with higher cohomology groups vanishing. The nontrivial H^1 detects the “loopiness” of the circle via cochains that measure how a 1-simplex wraps around the hole.
Spheres S^n: H^k(S^n; G) = 0 for 0 < k < n, with H^0 ≅ H^n ≅ G. These results are canonical and can be approached through various cochain models, each making the same invariants visible from different angles.

These computations illustrate how cochains translate geometric features into algebraic data, and how different spaces imprint distinctive cohomological fingerprints.

Foundations, interpretations, and debates

Cochains sit at the heart of a mature framework for extracting global invariants from local information. In practice, mathematicians choose models that align with their problem domain: differential geometers prefer de Rham cochains, while combinatorial topologists may favor simplicial or singular cochains. The development of cohomology theories under axiomatic systems (the Eilenberg–Steenrod framework) clarified when different constructions yield compatible invariants and how generalized cohomology theories extend these ideas.

One area of ongoing methodological discussion concerns the balance between concrete calculability and abstract generality. Singular cochains offer universality and are well-suited for proofs that only depend on basic properties of maps, while Čech and sheaf-theoretic approaches excel in settings where coverings and local-to-global principles dominate. The interplay between these viewpoints is a rich source of insights and occasional technical debates, but the central invariants remain consistent across models.

In physics and geometry, cochains and their cohomology classes provide a language for conserved quantities, fluxes, and topological constraints. The cup product and higher cohomology operations organize how local data can interact, with consequences in areas as varied as gauge theory, complex geometry, and the study of manifolds. See Cohomology and Cup product for broader context.