ncochain

In algebraic topology, an n-cochain with coefficients in an abelian group G is a function assigning to each singular n-simplex of a space X (or to each oriented n-simplex of a simplicial complex) a value in G. The set of all n-cochains is denoted C^n(X; G).

The cochains form an abelian group under pointwise addition. There is a coboundary operator δ^n: C^n(X; G) → C^{n+1}(X; G) defined via precomposition with the face maps and an alternating sum: (δf)(σ) = Σ_{i=0}^{n+1} (-1)^i f(σ ∘ ∂_i). For singular cochains one uses the singular face maps; for simplicial cochains, a similar formula holds with the face maps on simplices. A fundamental property is that δ^{n+1} ∘ δ^n = 0, making the cochains into a cochain complex.

The cohomology groups H^n(X; G) are defined as ker δ^n / im δ^{n-1}. They capture global algebraic information about X and can classify geometric and topological structures; for example, H^2 can classify line bundles in certain contexts. In degree 0, 0-cochains are functions on the vertices, and coboundary identifies locally constant functions.

Remarks: In the simplicial or singular setting, cochains can be viewed as Hom-objects dual to chains when G is abelian. When G is a ring, the cochain groups C^*(X; G) admit a cup product, giving a graded ring structure on the cohomology. Cochains provide a flexible computational framework for cohomology and its various applications in topology.