n1cochains

N−1 cochains, often referred to as (n−1)-cochains, are elements of the cochain group in a cochain complex used in algebraic topology. Given a topological space X or a simplicial complex K and a coefficient group or module A (typically abelian), the group of (n−1)-cochains, denoted C^{n−1}(X; A) or C^{n−1}(K; A), consists of all functions that assign to every oriented (n−1)-simplex the value in A. In the singular setting, an (n−1)-cochain is a function from the set of singular (n−1)-simplices in X to A.

The cochain groups form a graded abelian group under pointwise addition. A coboundary operator δ: C^{n−1} → C^n is defined so that for an (n−1)-cochain φ and an n-simplex σ, δφ(σ) is the alternating sum of φ over the (n−1)-faces of σ. Concretely, δφ(σ) = ∑_{i=0}^n (−1)^i φ(σ restricted to the i-th face). The fundamental property is δ∘δ = 0, which makes the sequence C^{0} → C^{1} → C^{2} → ... into a cochain complex.

Cohomology groups are defined as H^{n−1}(X; A) = ker δ: C^{n−1} → C^n / im δ: C^{n−2} → C^{n−1}. Cohomology classes capture global information about X and allow the use of cochains to represent geometric and topological features, such as in the description of functions on vertices (0-cochains) and their induced differences along edges (1-cochains). Cochains also support further structures like cup products, aiding the computation of richer invariants.