1cochain

A 1-cochain is a mathematical object used in cohomology theories, such as simplicial, singular, or de Rham cohomology, with coefficients in an abelian group G. In the context of a simplicial complex K, the group of 1-cochains C^1(K; G) consists of all functions that assign to every oriented 1-simplex (edge) [v0, v1] an element of G. Equivalently, C^1(K; G) ≅ Hom(C1(K), G), where C1(K) is the free abelian group generated by the oriented edges. A 1-cochain thus can be viewed as labeling each directed edge with a G-value; there is no requirement that the value on [v0, v1] determine the value on the reverse orientation [v1, v0].

The coboundary operator δ: C^1(K; G) → C^2(K; G) maps a 1-cochain to a 2-cochain. For a 2-simplex [v0, v1, v2], the value is typically given by (δf)([v0, v1, v2]) = f([v1, v2]) − f([v0, v2]) + f([v0, v1]), with signs determined by the chosen orientation. The 1-cocycles Z^1(K; G) are those 1-cochains f with δf = 0, while the 1-coboundaries B^1(K; G) are those arising as δg for some 0-cochain g ∈ C^0(K; G). The first cohomology group is H^1(K; G) = Z^1(K; G) / B^1(K; G).

In graph-theoretic terms, a 1-cochain assigns a value to each directed edge; a 1-coboundary corresponds to the edge-label differences of a vertex labeling, and a 1-cocycle encodes edge-labelings that are locally consistent around 2-simplices. The concept generalizes to other coefficient groups and to other cohomology theories.