k1cochain

K1-cochain is a term used in algebraic topology to denote a 1-cochain on a simplicial complex with coefficients in a field k. Concretely, if K is a simplicial complex and C1(K) is the free abelian group generated by the oriented 1-simplices of K, then the space of k1-cochains is the vector space C1^1(K; k) = Hom(C1(K), k). A k1-cochain assigns to each oriented edge [v0, v1] a value in k.

The cochain groups fit into a cochain complex with coboundary operators. The coboundary δ: C^0(K; k) → C^1(K; k) maps a 0-cochain (a function on vertices) to a 1-cochain via δf([v0, v1]) = f(v1) − f(v0). The next coboundary δ: C^1(K; k) → C^2(K; k) is defined on a 2-simplex [v0, v1, v2] by δf([v0, v1, v2]) = f([v1, v2]) − f([v0, v2]) + f([v0, v1]). The cohomology group H^1(K; k) is the kernel of δ: C^1 → C^2 modulo the image of δ: C^0 → C^1, capturing the 1-dimensional topological features of K.

Interpretations and examples: a k1-cochain is a coboundary if it lies in the image of δ from C^0, in which case it represents a discrete potential difference (a gradient). A 1-cochain is a cocycle if δ of it is zero, and cocycles modulo coboundaries define H^1(K; k). In graphs, H^1(K; k) counts independent cycles and relates to network flows and discrete differential forms.

See also: cochain, cohomology, simplicial complex, cochain complex, k-cochains.