ncoboundaries

N-coboundaries, or n-coboundaries, are a concept in cohomology theory referring to a specific subgroup of n-cochains that arise as the image of a coboundary operator. In a cochain complex (C^*, δ^*), each δ^n maps C^n to C^{n+1}. The n-coboundaries B^n are defined as the image of δ^{n-1}, that is, B^n = Im(δ^{n-1}) ⊆ C^n. They consist of those n-cochains that can be written as the coboundary of an (n−1)-cochain.

N-coboundaries play a key role alongside cocycles in the construction of cohomology groups. The n-cocycles Z^n are the cochains in ker δ^n, and by the property δ^n ∘ δ^{n-1} = 0, every n-coboundary is a cocycle, so B^n ⊆ Z^n. The n-th cohomology group is defined as the quotient H^n = Z^n / B^n, measuring the failure of cocycles to be coboundaries.

In practical terms, to compute n-coboundaries one examines a specific cochain complex, such as those arising from topological spaces (singular or simplicial cochains) or algebraic objects (modules over a ring). The coboundary maps are dual to boundary maps in homology, and B^n represents the “trivial” cohomology classes that come from lower degrees.

Summary: ncoboundaries are the image of the coboundary operator at degree n, forming a subgroup of n-cochains, contained in the cocycles, and they are essential for defining the cohomology groups H^n.