nboundaries

Nbounds, or n-boundaries, is a term used in algebraic topology to denote the n-dimensional boundaries within a chain complex. Given a chain complex C_* with boundary maps ∂n: Cn → Cn−1, the n-boundaries, denoted Bn, are the image of the next boundary map: Bn = ∂n+1(Cn+1) ⊆ Cn. Thus Bn consists of those n-chains that are the boundary of some (n+1)-chain.

Nbounds are related to and distinguished from n-cycles. The n-cycles Zn are defined as Zn = ker ∂n, the n-chains whose boundary is zero. The homology group in degree n is defined as Hn = Zn/Bn, which measures how many n-cycles fail to be boundaries. Therefore, Bn ⊆ Zn ⊆ Cn, and Hn captures information about the structure that persists across dimensions.

Properties and computation. Nbounds form a subgroup of Cn, and their computation is central to determining homology. In practice, Bn is obtained as the column space (image) of the matrix representing ∂n+1 with respect to chosen bases. This makes the relationship among Bn, Zn, and Hn amenable to linear-algebra methods, especially in cell complexes or simplicial complexes used in computational topology.

Examples. In a filled triangle (a 2-simplex), B1 is generated by the boundary of the triangle, and Zn captures cycles in the 1-skeleton. Since the filled interior bounds the 1-skeleton, H1 = Z1/B1 = 0. In a graph without 2-dimensional cells, B1 = 0, so H1 is isomorphic to Zn and can be nontrivial.

See also: chain complexes, boundary operators, cycles, homology, simplicial and singular homology.