DimC1

DimC1, sometimes written as dim C1 or dimF C1, is a notation encountered in algebraic topology and homological algebra to denote the dimension (rank) of the chain group C1 in a chain complex, or more generally the dimension of the vector space component labeled as C1 when coefficients lie in a field.

In the standard simplicial chain formalism, a chain complex (C*, ∂*) consists of vector spaces Ck for each dimension k, with boundary maps ∂k: Ck → Ck-1. The group C1 is the free vector space generated by the oriented 1-simplices (edges) of a chosen simplicial complex. Therefore dimF C1 equals the number of oriented 1-simplices in the complex, which is typically twice the number of undirected edges if each edge is represented in both orientations. For example, a triangle has three undirected edges and six oriented 1-simplices, giving dimF C1 = 6 over a field F.

DimC1 is often used alongside other dimensions in the chain complex, such as dimC0 for vertices and dimC2 for 2-simplices, and in computations of homology. The first homology group H1 depends on the kernels and images of boundary maps, with relationships governed by rank-nullity theorems: dim Im ∂2 ≤ dim Ker ∂1, and h1 = dim Ker ∂1 − dim Im ∂2.

Notes: the exact interpretation of dimC1 can vary with conventions (or with coefficients in different rings), so readers should check the specific definition used in a given text. See also chain complex, chain group, simplicial homology, and boundary map.