Chain and contrachain complexes

An oriented simplicial complex ‡ determines, for each dimension p, a chain group Cp and a boundary homomorphism ∂: Cp → Cp − 1 From these data the homology and contrahomology groups may be obtained. We now propose to confine attention to these purely algebraical concepts and accordingly define

3.1.1 Definition A chain complex C. = {Cp, ∂p}, is a collection of abelian groups Cp, one for each integer p, and of (right) homomorphisms ∂p : Cp → Cp − 1 such that ∂p ∂p − 1 = 0, for each p.

Dually, we define

3.1.1c Definition A contrachain complex, C = {Cp, δp} is a collection of abelian groups Cp, one for each integer p, and of (left) homomorphisms δp : Cp → Cp + 1 such that δp + 1δp = 0, for each p.

We shall generally write C for C, and shall often write ∂, δ for ∂p, δp.

It is clear how we may define the homology groups Hp (C) of the chain complex C; if Zp or Zp (C), the p-th cycle group, is the kernel of ∂p and Bp or Bp (C), the p-th boundary group, is the image of ∂p + 1, then Bp is a subgroup of the abelian group Zp and Hp (C) is the factor group Zp\Bp. The definition of Hp (C) is analogous; if the context makes it clear that a contrachain complex is in question, we may suppress the superscript dot, so that we may write Hp (C).