Cochain Complexes

Algebraic Topology · Axiom Academy

The dual construction to chain complexes in algebraic topology Given a chain complex (C n (X), ∂) and a coefficient group G, we define the n-th cochain group as the group of all homomorphisms from the n-th chain group to G. Elements of C n (X; G) are called n-cochains . They assign values in G to each n-dimensional chain. Just as the boundary operator ∂ maps chains to lower-dimensional chains, the coboundary operator δ maps cochains to higher-dimensional cochains. It is defined by dualizing the boundary operator. The coboundary operator increases dimension by one, moving "up" the cochain complex, dual to how the boundary operator decreases dimension. A fundamental property of the coboundary operator mirrors the property ∂² = 0 from chain complexes: applying δ twice always gives zero. Proof idea: For any n-cochain φ and any (n+2)-chain c, Since ∂² = 0 in the chain complex, we have ∂∂c = 0, so (δδφ)(c) = 0 for all c, proving δ² = 0. The property δ² = 0 means that im(δ n-1 ) ⊆ ker(δ n ), allowing us to define important subgroups. n-cocycles: Z n (X; G) = ker(δ n ) = φ ∈ C n : δφ = 0 n-coboundaries: B n (X; G) = im(δ n-1 ) = δψ : ψ ∈ C n-1 Cocycles are cochains with zero coboundary (they are "closed"), while coboundaries are cochains that are themselves coboundaries of lower-dimensional cochains (they are "exact"). 5. The Cochain Complex Structure Assembling all the cochain groups and coboundary operators, we obtain a cochain complex :

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