Cochain Complexes and Cohomology

Core Idea

A cochain complex is obtained by dualizing a chain complex: replacing each chain group C_n with the dual group Hom(C_n, G) (typically G = Z) and reversing the direction of the maps. The coboundary operator dⁿ goes "upward" from Cⁿ to Cⁿ⁺¹, and cohomology Hⁿ = ker(dⁿ)/im(d^n-1) measures the failure of cocycles to be coboundaries. While cohomology carries the same information as homology for spaces over a field, over the integers it carries strictly more information due to the universal coefficient theorem, and the cup product gives cohomology a ring structure that homology lacks.

Explainer

Cohomology is the dual theory to homology, obtained by applying the Hom functor to the chain complex. Given a chain complex C_* with boundary operators d_n : C_n -> C_{n-1}, and a coefficient group G (typically Z, Q, or Z/pZ), the cochain group C^n(X; G) = Hom(C_n(X), G) consists of all group homomorphisms from the n-th chain group to G. A cochain f in Cⁿ assigns an element of G to each singular n-simplex — it "evaluates" chains rather than being a chain itself. The coboundary operator dⁿ : Cⁿ -> Cⁿ⁺¹ is defined by d^n(f) = f compose d_{n+1}: it precomposes a cochain with the boundary map, pulling it up one dimension.

The fundamental property dⁿ⁺¹ compose dⁿ = 0 follows immediately from d compose d = 0 in the chain complex. This makes (C^*, d^*) a cochain complex — a sequence of abelian groups with maps going "upward" in dimension whose composition is zero. The n-th cohomology group is H^n(X; G) = ker(dⁿ) / im(d^n-1). Elements of ker(dⁿ) are called cocycles — cochains that vanish on boundaries. Elements of im(d^n-1) are called coboundaries — cochains that are the coboundary of a lower-dimensional cochain. Two cocycles represent the same cohomology class when they differ by a coboundary.

The relationship between homology and cohomology is governed by the universal coefficient theorem, which states (for integer coefficients) that H^n(X; Z) fits into a short exact sequence 0 -> Ext(H_{n-1}(X), Z) -> H^n(X; Z) -> Hom(H_n(X), Z) -> 0. When the homology groups are free abelian (no torsion), the Ext term vanishes and H^n(X; Z) = Hom(H_n(X), Z), the usual algebraic dual. Torsion in homology produces additional torsion in cohomology, shifted by one degree — this is the "extra information" that cohomology carries over the integers.

The deepest reason to study cohomology alongside homology is the cup product, which gives H^*(X; R) = direct sum H^n(X; R) the structure of a graded ring (when R is a commutative ring). This multiplicative structure is invisible from the homology side and provides a strictly finer topological invariant. Poincare duality — the statement that H^k(M) = H_{n-k}(M) for a closed oriented n-manifold M — is most naturally expressed in cohomological terms. Characteristic classes (Stiefel-Whitney, Chern, Pontryagin), which classify vector bundles, live in cohomology. Obstruction theory, which determines when maps with certain properties exist, is formulated cohomologically. Cohomology is not merely the "dual of homology" but a richer and more structured invariant in its own right.