Singular Cohomology

Core Idea

Singular cohomology H^n(X; G) is defined by applying Hom(-, G) to the singular chain complex of X. It inherits all the computational tools of singular homology (long exact sequences, Mayer-Vietoris, excision) in dualized form, with arrows reversed. Singular cohomology is the natural home for duality theorems, multiplicative structures, and obstruction theory. Its pairing with homology via the Kronecker pairing provides a bridge between the two theories.

Explainer

Singular cohomology with coefficients in an abelian group G is constructed by dualizing the singular chain complex: C^n(X; G) = Hom(C_n(X), G), the group of all homomorphisms from the singular n-chain group to G. A singular n-cochain assigns an element of G to each singular n-simplex in X. The coboundary dⁿ : Cⁿ -> Cⁿ⁺¹ is defined by (dⁿ f)(sigma) = f(d_{n+1} sigma), and the cohomology groups H^n(X; G) = ker(dⁿ)/im(d^n-1) measure the failure of cocycles (cochains vanishing on boundaries) to be coboundaries.

All the computational tools of singular homology have cohomological counterparts, obtained by applying Hom and using the functorial properties. There is a long exact sequence for pairs: ... -> H^n(X, A) -> H^n(X) -> H^n(A) -> Hⁿ⁺¹(X, A) -> ..., with arrows reversed relative to the homology version. There is a Mayer-Vietoris sequence for cohomology: ... -> H^n(X) -> H^n(A) direct sum H^n(B) -> H^n(A intersect B) -> Hⁿ⁺¹(X) -> ..., again with reversed arrows. Excision holds for cohomology (it is inherited from the chain-level excision). These tools are used to compute cohomology in exactly the same way as their homological counterparts, with the direction of maps reversed throughout.

The contravariance of cohomology — the fact that a map f : X -> Y induces f* : H^n(Y) -> H^n(X) in the reverse direction — is not a defect but a feature. It means cohomology classes are "pulled back" along maps, which is the correct behavior for quantities that assign values to cycles. A differential form on Y (in the de Rham setting) pulls back to a differential form on X via f; a characteristic class of a vector bundle pulls back to a characteristic class of the pullback bundle. Contravariance is natural for "measurement" or "evaluation" type quantities, which is why cohomology (not homology) is the correct framework for obstruction theory, characteristic classes, and sheaf theory.

The Kronecker pairing <,> : H^n(X; G) x H_n(X; Z) -> G, defined by evaluating a cocycle representative on a cycle representative, gives a natural bilinear map between cohomology and homology. When G = Z and the homology is free abelian, this pairing identifies H^n(X; Z) with Hom(H_n(X; Z), Z), the algebraic dual. When there is torsion, the universal coefficient theorem introduces a correction term from Ext. Over a field, cohomology and homology are perfectly dual, but over Z, the torsion information in cohomology comes from the torsion of H_{n-1} (shifted by one degree), giving cohomology a slightly different "view" of the same space.

The most important distinguishing feature of singular cohomology is that it carries a natural ring structure via the cup product, which will be developed in the next topic. This ring structure makes H^*(X) a graded ring, and the ring structure is a strictly finer invariant than the individual cohomology groups — spaces with isomorphic cohomology groups can have non-isomorphic cohomology rings. The cup product is the reason cohomology, rather than homology, is the primary algebraic tool in much of modern topology.