Cup products and cohomology rings

For a p-cochain $\varphi$ and a q-cochain $\psi$, evaluate their cup product on a singular (p+q)-simplex by evaluating the first on its front p-face and the second on its back q-face, then multiplying in R. The two faces meet at the vertex indexed p. This Alexander–Whitney convention is compatible with the earlier unsigned cochain differential $\delta\varphi=\varphi\partial$. The Leibniz rule supplies the degree sign that makes products of cocycles remain cocycles.

If one cocycle is changed by a coboundary, the Leibniz identity shows that the product changes by a coboundary. Therefore cup product descends to cohomology classes, where it has degree addition, associativity, and a unit in degree zero. Continuous maps preserve this product under pullback, making the induced cohomology homomorphism a graded ring map. These properties are stronger than preserving the separate additive groups; any proposed comparison of cohomology rings must respect both grading and multiplication.

For cohomology classes a and b of degrees p and q, the relation is $a\smile b=(-1)^{pq}b\smile a$. At the cochain level the two products need not agree exactly; a chain-homotopy argument produces the graded relation on cohomology. For an odd-degree class, this only forces twice its square to vanish. If the relevant target group has no two-torsion, the square is zero. Over characteristic two the sign alone imposes no vanishing conclusion.

The integer torus ring is an exterior algebra on two degree-one generators: each generator squares to zero, their product generates degree two, and reversing the product changes its sign. In contrast, real projective space with F₂ coefficients has nonzero powers of a degree-one generator up to its dimension. These examples show why a rule learned from real differential forms cannot be transferred without qualification to every coefficient ring. Coefficient choices alter both additive groups and the relations that can be deduced from graded signs.

A torus and the wedge of two circles with a two-sphere have the same integer cohomology groups in every degree: one copy in degrees zero and two, and two copies in degree one. Yet the torus has a nonzero product of its two independent degree-one classes. In the wedge, all products of degree-one classes vanish: restriction to each summand shows that no degree-two product is created. The additive groups alone therefore cannot establish a homotopy equivalence, while the ring provides an obstruction.

On an oriented manifold, evaluating a top-degree cup product on the fundamental class turns multiplication into intersection information. The chosen orientation fixes the sign of that evaluation. A continuous self-map's pullback must preserve cup products, so its action in degree one constrains its action in degree two. This is useful for checking degree calculations, but it should not be confused with a claim that every algebraic ring endomorphism necessarily comes from a continuous map. Realizability is a further topological problem.