Cup product
Hatcher §3.2 (pages 206–229). Mapped, not yet decomposed.
The cup product of cochains, defined on singular cochains by splitting a
simplex into front and back faces, descends to cohomology and makes
H∗(X; R) a graded ring, graded-commutative when R is commutative. The
ring structure is a genuinely finer invariant: S² ∨ S⁴ and ℂP² have the
same cohomology groups and different cohomology rings. The section computes
the rings of projective spaces, proves a Künneth formula for products, and
surveys spaces with polynomial cohomology.
Nothing here is upstream, and it depends on cohomology groups existing first. The graded-commutativity proof is the technical centre.