The Cup Product

Algebraic Topology · Axiom Academy

A fundamental ring structure on cohomology that makes it more powerful than homology 1. Definition of the Cup Product The cup product is a bilinear operation that takes two cohomology classes and produces a new one in higher dimension: Given α ∈ H p (X) and β ∈ H q (X), their cup product α ∪ β is an element of H p+q (X). This operation is defined at the cochain level and descends to cohomology. The cup product satisfies several important algebraic properties that make cohomology a graded-commutative ring: Bilinearity: The cup product is linear in each variable Associativity: (α ∪ β) ∪ γ = α ∪ (β ∪ γ) Unit Element: There exists 1 ∈ H⁰(X) such that 1 ∪ α = α ∪ 1 = α Graded Commutativity: The key property shown below The sign (-1) pq is crucial: it means the cup product is commutative only up to sign. When p or q is odd, swapping factors introduces a minus sign. This makes H * (X) = ⊕ H n (X) a graded-commutative ring . 3. Why Cohomology Has This Structure A natural question: why does cohomology admit a cup product, but homology doesn't have an analogous product? A p-cochain φ: Cₚ(X) → R is a linear functional Given two cochains φ and ψ, we can define their product by evaluating them on different faces of a simplex and multiplying the results This product operation is contravariant : it naturally goes "backward" along maps Homology is covariant (goes forward), so it lacks this multiplicative structure Let's look at concrete examples to build intuition:

This is the written version of the interactive lesson above. See the full Algebraic Topology course.