Dual invariants, the cup product, and cohomology rings
Cohomology is the algebraic dual of homology: instead of building cycles from chains, we assign values to chains and ask when those assignments are consistent. A cocycle is a function on chains that vanishes on boundaries — it “counts intersections” with a dual object.
Why bother with this dual viewpoint? Because cohomology carries strictly morestructure than homology. The cup product turns H*(X) into a graded ring, and this ring structure can distinguish spaces that homology alone cannot.
The punchline: two spaces can have identical homology groups yet completely different cohomology rings. The cup product detects how loops and surfaces are “linked” inside a space.
This triangulated torus (shown as a flat square with opposite edges identified) lets you draw both cycles and cocycles. Switch between modes: in cycle mode, click edges to trace a loop; in cocycle mode, click edges to assign them the value 1. The pairing 〈φ, c〉 counts how many edges of the cycle carry cocycle value 1 (mod 2).
Key insight: The Kronecker pairing 〈φ, c〉 is the bridge between cohomology and homology. A cocycle φ “evaluates” a cycle c by summing φ(e) over all edges e of c. This is the algebraic version of counting intersection points.
The cup product α ∪ β combines two cocycles into a higher-degree cocycle. On the torus T², the two 1-cocycles α and β (dual to the horizontal and vertical loops) have a nontrivial cup product that generates H². But on S¹ ∨ S¹ ∨ S² — which has the exact same homology groups — the cup product is zero!
| H&sup0; | ℤ |
| H¹ | ℤ ⊕ ℤ |
| H² | ℤ |
| α ∪ β | ≠ 0 (generates H²) |
| H&sup0; | ℤ |
| H¹ | ℤ ⊕ ℤ |
| H² | ℤ |
| α ∪ β | = 0 (trivial!) |
Key insight: The cup product detects “linking” — on the torus, the two generating loops intersect transversally in a single point, so α ∪ β ≠ 0. In the wedge sum, the loops live in separate circles and never interact, so the cup product vanishes. This is the fundamental example showing that cohomology rings are strictly stronger than homology groups.
The full cohomology ring H*(X; ℤ) organizes all cup products into a multiplication table. Select a space to see its generators and products. Nonzero products glow rose — these are the interesting ones! Click any cell to see its geometric meaning. Use compare mode to view two spaces side by side and see exactly how their ring structures differ.
Key insight: The cohomology ring is a graded-commutative ring: α ∪ β = (-1)&sup(deg α)(deg β) β ∪ α. For odd-degree classes this means α ∪ α = 0 (over ℤ), which is why the torus has an exterior algebra. For even-degree classes like the hyperplane class of ℂP², self-cup-products can be nontrivial — giving a polynomial ring truncated at the top dimension.