Cup Product and Ring Structure

The Cup Product

Cup Product

Let be a ring. The cup product is a map given by the following formula. For , , and a singular -simplex , we define: Here, is the restriction of to the “front face” (a -simplex), is the restriction to the “back face” (an -simplex), and the product is taken in the ring .

The cup product interacts with the coboundary operator via the following rule.

Coboundary Formula for Cup Product

For and , the following Leibniz-type rule holds:

An important consequence of this formula is that the cup product of two cocycles (, ) is again a cocycle. Furthermore, the product of a cocycle and a coboundary is a coboundary. This implies that the cup product descends to a well-defined product on cohomology:

The Cohomology Ring

The above properties mean that equipped with the cup product forms a ring:

Theorem

is a unital, associative, ring under the cup product. The unit is represented by the cocycle which maps every 0-simplex (a point in ) to the multiplicative identity . This is called the cohomology ring of with coefficients in .

Proof of Unital Property Let be the unit element, represented by the cocycle where for any point . Let be represented by the cocycle . For any -simplex with vertices , we have: Since is a 0-simplex, . Therefore, This shows that , so . A similar calculation shows , so it is a two-sided unit.

Induced Homomorphisms

Let be a continuous map. The induced map on cohomology is a homomorphism of graded rings. That is, it preserves the ring structure:

e.g. The cohomology ring of the torus with integer coefficients is isomorphic to the exterior algebra on two generators of degree 1. Recall our -complex structure for the torus: The chain complex is where , , and . So we have the homology groups: Dualize to get the cochain complex: where maps and is the zero map (we can think of and as the transpose of and ). Thus, the cohomology groups are:

If are generators for , then:

Graded Commutativity

The cohomology ring is not strictly commutative in general, but it satisfies a property called super-commutativity or graded commutativity.

Graded Commutativity of the Cup Product

If is a commutative ring, then for any cohomology classes and , the following relation holds:

Remark

An interesting consequence of graded commutativity is that for any element of odd degree in a graded commutative ring , we have This implies . If the ring has no 2-torsion (e.g., if the coefficient ring is or ), this means .

e.g. The cohomology rings of real and complex projective spaces are fundamental examples.

• The infinite complex projective space has a cohomology ring that is a polynomial ring on a single generator :
• For the finite-dimensional complex projective space , the ring is a truncated polynomial ring: with in degree , i.e., is the generator of .
• The infinite real projective space with coefficients also has a polynomial ring structure on a generator :
• For the finite-dimensional real projective space , the ring is aoh truncated polynomial ring: with in degree .

Moreover, the de Rham cohomology of a smooth manifold with real coefficients is a canonical example of a cohomology ring (see here).

Examples of Cohomology Rings

Exterior Algebra

The exterior algebra on generators over a ring , denoted , is an algebra where the generators anti-commute () and each generator squares to zero (). For example:

• is additively with .
• is additively with , , and .