---
tag: JX47MS1R1H
kind: concept
name: Cup product
msc: [55N45]
sources:
  - url: https://en.wikipedia.org/wiki/Cup_product
same_as:
  wikidata: Q3388699
  wikipedia: Cup product
---

## Summary

The cup product is a multiplication $H^k(X; R) \times H^l(X; R) \to H^{k+l}(X; R)$ on [[singular cohomology|6N7BC70CX0]] with coefficients in a commutative ring $R$.
It is associative, has a unit, and is graded commutative.
It makes cohomology into a ring, the [[cohomology ring|RDT4785M4D]], which carries much more information than the cohomology groups alone.

## Explanations

### L1 human

Cohomology classes can be multiplied, and the degrees add.
Take a $k$-cochain and an $l$-cochain, and a simplex of dimension $k + l$.
Test the first on the front face of dimension $k$, test the second on the back face of dimension $l$, and multiply the results.
On the torus, the two classes that measure winding around each circle multiply to a class that measures the whole surface.
On the wedge of two circles and a sphere the same groups appear, but every product of two degree $1$ classes is zero.
So the cup product tells the two spaces apart.