---
tag: 6N7BC70CX0
kind: concept
name: Singular cohomology
msc: [55N10]
sources:
  - url: https://en.wikipedia.org/wiki/Cohomology
---

## Summary

Singular cohomology assigns to a topological space $X$ and an abelian group $G$ the groups $H^n(X; G)$.
They are built from functions on singular simplices with values in $G$.
It is the dual of [[singular homology|M0Q5CFJ7ZH]], and it turns continuous maps into homomorphisms in the opposite direction.
Its extra structure, the cup product, makes $H^\bullet(X; R)$ a ring, which distinguishes spaces that have the same homology groups.

## Explanations

### L1 human

Homology builds things out of simplices; cohomology measures them.
A cochain is a rule that gives a number to every simplex, like a test that is applied to each piece of the space.
A cocycle is a test that gives zero on every boundary, so its value on a cycle depends only on the homology class.
For example, on a circle the rule "count how many times a loop winds around" is a $1$-cocycle, and it generates $H^1(S^1; \Z) \cong \Z$.