---
tag: YN280Q5G49
kind: concept
name: Bredon cohomology
msc: [55N91, 55P91]
sources:
  - url: https://en.wikipedia.org/wiki/Bredon_cohomology
same_as:
  wikidata: Q65089963
  wikipedia: Bredon cohomology
---

## Summary

Bredon cohomology is a cohomology theory for spaces with an action of a group $G$.
Its coefficients are not one group but a system of groups, one for each orbit type $G/H$.
It is computed from the cell structure of a $G$-CW complex and keeps track of the fixed point sets $X^H$ of all subgroups $H$.
It is the basic ordinary cohomology theory of equivariant homotopy theory, and the obstructions of equivariant obstruction theory lie in it.

## Explanations

### L1 human

When a group acts on a space, some points are moved freely and others are fixed by part of the group.
Borel's [[equivariant cohomology|C7MZ3SFGYF]] uses one coefficient group for the whole space.
Bredon cohomology looks separately at the points fixed by each subgroup and lets the coefficients depend on that subgroup.
For example, for a group acting on a single point, it returns the coefficient group attached to the whole group $G$.