---
tag: WA796ZCNRX
kind: concept
name: Classifying space of a group
msc: [55R35, 20J06]
sources:
  - url: https://en.wikipedia.org/wiki/Classifying_space
---

## Summary

A classifying space of a discrete [[group|G5S1M9]] $G$ is a path-connected space $BG$ whose [[fundamental group|S1XK3HXHFX]] is $G$ and whose higher [[homotopy groups|P976WYEFMB]] all vanish.
It is the [[Eilenberg–MacLane space|AJHEYJE5XY]] $K(G, 1)$, and among CW complexes it is unique up to [[homotopy equivalence|XZG6MTSWCN]].
Through $BG$, group theory and topology meet: the [[homology of the group|CFAG3VRXEA]] is the homology of the space $BG$.
For topological groups, the same symbol denotes a space that classifies principal $G$-bundles, see [[principal bundle|8E37E74XVP]].

## Explanations

### L1 human

A classifying space turns a group into a shape.
The loops of the shape behave like the elements of the group, and the shape has no other holes of any kind.
For the group $\Z$, the circle works: going around $k$ times is the element $k$.
For $\Z^2$, the torus works, with one kind of loop for each generator.