---
tag: S1XK3HXHFX
kind: concept
name: Fundamental group
msc: [55Q05, 57M05]
sources:
  - url: https://en.wikipedia.org/wiki/Fundamental_group
same_as:
  wikidata: Q662830
  wikipedia: Fundamental group
---

## Summary

The fundamental group $\pi_1(X, x_0)$ of a [[pointed space|T9DG8DDE75]] is the [[group|G5S1M9]] of loops at $x_0$ up to [[path homotopy|CCNKA8C711]], with concatenation of loops as product.
It is the first invariant of algebraic topology that detects one-dimensional holes.
Homotopy equivalent path-connected spaces have isomorphic fundamental groups.
For example, the plane has trivial fundamental group, while the [[circle|060W5CBZP6]] has fundamental group $\Z$.

## Explanations

### L1 human

Stand at a point of a space and walk along loops that come back to where you started.
Two loops count as the same if one can be deformed into the other while the start stays fixed.
Doing one loop after another is the product.
On a circle, a loop is classified by how many times it goes around, counted with sign, so the group is the integers.