---
tag: 2JFF5F3EGF
kind: concept
name: Loop space
msc: [55P35]
sources:
  - url: https://en.wikipedia.org/wiki/Loop_space
same_as:
  wikidata: Q2066070
  wikipedia: Loop space
---

## Summary

The loop space $\Omega X$ of a [[pointed space|T9DG8DDE75]] $(X, x_0)$ is the space of all loops in $X$ that start and end at $x_0$.
Its path components are the elements of the [[fundamental group|S1XK3HXHFX]] of $X$, and in general $\pi_n(\Omega X) \cong \pi_{n+1}(X)$.
Concatenation of loops makes $\Omega X$ an H-space.
The loop space is dual to the [[suspension|Q7RFNJH76Q]] in the sense of [[Eckmann–Hilton duality|1EK3355WMC]].

## Explanations

### L1 human

Every point of the loop space is a whole loop in $X$ based at $x_0$.
Two points of the loop space are close when the two loops stay close at every time.
Moving continuously in the loop space means deforming a loop.
For example, the loop space of the circle falls into pieces, one for each integer winding number.