---
tag: 545EV0W4CE
kind: concept
name: Stable homotopy theory
msc: [55P42, 55Q10]
sources:
  - url: https://en.wikipedia.org/wiki/Stable_homotopy_theory
same_as:
  wikidata: Q7595767
  wikipedia: Stable homotopy theory
---

## Summary

Stable homotopy theory is the part of homotopy theory that studies the phenomena that do not change under iterated [[suspension|Q7RFNJH76Q]].
By the [[Freudenthal suspension theorem|7DM2KSS2DB]], the homotopy groups of $\Sigma^k X$ stabilize as $k$ grows.
The stable values are often more regular than the unstable ones.
Its natural objects are [[spectra|1GHF4EKBXS]], which represent generalized cohomology theories, and its main computational problem is the [[stable homotopy groups of spheres|HGZAYPQBXW]].

## Explanations

### L1 human

Suspending a space makes it one dimension bigger, and suspending a map does the same to the map.
After enough suspensions, the set of homotopy classes of maps between two finite complexes stops changing.
Stable homotopy theory studies these stable answers.
For example, $\pi_{n+1}(S^n)$ is $\Z$ for $n = 2$ but $\Z/2$ for all $n \ge 3$, so the stable value is $\Z/2$.
In the stable world, maps can be added and fibre sequences behave like exact sequences, so many tools of algebra apply.