---
tag: 8G9QPMBT7R
kind: concept
name: Rational homotopy theory
msc: [55P62]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_homotopy_theory
same_as:
  wikidata: Q4391254
  wikipedia: Rational homotopy theory
---

## Summary

Rational homotopy theory studies spaces up to maps that induce isomorphisms on rational homotopy groups, so it ignores all torsion.
For simply connected spaces this coarser theory is completely algebraic.
By the work of Quillen and Sullivan, rational homotopy types correspond to commutative differential graded algebras over $\Q$, or to differential graded Lie algebras.
Concrete invariants such as [[Sullivan minimal models|3QQ2PBGY3N]] make many questions computable that are out of reach integrally.

## Explanations

### L1 human

The homotopy groups of even a sphere are very hard to compute, mostly because of their finite parts.
If one tensors everything with the rationals, the finite parts vanish and what remains is much simpler.
For example, $S^3$ rationally has only one homotopy group, $\Q$ in degree $3$, while $S^2$ rationally has $\Q$ in degrees $2$ and $3$.
Rational homotopy theory describes such answers with algebra: graded algebras with a differential, in place of spaces.