---
tag: VEGBD6Y61A
kind: definition
name: Rational homotopy theory
msc: [55P62]
defines: 8G9QPMBT7R
uses: [D0ZHVD797D, P976WYEFMB, M0Q5CFJ7ZH, XZG6MTSWCN, 5W95QFHXPR]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_homotopy_theory
---

## Definition

Consider simply connected [[CW complexes|5W95QFHXPR]].
A map $f \colon X \to Y$ between them is a *rational homotopy equivalence* if the maps $\pi_n(f) \otimes \Q \colon \pi_n(X) \otimes \Q \to \pi_n(Y) \otimes \Q$ on [[homotopy groups|P976WYEFMB]] are isomorphisms for all $n \ge 2$.
Two such spaces have the same *rational homotopy type* if they are joined by a finite chain of rational homotopy equivalences, in either direction.
Equivalently, their [[rationalizations|D0ZHVD797D]] $X_\Q$ and $Y_\Q$ are [[homotopy equivalent|XZG6MTSWCN]].
*Rational homotopy theory* is the study of simply connected spaces up to rational homotopy type.
In other words, it studies the homotopy category of simply connected spaces with the rational homotopy equivalences inverted.

## Remarks

The condition on $f$ is equivalent to: $H_n(f; \Q)$ is an isomorphism in [[singular homology|M0Q5CFJ7ZH]] for all $n$.

Example: the rationalization $S^3 \to K(\Q, 3)$ is a rational homotopy equivalence, since both sides have rational homotopy $\Q$ in degree $3$ and $0$ in other degrees.
So $S^3$ and $K(\Q, 3)$ have the same rational homotopy type.

Non-example: the map $S^2 \to K(\Q, 2)$ that represents a generator of $H^2(S^2; \Q)$ is an isomorphism on $\pi_2 \otimes \Q$.
It is not a rational homotopy equivalence: $\pi_3(S^2) \otimes \Q \cong \Q$, generated by the Hopf map, while $\pi_3 K(\Q, 2) = 0$.

Sullivan showed that this homotopy category, with finiteness conditions, is equivalent to a homotopy category of commutative differential graded algebras over $\Q$.
Quillen showed that it is also equivalent to a homotopy category of differential graded Lie algebras over $\Q$.