Rational homotopy equivalence

A map between simply connected spaces is a rational homotopy equivalence when it induces isomorphisms ${\pi }_n(X)\otimes \mathbb{Q}\to {\pi }_n(Y)\otimes \mathbb{Q}$ in every degree. Equivalently it induces an equivalence of their rationalizations, or a rational homology isomorphism by the rational Whitehead theorem. Spaces have the same rational homotopy type when they are connected by a zigzag of such maps. This is distinct from rational homotopy equivalence of algebraic cycles.

The rational homotopy type of a simply connected space is its equivalence class under zigzags of rational homotopy equivalences, or equivalently the homotopy type of its rationalization of a topological space. For the usual finite-type spaces, it is encoded contravariantly by the isomorphism class of the Sullivan minimal model. Isomorphic cohomology rings determine this type only when additional information, such as formality of a topological space, makes the cochain algebra equivalent to the cohomology algebra.