Rational space

In the simply connected homotopy category, a rational space is a simply connected CW complex whose positive homotopy groups are vector spaces over $\mathbb{Q}$. Multiplication by every nonzero integer on these groups is invertible. More generally one uses nilpotent spaces with localized fundamental group and higher homotopy groups; the simply connected version avoids local-coefficient twisting.

For simply connected $X$, rationalization of a topological space is a map $r:X\to X_{\mathbb{Q}}$ to a rational space inducing ${\pi }_n(X_{\mathbb{Q}})={\pi }_n(X)\otimes \mathbb{Q}$. It is also a rational homology equivalence. Its universal property is the bijection $r^{\ast }:[X_{\mathbb{Q}},Z]\to [X,Z]$ for every rational space $Z$. A Postnikov tower constructs it by replacing each fibre group with its rational tensor product and transporting the Postnikov invariant through the resulting cohomology isomorphism, then taking the homotopy inverse limit.

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.