Formal differential graded algebras and homomorphisms (Q1102577)

The Sullivan theory of rational homotopy associates to each topological space X a minimal model \(M_ X\). This is a commutative differential graded algebra over the field \({\mathbb{Q}}\) which describes completely the rational homotopy type of the space X. When there is a weak equivalence, in the category of C.D.G.A., between \(M_ X\) and \(H^*(X;{\mathbb{Q}})\) the space X is called formal. In the same way, one can define a formal map between two formal spaces. The main result of this paper states: if X and Y are both formal spaces then the natural function: \([X,Y]\to Hom(H^*(Y;{\mathbb{Q}}),H^*(X;{\mathbb{Q}}))\) restricted to the homotopy classes of formal maps is a bijection. This theorem has several consequences including results on the group of self-equivalences of a formal space.