Formal differential graded algebras and homomorphisms

Abstract

Let B be a differential graded algebra over the rationals (DGA), M a minimal DGA, [M,B] the homotopy classes of DGA maps M→B, and I:[M,B]→Hom(H*(M), H*(B)) the function which assigns the induced cohomology homomorphism to a homotopy class. Theorem. If M and B are formal, then I restricted to the homotopy classes of formal maps is a bijection. This theorem has several diverse consequences including results on the group of homotopy classes of homotopy equivalences of a formal DGA and results on the suspension Σ:[X,Y]→[ΣX,ΣY]co-H when X and Y are formal spaces. © 1988.