Singular Cochains and rational Homotopy Type

Abstract

Rational homotopy types of simply connected topological spaces have been classified by weak equivalence classes of commutative cochain algebras (Sullivan) and by isomorphism classes of minimal commutative $A_{infty}$-algebras (Kadeishvili).

We classify rational homotopy types of the space $X$ by using the (non-commutative) singular cochain complex, $C^{ast}(X,Q)$, with additional structure given by the homotopies introduced by Baues, ${E_{1,k}}$ and ${F_{p,q}}$. We show that if we modify the resulting $B_{infty}$-algebra structure on this algebra by requiring that its bar construction be a Hopf algebra up to homotopy, then weak equivalence classes of such algebras classify rational homotopy types.