Let $X$ be a topological space, and $Symp_\bullet (X)$ the simplicial set with simplices $$Symp_m(X) = \{ \textrm{singular } m- \textrm{simplices } \sigma: \Delta^m\to X\}$$ (and usual faces and degeneracies). Let $A^\bullet(X)$ be the complex of rational polynomial forms on $Symp_\bullet(X)$. If $k$ is a $\mathbb{Q}$-algebra, denote $A^\bullet_k(X)=A^\bullet(X)\otimes_\mathbb{Q}k$. Dennis Sullivan proved that:

The integration map $$I : A^\bullet_k(X)\to S^\bullet(X,k)$$ induces a $k$-algebra isomorphism on cohomology.

Here $S^\bullet(X,k)$ is the complex of singular $k$-valued cochains.

The theorem is one of the versions of a more powerful result, see D.Sullivan, Infinitesimal Computations in Topology, Theorem 7.1. In this form the theorem is stated in R.Hain's paper The de Rham homotopy theory of complex algebraic varieties I, section 2 (K-theory, 1987).

Let $X$ be a topological space, and $Symp_\bullet (X)$ the simplicial set with simplices $$Symp_m(X) = \{ \textrm{singular } m- \textrm{simplices } \sigma: \Delta^m\to X\}$$ (and usual faces and degeneracies). Let $A^\bullet(X)$ be the complex of rational polynomial differential forms on $Symp_\bullet(X)$. If $k$ is a $\mathbb{Q}$-algebra, denote $A^\bullet_k(X)=A^\bullet(X)\otimes_\mathbb{Q}k$. Dennis Sullivan proved that:

The integration map $$I : A^\bullet_k(X)\to S^\bullet(X,k)$$ induces a $k$-algebra isomorphism on cohomology.

Here $S^\bullet(X,k)$ is the complex of singular $k$-valued cochains.

The theorem is one of the versions of a more powerful result, see D.Sullivan, Infinitesimal Computations in Topology, Theorem 7.1. In this form the theorem is stated in R.Hain's paper The de Rham homotopy theory of complex algebraic varieties I, section 2 (K-theory, 1987).