Over the rationals or the reals, K.T. Chen described a concrete way of relating the bar construction of the CDGA algebra of differential forms on a manifold to the smooth singular cochains on the based loop space. It is dual to Adams' cobar construction and can be thought as a De Rham type theorem for the based loop space. The main statement is the following. Let $M$ be a connected smooth manifold and let $(\mathcal{A}(M), d, \wedge)$ be the CDGA of differential forms on $M$. Let $A$ to be a sub CDGA of $\mathcal{A}(M)$ such that $A^0=0$, $A^i=\mathcal{A}^i(M)$ for $i>1$, and $A^1$ is a complement of $d\mathcal{A}^0(M)$, i.e. $\mathcal{A}^1(M)=d\mathcal{A}^0(M) \oplus A^1$.

Consider the bar construction $(B(A), D)$. This is a DG commutative coassociative Hopf algebra with underlying vector space $T(sA)$, the tensor coalgebra on the shifted $A$, product given by shuffling monomials, coproduct given by deconcatenation of monomials, and differential given by extending $d + \wedge$ as a coderivation, as usual. Chen constructed a chain map \begin{eqnarray} \int: B(A) \to C^*(\Omega M) \end{eqnarray} inducing a map of Hopf algebras on cohomology, where $C^*(\Omega M)$ denotes the real smooth singular cochains on the based loop space of $M$. Moreover, if $M$ is simply connected, the above map induces an isomorphism on cohomology. The map is constructed by a process of iterated integration (or integration over a simplex) on a given monomial of differential forms on $M$. More precisely, given $[w_1|...|w_m] \in B(A)$, $\int w_1 ... w_m$ is the cochain that sends any smooth simplex $\sigma: \Delta^n \to \Omega M$ to the integral \begin{eqnarray} \int_{\Delta^m \times \Delta^n} \sigma_1^*(w_1)...\sigma_m^*(w_m) \end{eqnarray} where $\sigma_i: \Delta^m \times \Delta^n \to M$ is defined by $\sigma_i(t_1,...,t_m,s)=\sigma(s)(t_i)$. It is a beautiful theory exposed in several papers of Chen from the 70's and there are still paths to be explored.

Over the rationals or the reals, K.T. Chen described a concrete way of relating the bar construction of the CDGA algebra of differential forms on a manifold to the smooth singular cochains on the based loop space. It is dual to Adams' cobar construction and can be thought as a De Rham type theorem for the based loop space. The main statement is the following. Let $M$ be a connected smooth manifold and let $(\mathcal{A}(M), d, \wedge)$ be the CDGA of differential forms on $M$. Let $A$ be a sub CDGA of $\mathcal{A}(M)$ such that $A^0=0$, $A^i=\mathcal{A}^i(M)$ for $i>1$, and $A^1$ is a complement of $d\mathcal{A}^0(M)$, i.e. $\mathcal{A}^1(M)=d\mathcal{A}^0(M) \oplus A^1$.

Consider the bar construction $(B(A), D)$. This is a DG commutative coassociative Hopf algebra with underlying vector space $T(sA)$, the tensor coalgebra on the shifted $A$, product given by shuffling monomials, coproduct given by deconcatenation of monomials, and differential given by extending $d + \wedge$ as a coderivation, as usual. Chen constructed a chain map \begin{eqnarray} \int: B(A) \to C^*(\Omega M) \end{eqnarray} inducing a map of Hopf algebras on cohomology, where $C^*(\Omega M)$ denotes the real smooth singular cochains on the based loop space of $M$. Moreover, if $M$ is simply connected, the above map induces an isomorphism on cohomology. The map is constructed by a process of iterated integration (or integration over a simplex) on a given monomial of differential forms on $M$. More precisely, given $[w_1|...|w_m] \in B(A)$, $\int w_1 ... w_m$ is the cochain that sends any smooth simplex $\sigma: \Delta^n \to \Omega M$ to the integral \begin{eqnarray} \int_{\Delta^m \times \Delta^n} \sigma_1^*(w_1)...\sigma_m^*(w_m) \end{eqnarray} where $\sigma_i: \Delta^m \times \Delta^n \to M$ is defined by $\sigma_i(t_1,...,t_m,s)=\sigma(s)(t_i)$. It is a beautiful theory exposed in several papers of Chen from the 70's and there are still paths to be explored.

Over the rationals or the reals, K.T. Chen described a concrete way of relating the bar construction of the CDGA algebra of differential forms on a manifold to the smooth singular cochains on the based loop space. It is dual to Adams' cobar construction and can be thought as a De Rham type theorem for the based loop space. The main statement is the following. Let $M$ be a connected smooth manifold and let $(\mathcal{A}(M), d, \wedge)$ be the CDGA of differential forms on $M$. Let $A$ to be a sub CDGA of $\mathcal{A}(M)$ such that $A^0=0$, $A^i=\mathcal{A}^i(M)$ for $i>1$, and $A^1$ is a complement of $d\mathcal{A}^0(M)$, i.e. $\mathcal{A}^1(M)=d\mathcal{A}^0(M) \oplus A^1$.

Consider the bar construction $(B(A), D)$. This is a DG commutative coassociative Hopf algebra with underlying vector space $T(sA)$, the tensor coalgebra on the shifted $A$, product given by shuffling monomials, coproduct given by deconcatenation of monomials, and differential given by extending $d + \wedge$ as a coderivation, as usual. Chen constructed a chain map \begin{eqnarray} \int: B(A) \to C^*(\Omega M) \end{eqnarray} inducing a map of Hopf algebras on cohomology, where $C^*(\Omega M)$ denotes the real smooth singular cochains on the based loop space of $M$. Moreover, if $M$ is simply connected, the above map induces an isomorphism on cohomology. The map is constructed by a process of iterated integration (or integration over a simplex) on a given monomial of differential forms on $M$. More precisely, given $[w_1|...|w_m] \in B(A)$, $\int w_1 ... w_m$ is the cochain that sends any smooth simplex $\sigma: \Delta^n \to \Omega M$ to the integral \begin{eqnarray} \int_{\Delta^m \times \Delta^n} \sigma_1^*(w_1)...\sigma_m^*(w_m) \end{eqnarray} where $\sigma_i: \Delta^m \times \Delta^n \to M$ is defined by $\sigma_i(t_1,...,t_m,s)=\sigma(s)(t_i)$. It is a beautiful theory exposed in several papers of Chen from the 70's and there are still unexplored paths related to it.

Over the rationals or the reals, K.T. Chen described a concrete way of relating the bar construction of the CDGA algebra of differential forms on a manifold to the smooth singular cochains on the based loop space. It is dual to Adams' cobar construction and can be thought as a De Rham type theorem for the based loop space. The main statement is the following. Let $M$ be a connected smooth manifold and let $(\mathcal{A}(M), d, \wedge)$ be the CDGA of differential forms on $M$. Let $A$ to be a sub CDGA of $\mathcal{A}(M)$ such that $A^0=0$, $A^i=\mathcal{A}^i(M)$ for $i>1$, and $A^1$ is a complement of $d\mathcal{A}^0(M)$, i.e. $\mathcal{A}^1(M)=d\mathcal{A}^0(M) \oplus A^1$.

Consider the bar construction $(B(A), D)$. This is a DG commutative coassociative Hopf algebra with underlying vector space $T(sA)$, the tensor coalgebra on the shifted $A$, product given by shuffling monomials, coproduct given by deconcatenation of monomials, and differential given by extending $d + \wedge$ as a coderivation, as usual. Chen constructed a chain map \begin{eqnarray} \int: B(A) \to C^*(\Omega M) \end{eqnarray} inducing a map of Hopf algebras on cohomology, where $C^*(\Omega M)$ denotes the real smooth singular cochains on the based loop space of $M$. Moreover, if $M$ is simply connected, the above map induces an isomorphism on cohomology. The map is constructed by a process of iterated integration (or integration over a simplex) on a given monomial of differential forms on $M$. More precisely, given $[w_1|...|w_m] \in B(A)$, $\int w_1 ... w_m$ is the cochain that sends any smooth simplex $\sigma: \Delta^n \to \Omega M$ to the integral \begin{eqnarray} \int_{\Delta^m \times \Delta^n} \sigma_1^*(w_1)...\sigma_m^*(w_m) \end{eqnarray} where $\sigma_i: \Delta^m \times \Delta^n \to M$ is defined by $\sigma_i(t_1,...,t_m,s)=\sigma(s)(t_i)$. It is a beautiful theory exposed in several papers of Chen from the 70's and there are still paths to be explored.