bar construction and loop space cohomology

Question

Let $X$ be a topological space. Then its chain complex $C_{*}(X)$ is naturally a coalgebra (as explained here Does homology have a coproduct?). In particular if $X$ is simply connected we have that the homology of the cobar construction $\Omega C_{*}(X)$ of $C_{*}(X)$ is isomorphic to the homology of the pointed path space $\Omega X$ (Adam's theorem). I'm looking for reference about similar statements using the bar construction:

a) Consider the cochain complex $C^{*}(X)$ as a dg algebra equipped with the cup product. Assume that $X$ is simply connected. Then under which conditions the cohomology of the bar construction $BC^{*}(X)$ is isomorphic to the cohomology of the path space?

b) $C_{*}(X)$ is a coalgebra where the coproduct is the composition of the Alexander-Whitney map with the diagonal map. By taking the dual we get a dg algebra $C^{*}(X)$ with a product $\mu$. Let $B'C^{*}(X)$ be the bar construction of ($C^{*}(X)$, $\mu$). What is the relation between $BC^{*}(X)$ and $B'C^{*}(X)$?

Answer 1

You should read about the Eilenberg--Moore spectral sequence. John McCleary's book "A User's Guide to Spectral Sequences" is a good place to start. Another good reference is Larry Smith's paper in Transactions of the AMS "Homological algebra and the Eilenberg--Moore spectral sequence". In particular, the answer to your question (a) is always, provided $X$ has the homotopy type of a countable, simply-connected CW complex with finite type integral homology. This follows from the Theorem of Eilenberg--Moore (presented as Theorem 7.14 in McCleary and Theorem 3.2 of Smith) applied to the pullback diagram $$ \begin{array}{ccc} \Omega X & \to & PX\simeq\ast \newline \downarrow & & \downarrow \newline \ast & \to & X. \end{array} $$ Let's assume coefficients in a field $k$, so $C^\ast(X)=C^\ast(X;k)$. The key thing to realize is that the bar construction $BC^\ast(X)$ is a proper projective resolution of $k$ by $C^\ast(X)$-modules, and so its cohomology is $\operatorname{Tor}_{C^\ast(X)}(k,k)$, which by Eilenberg--Moore is isomorphic to $H^*(\Omega X)$.

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Answer 2

May be I misunderstood your question. $C^{\ast}(X)$ is an $E_{\infty}$-algebra. For pointed simply connected space $X$ there is an equivalent of $E_{\infty}$-algebra between $BC^{\ast}(X)$ and $C^{\ast}(\Omega X)$ under some finiteness conditions on $X$. The functor $B$ can be seen as the suspension functor (in the derived sense) in the category of augmented $E_{\infty}$-algebra and the functor $C^{\ast}(-)$ (under some assumption on spaces ) takes finite homotopy limits of spaces to homotopy colimits of $E_{\infty}$-algebras (I'm working over a field). some reference: math.univ-lille1.fr/~fresse/Bar-StructureUniqueness.pdf

It is better to see $C^{\ast}(X)$ as an $E_{\infty}$-algebra and not just as a differential graded algebra. As we see the interpretation can be made more comprehensive from homotopical view point.

Answer 3

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.