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 $B'C^{*}(X)$?
Best Answer
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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