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

  • $\begingroup$ isn't the cup product on $C^*(X)$ defined as the dual of the coproduction on $C_*(X)$ ? I don't quite understand the difference between your questions a) and b). $\endgroup$ Sep 7 '19 at 23:59

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 $H^*(\Omega X)$.


  • 2
    $\begingroup$ I was wondering if you assume some finiteness conditions on the space $X$ ? $\endgroup$
    – Ilias A.
    Sep 3 '15 at 19:44
  • $\begingroup$ @AmraniIlias: You are right, thank you. I've edited my answer to reflect this. $\endgroup$
    – Mark Grant
    Sep 4 '15 at 7:18
  • 3
    $\begingroup$ So the answer to (a) is always, but only sometimes? (That just strikes me as odd wording.) $\endgroup$ Sep 4 '15 at 7:21

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.


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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.