# Eilenberg–Moore equivalences for $C_*(\Omega M)$

Let $$M$$ be a nice connected topological space (I'm actually interested in manifolds) with base point $$p$$ and let $$\pi: E \to M$$ be a fibration. Then chains on the fiber $$F$$ at $$p$$, $$C_*(F)$$, become a dg-module over $$C_*(\Omega M)$$, the based loop space at $$p$$. Then if I'm interpreting Blumberg–Cohen–Teleman correctly (Open-closed field theories, string topology, and Hochschild homology line just after Lemma 2.5), there is an isomorphism $$C_*(E,k) \cong k\otimes^{L}_{C_*(\Omega M)} C_*(F)$$

I'm wondering if anyone can tell me a reference which explains this kind of formula in more detail? Perhaps, asked differently, let's say that I'm comfortable with Hatcher's book. I also can follow Moore's proof in Algèbre homologique et homologie des espaces classifiants which should correspond to the special case $$M=BG$$ using the fact that $$\Omega BG \simeq G$$.

• There's a fiber sequence $\Omega M \to F \to E$, which exhibits $E$ as the homotopy quotient of $F$ by an (homotopy) action of $\Omega M$; this is the action by monodromy about a loop. (For intuition, suppose $F$ is a space with an action of a group $G$. Then the homotopy orbits $F_{hG}$ sits in a fiber sequence $F \to F_{hG} \to BG$.) In modern language, your statement now follows from the fact that taking pointed suspension spectra preserves homotopy colimits.
– skd
Nov 15, 2021 at 4:10
• @skd Thanks for this! What you're saying seems like a very clean perspective. Where could I read more about it --- for example the fact that E becomes a "homotopy quotient" of F? I write this in quotes because I don't even really know the definition of homotopy quotient when the action is not strict. I suppose this must be fairly classical stuff but more modern textbook references would also be fine. Nov 15, 2021 at 14:22
• I don't know a reference off the top of my head, unfortunately. One way in which you can construct such an action is outlined in my answer at mathoverflow.net/a/361153. Another (equivalent) approach is to view the fibration E -> M as defining a homotopy-coherent functor p: Sing(M) -> Spaces ("take fibers") whose homotopy colimit is E. Now Sing(M) can be interpreted as B Sing(Loops M) since M is assumed to be connected, so this is giving a homotopy coherent action of Loops M on F = fiber(E -> M) whose homotopy quotient = colim(p) = E.
– skd
Nov 16, 2021 at 2:45