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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 $\Omega BG \simeq G$.

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    $\begingroup$ 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. $\endgroup$
    – skd
    Commented Nov 15, 2021 at 4:10
  • $\begingroup$ @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. $\endgroup$
    – onefishtwofish
    Commented Nov 15, 2021 at 14:22
  • $\begingroup$ 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. $\endgroup$
    – skd
    Commented Nov 16, 2021 at 2:45

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