Timeline for Eilenberg–Moore equivalences for $C_*(\Omega M)$
Current License: CC BY-SA 4.0
5 events
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| Nov 16, 2021 at 2:45 | comment | added | skd | 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. | |
| Nov 15, 2021 at 14:22 | comment | added | onefishtwofish | @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 4:10 | comment | added | skd | 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. | |
| Nov 15, 2021 at 3:50 | history | edited | LSpice | CC BY-SA 4.0 |
Names of papers
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| Nov 15, 2021 at 2:08 | history | asked | onefishtwofish | CC BY-SA 4.0 |