Timeline for Symmetric multiplication on homotopy quotient of topological space
Current License: CC BY-SA 3.0
7 events
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| Aug 30, 2016 at 17:54 | comment | added | user84144 | Mark, what you say sounds very sensible but I have to admit that I am confused by the gradings. If you look at Lurie's notes, the chain complex he takes for $E(\mathbb{Z}/2)$ is a free resolution of the cohomology of a point, so for instance it is concentrated in negative degrees, so the cochain complex of $X \times X \times E\mathbb{Z}/2$ wouldn't be what I call about $C \otimes C \otimes E(\mathbb{Z}/2)$). That's one of the reasons I am confused about this business! | |
| Aug 30, 2016 at 9:21 | answer | added | David C | timeline score: 5 | |
| Aug 30, 2016 at 8:13 | comment | added | Mark Grant | For the prequestion: I'm sure this map is induced by the diagonal $X\to X\times X\to (X\times X)\times_{\mathbb{Z}/2} E\mathbb{Z}/2$. So to describe what it does on cochains would require a cellular approximation of the diagonal, possibly with some extra equivariance properties. Anyway, some relevant references (none of which do exactly what you want) are Hatcher 4.L, Steenrod and Epstein's "Cohomology operations" and Bruner-May-McClure-Steinberger's "H_\infty ring spectra and their applications" | |
| Aug 30, 2016 at 0:19 | comment | added | user84144 | Also, while I greatly respect these higher-categorical notions, and have at least heard enough "slogans" about E_{infty} algebras to appreciate a little what they mean, I would prefer as much as possible to adhere to a pedestrian point of view here... I'd be really happy to write down, even for a simple cell complex, how to concretely write down a multiplication $D_2(C) \rightarrow C$ in terms of bare-bones generators. | |
| Aug 30, 2016 at 0:07 | comment | added | user84144 | Thanks! My intuition/guess was that this map should exist for formal reasons, but the short story is that I used it sort of carelessly in a calculation and got a nonsensical answer, so now I'm trying to make sure I understand every step meticulously. I think it is a tiny bit subtle to say what this map is, i.e. given two maps $EG \rightarrow X$ and an element of $C_*(EZ/2)$ how exactly to put this together into another map $EG \rightarrow X$ in an equivariant way. I have an example in mind which is messy to say here but which I would be happy to elaborate on over e-mail, if you would like. | |
| Aug 29, 2016 at 23:59 | comment | added | Denis Nardin | For the prequestion, take a look at this: mathoverflow.net/questions/210540/… . For the second question: I am sure it does if, instead of just a multiplication you work with E_∞-structures (the forgetful functor from E_∞-algebras to chain complexes preserves homotopy limits being a right Quillen adjoint). I suspect it is true also for just the symmetric power but I need to think about it more. | |
| Aug 29, 2016 at 23:26 | history | asked | user84144 | CC BY-SA 3.0 |