Timeline for Analogues of Sullivan Theory at a prime for coformality
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
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| Aug 11, 2023 at 10:26 | vote | accept | Andrea Marino | ||
| Aug 10, 2023 at 19:50 | comment | added | Andrea Marino | @connorMalin: thanks, but the (co)homological picture is known in my case (which is, by chance, closely related to little disk operads). | |
| Aug 10, 2023 at 17:52 | history | edited | Connor Malin | CC BY-SA 4.0 |
added 7 characters in body
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| Aug 10, 2023 at 17:44 | comment | added | Geoffroy Horel | Just one comment : in order to have an equivalence of homotopy theories one needs to use coefficients in $\overline{\mathbb{F}}_p$ and not in the $p$-adics. | |
| Aug 10, 2023 at 17:03 | comment | added | Connor Malin | @AndreaMarino I believe the $p$-adic analog of the commutative algebra story is pretty approachable. You might look into the rational computations which rely on the formality of the $E_n$-operad. I know there are partial results about formality with $F_p$-coefficients, so one might be able to produce analogs of those computations in low degrees. | |
| Aug 10, 2023 at 16:44 | comment | added | Andrea Marino | Hope I'll be there to see them :) for the moment, I can assume is a dead end considering my moderated skills. Thank you for your time. I'll wait a bit before accepting to see if there is some other interesting contribution popping up | |
| Aug 10, 2023 at 16:16 | comment | added | Connor Malin | @AndreaMarino It's usually a safe bet to assume no computation is easy in the $v_n$-local setting. For certain spaces (like spheres), there are cobar complexes which compute the spectral lie algebra associated to the space. I think the resulting spectral sequence ends up being the Goodwillie spectral sequence for the Bousfield Kuhn functor. Needless to say, not many computations have been done. I think we will eventually see people make some geometric computations using this machinery (specifically as applied to embedding calculus and orthogonal calculus), but its not there yet. | |
| Aug 10, 2023 at 16:02 | comment | added | Andrea Marino | Very nice!! Thanks. I was also wondering if such a model (as the Heuts model) somehow give an explicit way to compute the p-adic homotopy groups of spaces. In rational homotopy theory, this is quite effective. For example it has been used by Arone and Lambrechts to show the homotopy spectral sequence of knots in $\mathbb{R}^d, d \ge 4$, collapses. | |
| Aug 10, 2023 at 14:27 | history | answered | Connor Malin | CC BY-SA 4.0 |