Timeline for Analogues of Sullivan Theory at a prime for coformality
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Aug 17, 2023 at 9:56 | comment | added | Andrea Marino | Maybe the hard part is finding a decent cellular decomposition of the p-completon of $X$? Or is it the case that the cochain complex on $X_p$ can be found algebraically from the the cochain complex on $X$? | |
| Aug 17, 2023 at 9:32 | comment | added | Andrea Marino | Sorry I come back late, but I'm not sure what's wrong with the following argument. $C^*(S^n, \mathbb{F}_p)$ is quasi-isomorphic, thus homotopy equivalent (since we are over a field) to $H^*(S^n, \mathbb{F}_p)$ by formality of spheres. The map $C^*(S^n, \mathbb{F}^p) \to H^*(S^n, \mathbb{F}_p)$ can be used by homotopy transpher to induce a $E_{\infty}$ structure on $C^*(S^n, \mathbb{F}_p)$ which I expect to be equivalent the classical one. At this point the set $[C^*(S^n, \mathbb{F}_p), C^*(X, \mathbb{F}_p)] $ should be equivalent to sthing like zero-square elements of order $n$ up to heq. | |
| Aug 11, 2023 at 10:26 | vote | accept | Andrea Marino | ||
| Aug 11, 2023 at 6:50 | comment | added | Geoffroy Horel | The statement is that $p$-completed homotopy groups of simply connected finite type spaces are the homotopy groups of the $p$-completion (i.e. the Bousfield localization at $H_*(-,\mathbb{F}_p)$-iso), so in your formula, you should replace $X$ by $X_p$ on the left hand side (it does not matter on the right hand side). For question 2) the answer is yes, and for question 3), finding a presentation of $C^*(S^n)$ as an $E_\infty$-algebra is very difficult (and more or less equivalent to computing homotopy groups of $S^n$). | |
| Aug 10, 2023 at 18:36 | comment | added | Andrea Marino | I understand. Does this means we can theoretically compute homotopy groups of a simply connected p-complete space $X$ as $[S^n_{p}, X] \simeq [ C^*(S^n_{p}, \bar{\mathbb{F}}_p), C^*(X, \bar{\mathbb{F}}_p]_{E_{\infty}}$, up to taking into account the pointed-ness of homotopies? These are actually three questions: 1) Are p-completed homotopy groups represented by an object, possibly the p-completed sphere? 2) What you stated implies the bijection on maps up to homotopy? 3) Does the representing object has a nice presentation? | |
| Aug 10, 2023 at 18:22 | history | became hot network question | |||
| Aug 10, 2023 at 17:42 | comment | added | Geoffroy Horel | Yes the p-completed homotopy groups are theoretically determined by Mandell's model. Indeed Mandell shows that the homotopy theory of p-complete finite type nilpotent homotopy types embeds fully faithfully in $E_\infty$-agebras over $\overline{\mathbb{F}}_p$ (note that it is important to use the algebraic closure otherwise the statement is incorrect). However, as far as I know, no homotopy groups was ever computed using this fact. | |
| Aug 10, 2023 at 16:57 | history | edited | LSpice | CC BY-SA 4.0 |
Typos
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| Aug 10, 2023 at 16:02 | history | edited | Andrea Marino | CC BY-SA 4.0 |
deleted 1 character in body
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| Aug 10, 2023 at 14:27 | answer | added | Connor Malin | timeline score: 7 | |
| Aug 10, 2023 at 10:17 | history | asked | Andrea Marino | CC BY-SA 4.0 |