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In rational homotopy theory, one can study the rational homotopy and cohomology categories via an algebraic structure, respectively the rational Lie Algebra model and the Sullivan cdga model.

If I am not wrong, over a prime there is a similar theorem for $\mathbb{F}_p$ cohomology, that is Mandell's Theorem: under adequate assumptions, $\mathbb{F}_p$ cohomology together with its $E_{\infty}$ structure does a job analogous to Sullivan cdga in rational homotopy theory.

I am definitely not confident in this field, but I don't understand if $p$-homotopy groups $\mathbb{F}_p \otimes \pi_n$ (or some $p$-adic friend) are determined by Mandell's model or not, and what is the analog of the Lie model.

What I want in the end is a generalization of the notion of coformality to the $p$-adic world, which I recall: $X$ is said to be coformal if there exists a (zig-zag of) quasi-isomorphism connecting the rational homotopy groups of $\Omega X$ with the free Lie Algebra on $C^* X$.

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  • $\begingroup$ 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. $\endgroup$
    – Geoffroy Horel
    Commented Aug 10, 2023 at 17:42
  • $\begingroup$ 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? $\endgroup$
    – Andrea Marino
    Commented Aug 10, 2023 at 18:36
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    $\begingroup$ 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 $S^n$). $\endgroup$
    – Geoffroy Horel
    Commented Aug 11, 2023 at 6:50
  • $\begingroup$ 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. $\endgroup$
    – Andrea Marino
    Commented Aug 17, 2023 at 9:32
  • $\begingroup$ 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 $X$? $\endgroup$
    – Andrea Marino
    Commented Aug 17, 2023 at 9:56

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Mandell shows that, under some hypotheses, the $\mathbb{F}_p$-cochains detect the $\mathbb{F}_p$-homotopy type in the sense that there is such an equivalence $X \simeq Y$, if and only if, there is an equivalence of $E_\infty$-algebras $C_*(X; \mathbb{F}_p) \simeq C_*(X;\mathbb{F}_p)$. This is different than saying that $\mathbb{F}_p$-homotopy theory is equivalent to the homotopy theory of $E_\infty$-algebras in $\mathbb{F}_p$-cochain complexes because it says nothing about essential surjectivity or equivalence of mapping spaces. Nonetheless, it is still reasonable to call a space $p$-formal if there is an equivalence of $E_\infty$ algebras $C_*(X;\mathbb{F}_p) \simeq H_*(X;\mathbb{F}_p)$, and this is a useful, if rarely satisfied, condition. As an aside, Mandell showed this result actually be improved to an equivalence of homotopy theories if one instead takes coefficients in $\bar{\mathbb{F}}_p$.

Mandell also shows that our only guess for the Lie model of $\mathbb{F}_p$-homotopy theory, the Koszul dual of $C_*(X;\mathbb{F}_p)$ is actually contractible. I expect this result implies there is not an easy way to detect homotopy groups from the $E_\infty$-algebra model.

All is not lost! The correct way to model $p$-torsion information with Lie algebras was demonstrated by Heuts. Using chromatic homotopy theory, one can construct the $v_n$-localization of spaces and spectra. These localizations can see $p$-torsion information when $n>0$ and when $n=0$ it coincides with rationalization. Heuts showed that $v_n$-local spaces are modeled by Lie algebras in $v_n$-local spectra. I don't think there is a definition of coformality in this context, but I would be very interested to see one.

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  • $\begingroup$ 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. $\endgroup$
    – Andrea Marino
    Commented Aug 10, 2023 at 16:02
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    $\begingroup$ @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. $\endgroup$
    – Connor Malin
    Commented Aug 10, 2023 at 16:16
  • $\begingroup$ 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 $\endgroup$
    – Andrea Marino
    Commented Aug 10, 2023 at 16:44
  • $\begingroup$ @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. $\endgroup$
    – Connor Malin
    Commented Aug 10, 2023 at 17:03
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    $\begingroup$ 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. $\endgroup$
    – Geoffroy Horel
    Commented Aug 10, 2023 at 17:44

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