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I'm looking to understand the state of the art for $p$-adic (unstable) homotopy theory of non-simply connected (non-nilpotent!) spaces. Ideally, I'd also like integral versions, e.g. things like Mandell and Yuan's theorems, but without any restriction on fundamental group (the spaces I care about are all $K(\pi,1)$s). Squinting at the rational homotopy theory literature in the non-simply connected case (e.g. Brown-Szczarba, or Gomez-Tato-Halperin-Tanre), I can imagine that what I want may be "known to experts", in which case pointers toward that, or precise statements of what's expected are most appreciated. Thanks!

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  • $\begingroup$ If the spaces you care about are $K(\pi,1)$'s, then all their higher homotopy groups are trivial and you're just left with group homology (or I guess you could also look at extraordinary cohomology theories). The more classical rational homotopy theory literature will tell you a lot about about the Malcev completion of your fundamental group, but I suppose you already know that. $\endgroup$
    – Thomas
    Aug 5 at 20:36
  • $\begingroup$ I want to avoid Malcev completion (the information I care about is contained in the intersection of all the groups in the derived series). At a basic level, I'm trying to understand how much of the (category of) groups I can access through the ($E_\infty$) ring structure on the cochains (or the cocommutative coalgebra structure on the chains). $\endgroup$ Aug 5 at 20:47
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    $\begingroup$ I think that a lot of what you are looking for can be found in the work of Manuel Rivera. I don't know if he is active on MO, but his papers (esp the ones about cochains and the fundamental group) are on his webpage: riveramanuel.com $\endgroup$
    – Thomas
    Aug 5 at 21:23
  • $\begingroup$ Thanks, Thomas! I just found his 2022 Transactions paper with Wierstra and Zeinalian which is very close to what I want. The main gap between what they prove and what's in e.g. Yuan is the consideration of how these algebraic models capture mapping spaces. But it's definitely in the right direction. $\endgroup$ Aug 5 at 22:44
  • $\begingroup$ I'd say that since Bousfield-Kan the only big step was the proof of Bousfield's conjecture that free group is "bad" for both rational and mod p homology (i. e. map to a pronilpotent/pro-p completion is not a homology equivalence) due to R. Mikhailov and S. Ivanov few years ago. $\endgroup$
    – Denis T.
    Aug 6 at 19:47

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