# A $p$-adic homotopy theory for non-simply connected spaces?

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!

• 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. Aug 5 at 20:36
• 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). Aug 5 at 20:47
• 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 Aug 5 at 21:23
• 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. Aug 5 at 22:44
• 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. Aug 6 at 19:47