# 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 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 probably can add a little. You can play some interesting game with localisations being embedded in completions in some "nice" way and try to understand how "small" things (colimits) sit inside "wild" things (completions) and how much latter reduces to former. There are quite a lot of things written on integral or integral-to-rational group-theoretic side (Vogel localisation, Baumslag rationalisation, Teichner-Orr-Igusa works on transfinite concordance invariants). But! You need to thread a very thin line between situation where you live in pre-transfinite nicely filtered world and lose... Aug 11 at 19:11
• everything in intersection of l. c. sequence, and one where you just end up with faithful (or close to it) functor from groups to algebras of some kind; and that is pretty much what you get from works of Rivera, Zeinalian and Casacuberta. (precisely what you get from their models is various flavours of fibrewise homological localisation of spaces; you leave the group itself completely intact and simplify a bit the k-invariant which glues fundamental group to the universal covering) Aug 11 at 19:18