# Problems in homotopy theory Stable homotopy theory

• Assuming we know ${L}_{K\left(n\right)}X$, give an effective procedure—or even a systematic set of examples—for computing ${L}_{n-1}{L}_{K\left(n\right)}X$; equivalently, given some way to understand the $n$th monochromatic layer ${M}_{\mathrm{nY}}$ where $Y$ is $K\left(n\right)$-local. As a point of entry, first study ${L}_{K\left(n-1\right)}{L}_{K\left(n\right)}X$.

Some preliminary algebraic work has been done in the papers of Torii (MR2004428), which uses work of Gross to study what happens to deformations of formal groups. For example, given a height $n$ formal group $G$ over an algebraic extension $𝔽$ of ${𝔽}_{p}$, we can take the deformation of $G$ to $𝔽\left[\phantom{\rule{-0.1667 em}{0ex}}\left[{u}_{n-1}\right]\phantom{\rule{-0.1667 em}{0ex}}\right]$ and then consider this deformation as a height $n-1$ formal group over $𝔽\left[\phantom{\rule{-0.1667 em}{0ex}}\left[{u}_{n-1}\right]\phantom{\rule{-0.1667 em}{0ex}}\right]\left[{u}_{n-1}^{-1}\right]\right]$. We might examine and reinterpret this algebra in homotopy theory and then extend this to a calculation and interpretation of

$\left({E}_{n-1}{\right)}_{*}{E}_{n}={\pi }_{*}{L}_{K\left(n-1\right)}\left({E}_{n-1}\wedge {E}_{n}\right)$(E_{n-1})_\ast E_n = \pi_\ast L_{K(n-1)}(E_{n-1} \wedge E_n)

where ${E}_{s}$ is Morava $E$-theory. This is surely naive. Write ${C}_{n}$ for the cyclic group. Then the formal group over $\left({E}_{n}{\right)}_{*}$ is the formal spectrum of

${\pi }_{0}F\left({\mathrm{ℂℙ}}^{\infty },{E}_{n}\right)=\mathrm{lim}{\pi }_{0}F\left({\mathrm{BC}}_{{p}^{n}},{E}_{n}\right)$\pi_0F(\mathbb{CP}^\infty,E_n) = \lim \pi_0 F(BC_{p^n},E_n)

and we probably should consider the inverse system

${\pi }_{0}{L}_{K\left(n-1\right)}F\left({\mathrm{BC}}_{{p}^{n}},{E}_{n}\right)$\pi_0L_{K(n-1)}F(BC_{p^n},E_n)

which is a $p$-divisible group. Thus $p$-divisible groups appear again.

• The Chromatic Splitting Conjecture asserts that the map $j:{L}_{n-1}X\to {L}_{n-1}{L}_{K\left(n\right)}X$ in the chromatic fracture square is a split inclusion—and much more besides. Can even this simple statement be verified, at least for some $X$?

• Revisit the Chromatic Splitting Conjecture. If it’s true, an initial case to study would be when $n$ is very small with respect to the prime $p$. Are there cases where it might not true? For example, if $n$ is not a unit in $p$, the reduced norm map ${𝔾}_{n}\to {ℤ}_{p}$ from the Morava stabilizer group doesn’t have a central splitting. If ${𝔾}_{n}^{1}$ is the kernel of the reduced norm, there is then a Lyndon-Serre-Hochschild Spectral Sequence

${H}^{s}\left({ℤ}_{p},{H}^{t}\left({𝔾}_{n}^{1},{E}_{*}X\right)\right)⟶{H}^{s+t}\left({𝔾}_{n},{E}_{*}X\right)\right)$H^s(\mathbb{Z}_p,H^t({\mathbb{G}}_n^1,E_\ast X)) \longrightarrow H^{s+t}({\mathbb{G}}_n,E_\ast X))

and the action of ${ℤ}_{p}$ can be very complicated. Does that twist the cohomology so that the CSS becomes difficult? Recent work of Beaudry might be applicable.

• (P. Goerss) Calculations in $K\left(2\right)$-local homotopy theory. Much has been done, but more can be understood. For example, use Olivier’s thesis (Strasbourg 2013) to understand the Shimomura school’s calculations at large primes.

• (P. Goerss) Structured ring spectra $p$-divisible groups. A map $ℳ\to {ℳ}_{\mathrm{fg}}$ from an algebraic stack to the moduli stack of formal groups is realizable if there is a sheaf $𝒪$ of ${E}_{\infty }$-ring spectra on $ℳ$ (in an appropriate topology) with ${\pi }_{*}𝒪\cong {\omega }^{\otimes */2}$, where $\omega$ is the sheaf of invariant differentials.

• It has been asserted that it’s not possible to realize ${ℳ}_{\mathrm{fg}}$ itself, mostly because the only sensible topology is the $\mathrm{fpqc}$-topology. It would nice to have this fact proved and recorded, if it’s true.

• (N. Naumann) There is an argument that it is not possible to adjoin a $p$‘th root of unity to the $p$-adic $K$-theory spectrum as an ${E}_{\infty }$ ring spectrum (which seems likely to be originally due to Hopkins). You can view this as a special case of adjoining a Drinfel’d level structure to the universal deformation space of height 1. As such it generalizes to all heights, and I wrote that up in 2009 for Matt Ando.

• (T. Lawson) Whether or not the realization problem can be solved in a much weaker setting (e.g. merely on the spectrum level) is also something that would be worth proving.

• Results of Lurie give realization criteria for 'etale maps $ℳ\to {ℳ}_{p}\left(n\right)$ where ${ℳ}_{p}\left(n\right)$ is the moduli stack (over ${ℤ}_{p}\right)$ for $p$-divisible groups of height $n$ with formal part of dimension 1. Explore the geometry of the map ${ℳ}_{p}\left(n\right)\to {ℳ}_{\mathrm{fg}}$ to the moduli stack of formal groups. Understand the resulting descent problem as well. The map is not representable if $n>1$.

• All ${E}_{\infty }$-ring spectra come equipped with power operations. Lurie’s result and its antecedent, the Hopkins-Miller theorem for Morava $E$-theories, produce ${E}_{\infty }$-ring spectra but make no mention of power operations; thus it would seem the power operations are canonically dictated by the geometry of $p$-divisible groups. Explain this. Recent work of Rezk would be a place to start; this suggests analysis of the subgroup structure of $p$-divisible groups is important.

• Shimura varieties. One of the important features of the Hopkins-Miller theory of topological modular forms is that one gets sheaf of ${E}_{\infty }$-ring spectra on the compactified Deligne-Mumford moduli stack of elliptic curves. Are there good compactifications of Shimura varieties that would apply to the Behrens-Lawson theory of topological automorphic forms? Work of Kai-Wen Lan might be applicable.

• (P. Goerss) Gross-Hopkins duality. Brown-Comenetz duality and its variants are something of a curiosity in stable homotopy theory, but it is an insight of Hopkins that in the $K\left(n\right)$-local category it is much more like Serre-Grothendieck duality.

• Flesh out that statement. Some work has been done by Hopkins himself, and more by Devinatz, but this is just the start.

• If possible, compare Gross-Hopkins duality to the Poincar'e duality structure of the Morava stabilizer group. Again some work has been done by people looking at Shimomura’s calculation, most lately by Behrens. Develop a larger theory that works, if possible, at primes where there’s torsion in the Morava stabilizer group.

• Think about the Behrens $Q\left(\ell \right)$ spectra in this context.

• I don’t know this for a fact, but Lurie’s theory probably gives a notion of Serre-Grothendieck duality in derived algebraic geometry. Make this precise and concrete; that is, give calculations. Work of Stojanoska should point the way.

• (D. Isaksen) Compute the ${E}_{2}$-term of the Adams-Novikov spectral sequence in as large a range as possible, by hand or by machine.

• (A. Salch) If $E$ is a homotopy commutative ring spectrum, then a complex orientation on $E$ is a class in ${E}^{2}\left(\mathrm{BU}\left(1\right)\right)$ with an appropriate property. Suppose $G$ is some gadget of some kind that has an underlying 1-dimensional formal group law. (For example, $G$ could be 1-dimensional formal $A$-modules, for some particular commutative ring $A$.) You can ask whether there exists some space $X$ equipped with a map $\mathrm{BU}\left(1\right)\to X$ such that, for any complex orientation $c$ in ${E}^{2}\left(\mathrm{BU}\left(1\right)\right)$, choices of the structure of a $G$-object on the formal group associated to $c$ are in bijection with lifts of $c$ to an element in ${E}^{2}\left(X\right)$.

The answer to this is probably “no” for most choices of $G$. But it would be good to figure out when it’s yes and when it’s no.

Then you can mimic the story of formal groups but with X in the place of BU(1).

• (A. Salch) Go back to the Mahowald-Ravenel-Shick paper about their attempted counterexample to the ${v}_{2}$-telescope conjecture and explain why they couldn’t quite finish what they were doing. As a side effect this would lead to a lot of good side material. Hoping to complete what Mahowald-Ravenel-Shick left unfinished is probably too much to hope for, though.

• (A. Salch) Barry Walker’s thesis on ${E}_{\infty }$ complex orientations of $\mathrm{KU}$ had a lot of very good stuff. Finding a way to do any of that stuff at higher heights would be great. (Ask Matt Ando and/or Charles Rezk about this.)

• (A. Salch) We now know, after Johnson-Noel, that the usual ring map $\mathrm{MU}\to \mathrm{BP}$ isn’t an ${E}_{\infty }$-ring map for small primes ($p<19$ or something like that). Basterra and Mandell showed, however, that it is an ${E}_{4}$-map. It would be interesting to know, for a given prime $p$, what’s the largest integer $i$ such that the usual map $\mathrm{MU}\to \mathrm{BP}$ is an ${E}_{i}$-ring map.

• (J. Noel) Since the $p$th space of the ${E}_{n}$ operad is $n-2$-connected, the associated total cyclic $p$th ${E}_{n}$ operation agrees with the power operations calculated in Johnson-Noel through the range coming from the $n-2$-skeleton. So the obstructions listed there give upper bounds on $n$. In particular at the prime $2$ a universal $2$-typical orientation is at most ${E}_{13}$. I think for primes greater than $3$ this bound is approximately $10{p}^{2}-16*p+7$. Lazarev also showed the universal orientation was ${E}_{1}$. These give some crude bounds. I think a first shot in this direction would be showing Basterra-Mandell’s ${E}_{4}$ ring structure admits a universal $p$-typical ${E}_{4}$ orientation (they already construct the splitting).
• (J. Wu) For which spaces X are the stable homotopy groups summands of its unstable homotopy groups?

Namely, for each ${\pi }_{n}^{s}\left(X\right)$, there exists an unstable homotopy group ${\pi }_{m}\left(X\right)$ so that ${\pi }_{n}^{s}\left(X\right)$ is a summand of ${\pi }_{m}\left(X\right)$.

Here we do not require that $m$ is the same as $n$.

Counter example: If $X$ is a sphere, the answer is no. However there are some complexes $X$ having the property that its stable homotopy groups are summands of its unstable homotopy groups.

Methodology (from what we have known) for solving the question for certain special spaces $X$: If there exists a sequence of positive integers ${n}_{k}$ with ${n}_{k}$ tending to infinity such that the loop space $\Omega {\Sigma }^{{n}_{k}}X$ is a retract of $\Omega X$, then one can see that every stable homotopy group of $X$ becomes a summand of one of its unstable homotopy groups. Technically one needs to search those retracts in the loop space $\Omega X$ given as the loops on the suspensions of $X$. This methodology works for special complexes $X$ through computations. For more general cases, it seems not easy.

• The telescope conjecture?.

• The generating hypothesis?.

• The chromatic splitting conjecture?.