# Problems in homotopy theory Problems from the E-theory seminar

This is currently a placeholder for questions raised by the E-theory seminar going on at MIT. The short version of this would be to simply go through and write down everything that has a bold-block “Question” or “Conjecture” next to it, but I think that there are enough little things that it’s worth going through in more detail.

(This is incomplete.)

• When is the map from the $K\left(n\right)$-local Picard group to the algebraic Picard group surjective or injective? What can one say about finiteness, or finite generation, or completeness, of the source, target, and kernel?
• Do the lower Bousfield-Kuhn functors ${\Phi }_{t}$, applied to the total unstable power operation for ${E}_{n}$, act nontrivially? Does this give some filtration of the total power operation on ${E}_{n}$?
• Does the “inertia groupoid” functor ${\Lambda }_{G}$ respect ${E}_{\infty }$ structures?
• Can Marsh’s computations of the $E$-theory of linear algebraic groups over finite fields be expressed algebro-geometrically? (They appear to be connected to symmetric powers of formal schemes.)
• Ando has a program for studying ${E}_{\infty }$ orientations of $E\left(2\right)$-local $\mathrm{TMF}$ by finding a presentation of $\mathrm{MU}$ in this category, and then finding the space of ${E}_{\infty }$ maps $\mathrm{MU}\to \mathrm{TMF}$. (This may already or imminently be being considered by a graduate student, and this problem has been considered by Jan-David Möllers in his thesis.)
• (H. Miller) How does the Morava stabilizer group ${𝕊}_{n}$ act on the formal group ${C}_{t}\otimes 𝔾$ appearing in Nathaniel Stapleton’s character theory? (It then “acts” on ${ℚ}_{p}/{ℤ}_{p}$^{n-t}, but also acts on the base scheme.)
• (N. Stapleton) Does going down in height by one give the determinant representation of ${𝕊}_{n}$?
• (S. Glasman) What happens when $n=1$, $t=0$? In this case the ring ${C}_{t}$ is a maximal ramified extension of ${ℚ}_{p}$.
• (H. Miller) Is this character theory the unit of an adjunction between “height $n$ stuff” and “height $t$ stuff”, for some definition of “stuff”?
• Is there a nice category of spaces over $\pi$-finite spaces with fibers finite CW-complexes? In particular, can one make sense of the intertia groupoid in this category?
• What spectra can be decomposed $K\left(n\right)$-locally using attaching maps along exotic elements of the $K\left(n\right)$-local Picard group?
• Is there an analogous notion of “vector bundle” which is classified by analogues of the spaces $\mathrm{BU}\left(m\right)$, which come equipped with associated bundles which are fibered in determinantal spheres?
• What is the topological Hochschild homology of the Lubin-Tate spectrum ${E}_{n}$? The $K\left(n\right)$-localization is ${E}_{n}$ itself (G. Horel) and $\mathrm{THH}\left(\mathrm{KU}\right)\simeq \mathrm{KU}\wedge \Sigma {\mathrm{KU}}_{ℚ}$ (McClure-Staffeldt).
• Does $\mathrm{THH}\left({E}_{n}\right)$ classify some sort of deformations of the Honda formal group law along with an automorphism? (It is claimed that ${E}_{n}$ doesn’t have a moduli-theoretic interpretation in the terms of derived algebraic geometry.)
• Is the Dennis trace map from the algebraic $K$-theory of the group algebra ${E}_{n}\left[G\right]$ to ${E}_{n}\wedge {\Sigma }_{+}^{\infty }\mathrm{LBG}$ related to chromatic redshift?
• Is there a surjection from the exotic elements of the $K\left(p-1\right)$-local Picard group onto $ℤ/p$, as at $n=1$ and $n=2$? Does this split?
• (Hughes, Lau, Peterson) The quotient of the mod-2 cohomology of $\mathrm{BU}⟨2k⟩$ by images of the classes in odd degree by the elements in the Steenrod algebra has an algebro-geometric interpretation, in terms of a symmetric power of a scheme of divisors.
• How does the dual Steenrod algebra act on the Cartier dual scheme to the symmetric power from the previous question?
• Does $\mathrm{BU}⟨2k⟩$ refine to a space $X\left(k\right)$ whose $E$-cohomology realizes this symmetric power?