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(n)-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 Φ 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 Λ G respect E 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 orientations of E(2)-local TMF by finding a presentation of MU in this category, and then finding the space of E maps MUTMF. (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 t𝔾 appearing in Nathaniel Stapleton’s character theory? (It then “acts” on p/ 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 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 π-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(n)-locally using attaching maps along exotic elements of the K(n)-local Picard group?
  • Is there an analogous notion of “vector bundle” which is classified by analogues of the spaces BU(m), 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(n)-localization is E n itself (G. Horel) and THH(KU)KUΣKU (McClure-Staffeldt).
  • Does THH(E n) 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[G] to E nΣ + LBG related to chromatic redshift?
  • Is there a surjection from the exotic elements of the K(p1)-local Picard group onto /p, as at n=1 and n=2? Does this split?
  • (Hughes, Lau, Peterson) The quotient of the mod-2 cohomology of BU2k 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 BU2k refine to a space X(k) whose E-cohomology realizes this symmetric power?