Miscellaneous problems

This is part of an algebraic topology problem list, maintained by Mark Hovey.

  1. Build MU from the moduli stack of formal groups. This has got to be doable somehow, though it is an old problem (I first heard it in Ravenel's green book). Note that we have some more tools now--the moduli stack is, I think, just a space in Voevodsky's category, so maybe it is an infinite loop space there?

  2. Classify all possible Bousfield classes of E-infinity ring spectra. I know very little about this problem. Note that the Spanier-Whitehead dual of the suspension spectrum of a space X is an E-infinity ring spectrum, with multiplication dual to the diagonal map. But you are forgetting the disjoint basepoint!! When you add, as you must, a disjoint basepoint to X, you find that the E-infinity ring spectrum has the Bousfield class of the sphere. Also, I once heard somebody--Jim McClure?--say that if you start with, say, MU, and you mod out by p, you find that you must also mod out by all the higher v's too in order to get an E-infinity ring spectrum. Therefore, I conjecture that if E is an E-infinity ring spectrum that kills a nontrivial finite spectrum X, then E has the Bousfield class of E(n) for some n. (I must be p-local here). We know these Bousfield classes do occur, since Morava E-theory is an E-infinity ring spectrum. (Goerss-Hopkins).

  3. This one is due to Mike Hopkins. Generalize the whole Thom spectrum business as follows. Take an A-infinity ring spectrum E. Look at the space of A-infinity self equivalences of E. I think this has a classifying space B, because of composition of self equivalences. Given a map X --> B you should be able to construct a Thom spectrum, and it should be some kind of half-smash product of E and X. Mike had a more explicit description, which I seem to have forgotten.

  4. Mahowald's root invariant. Recall the root invariant is a self-map of the stable homotopy groups of spheres which I believe is a differential in some spectral sequence, so the root invariant R(x) of x is not a single homotopy class, but a collection. It is known that R(x) is in degree at least twice the degree of x; the Bredon-Loffler conjecture asserts that it is in at most 3 times the degree of x. In general, the root invariant is supposed to convert v_n -periodic elements to v_n+1 periodic elements, so the root invariant of an alpha is supposed to be a beta, but nothing certain is known about this except in fairly low degrees. Hal Sadofsky's papers would be a place to start.

  5. Related to the root invariant is Tate cohomology. Given a spectrum X, we think of it as an equivariant Z/p spectrum with trivial action. We then take its Tate cohomology a la Greenlees-May, and then take the fixed point spectrum. This construction kills Morava K-theory K(n), and takes E(n) to something like E(n-1) (but its rather complicated--see Ando-Morava-Sadofsky). When applied to a finite spectrum X, we recover the p-completion of its desuspension--this is Lin's theorem at p=2. What do we get when we apply it to L_n X? If X is type n, we get nothing. If X is type n-1, we are supposed to get 2 copies of L_n-1 X, one copy desuspended once and one copy desuspended twice. This conjecture is due to Hopkins and Mahowald, I think, and seems to be related to the chromatic splitting conjecture. Hal Sadofsky has probably thought the most about this.

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Mark Hovey
Department of Mathematics
Wesleyan University