This is part of an algebraic topology problem
list, maintained by Mark Hovey.
- 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?
- 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).
- 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.
- 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.
- 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.
Department of Mathematics