Axiomatic stable homotopy theory
This is part of an algebraic topology problem
list, maintained by Mark Hovey.
The general reference here is my memoir of the AMS with Palmieri and
Strickland on axiomatic stable homotopy theory.
- In our memoir, we give a conjecture for the thick subcategories in
a Noetherian stable homotopy category C--they should be in 1-1 correpondence
with subsets of Spec pi_* S closed under specialization. Prove this
conjecture. (This seems out of reach to me, but you never know).
- Characterize the stable homotopy category up to equivalence. This
has been done for categories that are homotopy categories of model
categories by Schwede, I think, and a lot is known in general. Margolis
characterized the quotient category when phantoms are killed, and
Christensen-Strickland characterized the phantom subcategory. So the
category is determined up to a square 0 extension, but maybe there are
tons of different such square 0 extensions.
- Show that there is only a set of localizing subcategories. It is
known that there is only a set of Bousfield classes (Ohkawa; Strickland
simplified his proof a bit, and then Dwyer and Palmieri have an even
simpler proof). Of course, we expect there to be a 1-1 correspondence
between Bousfield classes, localizing subcategories, and colocalizing
subcategories, but perhaps that is out of reach.
- In one of Bob Thomason's last papers, he determined the thick
subcategories of finite objects in the derived category of a scheme.
For the derived category of a ring R, they are in 1-1 correspendence
with unions of closed sets X_i in Spec R such that Spec R - X_i is
quasi-compact. How about localizing subcategories? What about applying
his ideas to more general non-Noetherian stable homotopy categories?
- John Palmieri has determined the E_2 term of the Adams spectral
sequence up to nilpotence--at least he has found a computable ring which
is f-isomorphic to the E_2 term. I think this is only at p=2. Will his
methods work in other specific non-Noetherian stable homotopy
categories? Haynes Miller suggested trying to use John's methods to
calculate group cohomology of GL(infinity) up to f-isomorphism.
- My general feeling about stable homotopy categories is that they
are like commutative rings. Follow this up; define Spec C for example,
for a stable homotopy category C. Several people have had an idea like
this; Jack Morava said something to me about it that I forgot.
- The equivariant stable homotopy category is not treated very well
in our memoir. That is, we assume that the generators have to be
dualizable. This is not true unless you use the complete universe.
Peter May tried to talk us out of this at the time, and I think he was
right. So try to understand what happens when the generators are not
dualizable. Is there some other condition that does hold in the
equivariant stable homotopy category over an incomplete universe that
replaces this? There is a notion of weakly dualizable, for example,
which is just that D^2 X = X. Maybe the generators are weakly
- From an axiomatic point of view, I don't understand Grojnowski's
equivariant elliptic cohomology. This theory takes values in an abelian
category that is not modules over a ring--I think it is sheaves over a
scheme or something. Is there some way to understand homology theories
that land in, say, Grothendieck categories, from an axiomatic point of
- Suppose G is a self-equivalence of the stable homotopy category.
Must G be some iterate of the suspension functor? If G commutes with the
suspension, I can prove that GS^0 = S^n for some n, but this is almost
all that I know. One could ask the same question for the K(n) local
category or the E(n) local category.
- What is the endomorphism ring of the identity functor on the stable
homotopy category? The ring Z splits off this ring, including by
multiples of the identity, and projecting off by observing what the
natural transformation does to the identity map of S^0. If we assume
the generating hypothesis, any element of the kernel is a natural
self-phantom map of X for all X. This must force it to be 0, but why?
And do we really need the generating hypothesis?
Department of Mathematics