The telescope & chromatic splitting conjectures
Tobi Barthel
April 26, 2013
There’s a general feeling that algebraic topology right now lacks conjectures. This talk is about conjectures that aren’t quite new – they’re about 20 years old – but they’re still highly interesting.
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Motivation
Conjecture 1 (Generating hypothesis, Freyd 1966). Let f : X → Y be a map of finite CW-spectra with
π∗ f = 0. Then f is nullhomotopic.
This is still open – as you are all by now familiar, it’s pretty hard to compute homotopy groups of finite
complexes. One thing that would follow if it were true is that the functor π∗ : [X, Y ] → Homπ∗ S (π∗ X, π∗ Y )
is fully faithful. In particular, Devinatz and Hopkins investigated this where Y is the sphere. Suppose
f : X → S is not null. If f is of infinite order, this is easy, so suppose f has order n. We recall that there’s
a spectrum I with [X, I] = Hom(π0 X, Q/Z) for X finite. Smashing with I, we get a triangle
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n ∈π0 I=Q/Z
S
/ I ∧X
1∧f
/, I ∧ S.
If I is a colimit of finite spectra Iα , then for some α, this diagram factorizes as
1
n ∈π0 I=Q/Z
S
/ I ∧X
O
1∧f
#
Iα ∧ X
/, I ∧ S
O
/ Iα ∧ S,
and smashing with DIα gives a map DIα → X → S of order n. It suffices to prove the conjecture for these
‘universal examples’.
In order to deal with this, we need to actually construct models for the Iα , and then prove the generating
hypothesis for these examples. By the chromatic convergence theorem, it suffices to work En -locally. The
telescope conjecture and chromatic splitting conjecture have something to say about this. By the way, this
has been done at height 1:
Theorem 2 (Devinatz). The generating hypothesis holds K-locally, using the above method.
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Localization and the telescope conjecture
Throughout, we localize at a prime p.
Theorem 3 (Bousfield 1979). If E is a spectrum, there’s a localization functor LE with respect to E, and
indeed a functor sending X to a cofiber sequence CE X → X → LE X with LE X an E-localization and CE X
an E-acyclicization.
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The proof relies on building a small set of generators for the category of E-acyclics. It’s an open problem
whether every localization functor (a functor assigning to every X a terminal object of some category of
acyclics over X, with LX its cofiber) arises as localization with respect to some spectrum E.
Definition 4. A localization functor is finite if there is a collection Xα of finite spectra that generate its
category ker L of acyclics.
Proposition 5 (Ravenel, Miller, Mahowald-Sadofsky).
1. Finite localizations exist on the category of spectra, and correspond to thick subcategories: Lfn =
LTel(0)∨···∨Tel(n) , where Tel(i) is a type i spectrum with a vi -self map inverted.
2. Each Lfn is smashing with something.
3. If X is type n, then Lfn X = Tel(X).
4. There are natural transformations 1 → Lfn → Ln .
We won’t go through the proof, but the idea of the first part is roughly that a finite localization is
determined by a collection of finite acyclic objects, which one shows generate a thick subcategory.
Bousfield asked the question (in different language): is every smashing localization functor finite? This
leads us to investigate smashing localization functors. If L is smashing, then it’s in fact true that for each
n, LK(n) is either 0 or K(n), and moreover if LK(n) = 0 then LK(m) = 0 for all m ≥ n. Thus there are
three cases:
• LK(n) = 0 for all n, in which case L = 0.
• LK(n) ' K(n) for all n, in which case L is the identity.
• For some n, LK(n) = K(n) and LK(n + 1) = 0. Then ker Lfn ⊆ ker L ⊆ ker Ln .
This leads us to
Conjecture 6 (Telescope conjecture, Ravenel 1977). Lfn = Ln , implying that all smashing localization
functors are finite and can be identified by their effects on the K(n).
Proposition 7 (Weak version). The following are equivalent for X finite of type n:
1. Tel(X) = Ln X.
2. hTel(X)i = hK(n)i.
3. If Lfn−1 = Ln−1 , then Lfn = Ln .
4. The ANSS for Tel(X) converges to π∗ Tel(X).
In particular, a single example or counterexample that is finite of type n is enough, by the thick subcategory theorem.
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What’s known?
For n = 0, this is easy.
Bousfield showed that this is true for n = 1, by computing the L1 -localization and telescope of V (0). This
was based on work of Mahowald and Miller, who computed the homotopy groups of Tel(V (0)) at respectively
p = 2 and p > 2.
For n ≥ 2, Ravenel did the same thing for V (1) and p ≥ 5 in about 1990, showing that the conjecture
was false in these cases. In 1992, however, he started having doubts, and never published the paper. In 1995,
Mahowald-Ravenel-Shick constructed a type n spectrum Y (n), which is not finite, such that π∗ Ln Y (n) is
finitely generated over R(n)∗ = K(n)∗ [vn+1 , . . . , v2n ], but π∗ Lfn Y (n) is only finitely generated over R(n)∗ if
there is a ‘bizarre pattern of differentials’ in the ANSS. Thus, if we can establish that no such differentials
exist, we can disprove the telescope conjecture at height 2. Currently, the status of the conjecture here is
unknown, but believed false. (Mark Mahowald really wants someone to work on this!)
Finally, the telescope conjecture is known to be true after smashing with BP .
4. THE CHROMATIC SPLITTING CONJECTURE
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The chromatic splitting conjecture
There’s both a qualitative and a quantitative version of this. The qualitative version is weaker, and we’ll
focus on it here. There’s a map of fiber sequences, where the right square is a pullback:
/ Ln X
F (Ln−1 S, Ln X)
(1)
∼
F (Ln−1 S, Ln X)
/ LK(n) X
(2)
x
/ Ln−1 X m
/ Ln−1 LK(n) X
(3)
If (1) exists, then (2) is nullhomotopic, and thus a splitting (3) exists.
Conjecture 8 (Chromatic splitting conjecture, Hopkins, early 90s). If X is a p-complete finite spectrum,
then (1) exists.
There’s a stronger version which also says that F (Ln−1 S, Ln X) decomposes further. This is likely to be
false for small primes.
Example 9. Take n = 1. Then πi LK(1) S = Zp for i = 0, 1, Zps+1 for i = (p − 1)ps m with p - m, and 0
otherwise. Thus πi L0 LK(1) S = Qp for i = 0 and −1 and 0 otherwise; of course π∗ L0 S is Qp in degree zero.
One can then see that L0 LK(1) S splits as L0 S ∨ L0 S −1 .
Example 10. Take n = 2 and p > 2. Then L1 LK(2) S ' L1 (S 0 ∨ S −1 ) ∨ L0 (S −3 ∨ L−1 ), which is evidence
for the conjecture though not a complete proof in this case.
Remark 11 (Devinatz). The chromatic splitting conjecture is false for X = BP .
Here are some consequences of this weak form of the conjecture.
Q
1. If X is finite, then X → n LK(n) X is split injective. Thus we don’t even need to glue the monochromatic layers of X back together to get X back.
2. The compositions LK(n+1) X → Ln X → LK(n) X, where the first map is (1) above, give a map X →
limn LK(n) X. The CSC implies that this is an equivalence. (This is due to Hopkins and Hovey.)
3. The homotopy groups of π∗ LK(n) S are finitely generated over Zp . (This isn’t published, but should
be immediate from a paper of Devinatz.)
Theorem 12 (Minami 2003). If X is a robust spectrum (a class that includes the finite spectra and BP ),
and m and k are positive integers satisfying m − k ≥ n + s0 + 1 where s0 is the vanishing line constant for
the ANSS for E(n), then the map Lm X → Ln X factors through LK(k+1)∨···∨K(m) X.