In the previous post, I described the Sullivan conjecture and gave a vague outline of its proof by Miller. Shortly after Miller’s paper, various applications of the theorem to other problems in homotopy theory were discovered.

The intuition here is that there are two ways a space might appear to be finite: one is that its homotopy groups might be bounded and the other is that its homology groups might be bounded. It’s generally very hard for the two to happen at once, at least in the simply connected case. For instance, Eilenberg-MacLane spaces — the basic examples of spaces with bounded homotopy groups — have very messy cohomology. Similarly, finite complexes — spaces with bounded homology — generally have very complicated homotopy groups.

The purpose of this post is to explain a proof of the following result, conjectured by Serre:

 

Theorem 5 (McGibbon, Neisendorfer) Let {X} be a simply connected finite complex such that {\widetilde{H}_*(X; \mathbb{Z}/p) \neq 0}. Then {\pi_i X} contains {p}-torsion for infinitely many {i}.

 

As we’ll see, this result can be proved using the Sullivan conjecture.

 

1. The analog in stable homotopy

Let {X} be a finite {p}-local spectrum. Then one has a similar result:

Theorem 6 If {X \neq 0}, then {\pi_i X \neq 0} for infinitely many {i}.

Since the homotopy groups {\pi_i X\otimes\mathbb{Q} = H_i(X; \mathbb{Q})} are bounded, it follows that {\pi_* X} must have an infinite amount of {p}-torsion.

Let’s describe a short proof of this result, which I learned from Ravenel’s paper “Localization with respect to certain periodic homology theories.” Without loss of generality, we can replace {X} by {X/p}, that is the smash product of {X} with the mod {p} Moore spectrum. This is still nonzero, because

\displaystyle H_*(X/p; \mathbb{Z}/p) = H_*(X; \mathbb{Z}/p) \otimes H_*(S/p; \mathbb{Z}/p) \neq 0.

In fact, more generally, the smash product of two nonzero finite {p}-local spectra is always nonzero.

There’s a bit of technology that one has to now invoke. Let’s say that a {p}-local spectrum is harmonic if it is local with respect to the wedge {\bigvee_{n \geq 0} K(n)} of the Morava {K}-theories, and dissonant if it is annihilated by that wedge of Morava {K}-theories. Here the Morava {K}-theories are complex-orientable ring spectra with {K(n)_* = \mathbb{F}_p[v_n^{\pm 1}]} whose formal group law is of height exactly {n}. The definition of being local states that there are no nontrivial maps from a dissonant spectrum to a harmonic one. In particular, a spectrum that is both dissonant and harmonic is zero.

The following two examples now imply the theorem:

 

Example 1: If {X} is a torsion, bounded-above spectrum (i.e. {\pi_i X = 0 } for {i \gg 0}), then {X} is dissonant. In fact, any such {X} lies in the smallest subcategory of spectra closed under colimits and translations and containing {H \mathbb{Z}/p}, so it suffices to show that {H \mathbb{Z}/p} is dissonant. This is a consequence of the fact that no formal group law over a ring with {p=0} can have height {\infty} and {n} at the same time.

Example 2 If {X} is any finite spectrum, then {X} is harmonic. This is a little harder to prove, and probably deserves its own post. It’s the non-formal input that goes into the proof of Theorem 6 above.

 

One concludes with the following (easier!) stable analog of the Sullivan conjecture:

Theorem 7 Let {X} be a finite spectrum. Then the spectrum of maps

\displaystyle X^{H \mathbb{F}_p}

is contractible. More generally, this is true with {H \mathbb{F}_p} replaced with any torsion, bounded-above spectrum.

 

In particular, it follows that {X} itself cannot be a bounded-above torsion spectrum if {X} is finite. This proves the theorem above. It follows as a consequence that the stable stems {\pi_*(S^0)_{(p)}} exist in infinitely many dimensions (that could be seen very concretely using something like the {J}-homomorphism, though).

The “harmonic” technology is also useful because a theorem of Hopkins and Ravenel states that every suspension spectrum is harmonic. As a consequence, it follows that the stable homotopy groups of any torsion space cannot be bounded above.

 

2. The McGibbon-Neisendorfer result

 Let’s now describe the unstable McGibbon-Neisendorfer theorem. This is a little trickier to prove, because the unstable category doesn’t behave quite as nicely as the stable one. For example, the rational homotopy groups of a finite complex don’t have to be concentrated in finitely many dimensions.

However, let’s suppose now that {X} is a finite, simply connected complex. Let’s restate the McGibbon-Neisendorfer result.

Theorem 8 Suppose {\widetilde{H}_*(X; \mathbb{Z}/p) \neq 0}: that is, the localization {X_{(p)}} is not contractible. Then {\pi_i X} contains {p}-torsion for infinitely many {i}.

To prove this, let’s suppose that there is no {p}-torsion in {\pi_m X} for {m > n}. (Here {n \geq 2}.) In particular, the homotopy groups of {X_{(p)}} in degrees {> n} are free {\mathbb{Z}_{(p)}}-modules. We want to get a contradiction from this.

There are two cases:

  • {\pi_m X_{(p)} = \ast} for {m \gg 0} (that is, {X} is bounded above).
  • {n} can be chosen as above such that {\pi_n X} has a nontrivial torsion-free summand.

We’ll derive a contradiction from each of these cases.

3. The first case

In order to prove the McGibbon-Neisendorfer result, we also need to consider the case of a complex {X} whose {p}-localized homotopy groups are bounded above.

Proposition 9 Suppose {X} is a finite simply connected complex and {\pi_*(X)_{(p)}} is bounded above. Then {X_{(p)} = \ast}.

 

Proof: To do this, let {\pi_n X_{(p)}} be the largest nonzero {p}-localized homotopy group. If {\pi_n X_{(p)}} contains a torsion-free summand, then we are actually in the situation of the first case. Let’s suppose instead that {\pi_n X_{(p)}} is all torsion. Then the connected component of { \Omega^{n-1} X_{(p)} } is a {K(G, 1)} for {G} a {p}-torsion abelian group. This contradicts the Sullivan conjecture.

\Box

To elaborate, the Sullivan conjecture states that the space of maps

\displaystyle \mathrm{map}_*(B G, X_{(p)}) \simeq \mathrm{map}_*(BG, X)

is contractible. This means not only that any map {BG \rightarrow X_{(p)}} is nullhomotopic, but that the same is true with {X_{(p)}} replaced by any iterated loop space thereof.

 

4. The second case

The second case is the more difficult part of the McGibbon-Neisendorfer theorem. We’ll state it as a result in its own right:

 

Proposition 10 Suppose {X} is a finite simply connected complex and there exists {n} such that {\pi_m X_{(p)}} is torsion-free for {m > n} and {\pi_n X_{(p)} \otimes \mathbb{Q} \neq 0}. Then {X_{(p)} \simeq \ast}.

 

Let’s replace {X} by the localization {X_{(p)}}. Let {W} be the universal cover of {\Omega^{n-2} X}. Then we have an equivalence:

\displaystyle \tau_{\leq 2}W \simeq K( \Pi, 2),

for {\Pi = \pi_n X}. and we thus produce a map

\displaystyle W \rightarrow \tau_{\leq 2} W \rightarrow K(\Pi, 2) \rightarrow K(\mathbb{Z}_{(p)}, 2).

The claim is that this map has a section. In order to prove this, choose a Postnikov tower for {W},

\displaystyle W \rightarrow \dots \rightarrow \tau_{\leq 3} W \rightarrow \tau_{\leq 2} W.

We have a map {K(\mathbb{Z}_{(p)}, 2) \rightarrow \tau_{\leq 2} W} which we want to lift up this Postnikov tower, which is a tower of principal fibrations with fibers of the form {K(G, n)} for {G} torsion-free: that’s a consequence of the fact that {\pi_* (X)} is torsion-free above dimension {n}. The obstruction diagram in lowest degree looks like:

Screenshot-53

For example, to lift {K(\mathbb{Z}_{(p)}, 2)} to {\tau_{\leq 3} W}, we have to check that the composite to {K(\pi_3 W, 4)} is trivial.

 

Lemma 11 The obstructions to lifting {K(\mathbb{Z}_{(p)}, 2)} up the Postnikov tower are trivial.

To check this, observe that the obstructions are in fact trivial after rationalizing. That’s a consequence of the fact that {W}, as a simply connected {H} space, becomes a product of Eilenberg-MacLane spaces after rationalizing, and so its {k}-invariants vanish. That is, the map {W \rightarrow \tau_{\leq 2} W \simeq K(\Pi, 2)} already has a section after rationalizing.

But the obstructions live in the cohomology of {K(\mathbb{Z}_{(p)}, 2) = \mathbb{CP}^\infty_{(p)}} with torsion-free coefficients (namely, coefficients in {\pi_* W}). It follows that if they vanish after rationalization, they were zero to begin with.

The upshot of this discussion is that we had a map

\displaystyle W \rightarrow \tau_{\leq 2} W \rightarrow K(\mathbb{Z}_{(p)}, 2)

which was nontrivial (in fact, it was nonzero on {\pi_2}). We produced a section {s: K(\mathbb{Z}_{(p)}, 2) \rightarrow W} of this map using obstruction theory.

The next lemma provides a contradiction, and proves the McGibbon-Neisendorfer theorem.

Lemma 12 Let {W} be the universal cover of an iterated loop space of a finite {p}-local complex. Consider any map {f: W \rightarrow K(\mathbb{Z}_{(p)}, 2)}. Then it cannot have a section.

Proof: In fact, if {f} had a section {s: K(\mathbb{Z}_{(p)}, 2) \rightarrow W}, the map

\displaystyle K(\mathbb{Z}/p, 1) \stackrel{\beta}{\rightarrow} K(\mathbb{Z}_{(p)}, 2) \stackrel{s}{\rightarrow} W

is nontrivial, since the composite with {f} is the Bockstein {K(\mathbb{Z}/p, 1) \rightarrow K(\mathbb{Z}_{(p)}, 2)}. It follows that we get a nontrivial map from {B\mathbb{Z}/p} to {W}. This contradicts the Sullivan conjecture. In fact, if {W = \tau_{\geq 1} \Omega^k X} for {X} a finite {p}-local complex, then the Sullivan conjecture states that the space of pointed maps

\displaystyle B\mathbb{Z}/p \rightarrow \Omega^k X

is nullhomotopic, and in particular any pointed map {B\mathbb{Z}/p \rightarrow \Omega^k X} is contractible. But maps {B \mathbb{Z}/p \rightarrow W} are the same thing as maps {B \mathbb{Z}/p \rightarrow \Omega^k X} that induce zero on {\pi_1}, so any map {B \mathbb{Z}/p \rightarrow W} has also to be null. This implies the lemma. \Box

 

 
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