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 be a simply connected finite complex such that . Then contains -torsion for infinitely many .

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

**1. The analog in stable homotopy**

Let be a finite -local spectrum. Then one has a similar result:

Theorem 6If , then for infinitely many .

Since the homotopy groups are bounded, it follows that must have an infinite amount of -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 by , that is the smash product of with the mod Moore spectrum. This is still nonzero, because

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

There’s a bit of technology that one has to now invoke. Let’s say that a -local spectrum is **harmonic** if it is local with respect to the wedge of the Morava -theories, and **dissonant** if it is annihilated by that wedge of Morava -theories. Here the Morava -theories are complex-orientable ring spectra with whose formal group law is of height exactly . 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 is a torsion, bounded-above spectrum (i.e. for ), then is dissonant. In fact, any such lies in the smallest subcategory of spectra closed under colimits and translations and containing , so it suffices to show that is dissonant. This is a consequence of the fact that no formal group law over a ring with can have height and at the same time.

**Example 2** If is any finite spectrum, then 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 7Let be a finite spectrum. Then the spectrum of maps

is contractible. More generally, this is true with replaced with any torsion, bounded-above spectrum.

In particular, it follows that itself cannot be a bounded-above torsion spectrum if is finite. This proves the theorem above. It follows as a consequence that the stable stems exist in infinitely many dimensions (that could be seen very concretely using something like the -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 is a finite, simply connected complex. Let’s restate the McGibbon-Neisendorfer result.

Theorem 8Suppose : that is, the localization is not contractible. Then contains -torsion for infinitely many .

To prove this, let’s suppose that there is no -torsion in for . (Here .) In particular, the homotopy groups of in degrees are free -modules. We want to get a contradiction from this.

There are two cases:

- for (that is, is bounded above).
- can be chosen as above such that 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 whose -localized homotopy groups are bounded above.

Proposition 9Suppose is a finite simply connected complex and is bounded above. Then .

*Proof:* To do this, let be the largest nonzero -localized homotopy group. If contains a torsion-free summand, then we are actually in the situation of the first case. Let’s suppose instead that is all torsion. Then the connected component of is a for a -torsion abelian group. This contradicts the Sullivan conjecture.

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

is contractible. This means not only that any map is nullhomotopic, but that the same is true with 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 10Suppose is a finite simply connected complex and there exists such that is torsion-free for and . Then .

Let’s replace by the localization . Let be the universal cover of . Then we have an equivalence:

for . and we thus produce a map

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

We have a map which we want to lift up this Postnikov tower, which is a tower of principal fibrations with fibers of the form for *torsion-free*: that’s a consequence of the fact that is torsion-free above dimension . The obstruction diagram in lowest degree looks like:

For example, to lift to , we have to check that the composite to is trivial.

Lemma 11The obstructions to lifting 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 , as a simply connected space, becomes a product of Eilenberg-MacLane spaces after rationalizing, and so its -invariants vanish. That is, the map already has a section after rationalizing.

But the obstructions live in the cohomology of with torsion-free coefficients (namely, coefficients in ). 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

which was nontrivial (in fact, it was nonzero on ). We produced a section of this map using obstruction theory.

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

Lemma 12Let be the universal cover of an iterated loop space of a finite -local complex. Consider any map . Then it cannot have a section.

*Proof:* In fact, if had a section , the map

is nontrivial, since the composite with is the Bockstein . It follows that we get a nontrivial map from to . This contradicts the Sullivan conjecture. In fact, if for a finite -local complex, then the Sullivan conjecture states that the space of pointed maps

is nullhomotopic, and in particular any pointed map is contractible. But maps are the same thing as maps that induce zero on , so any map has also to be null. This implies the lemma.

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