Genetics of homotopy theory and the Adams conjecture

The purpose of this post is to describe Sullivan’s proof of the Adams conjecture via algebraic geometry; the conjecture and its motivation were described in the previous post (from where the notation is taken). The classical reference is Sullivan’s paper “Genetics of homotopy theory and the Adams conjecture,” and the MIT notes on “Geometric topology.”

1. First step: completion at a prime

Sullivan’s proof of the Adams conjecture is based on interpreting the Adams operations via a surprising Galois symmetry in the (profinitely completed) homotopy types of classifying spaces. Let’s work in the complex case for simplicity. Our goal is to show that the composite

$\displaystyle BU(n) \stackrel{\psi^k - 1}{\rightarrow } BU \stackrel{J}{\rightarrow} B \mathrm{gl}_1(S)[1/k]$

is nullhomotopic. (The map $J$ was defined in the previous post.)

Since the homotopy groups of ${B \mathrm{gl}_1(S)[1/k]}$ are finite, it will follow (by the Milnor exact sequence) that we can let ${n \rightarrow \infty}$ and conclude that the map

$\displaystyle BU \stackrel{\psi^k - 1}{\rightarrow } BU \stackrel{J}{\rightarrow} B \mathrm{gl}_1(S)[1/k]$

is nullhomotopic (i.e., there are no phantom maps into a spectrum with finite homotopy groups).

Using again the finiteness of the homotopy groups of ${B \mathrm{gl}_1(S)[1/k]}$ , we can get a splitting

$\displaystyle B \mathrm{gl}_1(S)[1/k] = \prod_{p \nmid k} \widehat{ B \mathrm{gl}_1(S)}_p$

into the respective profinite completions. There is a well-behaved theory of profinite completions for connective spectra, or for sufficiently nice (e.g. simply connected with finitely generated homology) spaces, which will be the subject of a different post.

It suffices therefore to prove that for a prime ${p \nmid k}$ , the map

$\displaystyle \widehat{BU(n)}_p \stackrel{\psi^k-1}{\rightarrow} \widehat{BU}_p \stackrel{J}{\rightarrow} \tau_{\geq 2}\widehat{ B \mathrm{gl}_1(S)}_p \simeq \tau_{\geq 2} B \mathrm{gl}_1(\widehat{S}_p)$

is nullhomotopic. Here ${B \mathrm{gl}_1( \widehat{S}_p)}$ is the space that classifies families of ${p}$ -adically completed copies ${\widehat{S}_p}$ of the sphere spectrum; it has the property that

$\displaystyle \pi_1 B \mathrm{gl}_1( \widehat{S}_p) = \mathbb{Z}_p^{\times},$

but the higher homotopy groups are that of ${\widehat{B \mathrm{gl}_1(S)}_p}$ . In any event, from all this, we get a map

$\displaystyle J (\psi^k -1): \widehat{BU(n)}_p \rightarrow B \mathrm{gl}_1( \widehat{S}_p)$

classifying a “completed” stable spherical fibration over ${\widehat{BU(n)}_p}$ . Since ${BU(n)}$ is simply connected, our goal is to show that it is nullhomotopic.

This is the first step in Sullivan’s argument, which is to complete at a given prime.

2. Outline of the argument

Let’s unwind this a bit further. We want to show that the two maps ${\psi^k, 1: \widehat{BU(n)}_p \rightarrow B \mathrm{gl}_1( \widehat{S}_p)}$ are homotopic. But we can interpret these in terms of (completed) spherical fibrations. The map

$\displaystyle J: \widehat{BU(n)}_p \rightarrow B \mathrm{gl}_1( \widehat{S}_p)$

classifies the completed spherical fibration over ${\widehat{BU(n)}_p}$ associated to fibration

$\displaystyle \widehat{BU(n-1)}_p \stackrel{\iota}{\rightarrow} \widehat{BU(n)}_p$

given by ${p}$ -completing the inclusion map ${\iota}$ . The map ${J \circ \psi^k}$ classifies the pull-back of the completed spherical fibration over ${\widehat{BU}_p}$ under ${\psi^k: \widehat{BU(n)}_p \rightarrow \widehat{BU}_p}$ .

The key observation of Sullivan that leads to a proof of the Adams conjecture is:

Theorem 1 Suppose ${p \nmid k}$ . Then the map ${\psi^k: \widehat{BU(n)}_p \rightarrow \widehat{BU}_p}$ (canonically) factors through ${\widehat{BU(n)}_p}$ , giving an unstable Adams operation

$\displaystyle \psi^k: \widehat{BU(n)}_p \simeq \widehat{BU(n)}_p$

(which is a homotopy equivalence). These Adams operations are compatible with one another as ${k}$ varies.

Note that, integrally, the maps ${\psi^k}$ are endomorphisms of ${BU}$ , not of ${BU(k)}$ . In particular, we have a commutative diagram

The Adams conjecture is in fact a consequence of these observations:

It shows that the spherical fibration over ${\widehat{BU(n)}_p}$ classified by ${J \circ \psi^k}$ is exactly the pull-back of ${\iota: \widehat{BU(n-1)}_p \rightarrow \widehat{BU(n)}_p}$ along ${\psi^k: \widehat{BU(n)}_p \rightarrow \widehat{BU(n)}_p}$ .
The above diagram (and the fact that the horizontal maps are homotopy equivalences!) shows that the two ${p}$ -completed spherical fibrations over ${\widehat{BU(n)}_p}$ (defined by ${J}$ and ${J \circ \psi^k}$ ) are homotopy equivalent.

In particular, it follows that ${J(\psi^k-1) \simeq 0}$ .

3. Etale homotopy theory and unstable Adams operations

This post will take étale homotopy theory and its basic properties as a black box:

There is a functor ${\mathrm{Et}: \mathrm{Sch} \rightarrow \mathrm{Pro}(\mathrm{Spaces})}$ that assigns to a scheme a pro-object in spaces (well-defined up to homotopy in pro-spaces).
The cohomology of the associated pro-space is the étale cohomology of the scheme.
Let
$\displaystyle \lim: \mathrm{Pro}(\mathrm{Spaces}) \rightarrow \mathrm{Spaces}$

be the functor that sends a pro-object to its homotopy limit. Then, given a variety ${X/\mathrm{Spec}\overline{\mathbb{Q}}}$ , we have a homotopy equivalence

$\displaystyle \widehat{ ( X(\mathbb{C}))} \simeq \lim \mathrm{Et}( X)$

between the profinite completion of the space ${X(\mathbb{C})}$ of complex points and the (homotopy limit of the) étale homotopy type of ${X}$ .

This is essentially all we need to know. The important is the following:

Theorem 2 Let ${X}$ be a variety defined over ${\mathbb{Q}}$ . Then there is an action of ${\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$ on the profinite completion ${( X(\mathbb{C})^{\widehat{}}}$ .

This is a remarkable fact: the elements of the Galois group ${\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$ do not in any sense induce continuous automorphisms of ${X}$ . However, étale homotopy theory implies that they do act on the profinite completion.

Let’s consider a basic example: ${\mathbb{P}^1}$ . In this case, we find that we have a ${\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$ action on ${\widehat{\mathbb{CP}^1}}$ . What does it look like? We can determine the action in cohomology: the étale cohomology of ${\mathbb{P}^1_{\overline{\mathbb{Q}}}}$ with ${\mathbb{Z}/n}$ -coefficients can be computed from the exact sequence (of étale sheaves)

$\displaystyle 0 \rightarrow \mu_n \rightarrow \mathbb{G}_m \stackrel{n}{\rightarrow} \mathbb{G}_m \rightarrow 0 .$

To work with ${\mu_n}$ (the sheaf of $n$ th roots of unity) is equivalent to working with ${\mathbb{Z}/n}$ : they are the same as étale sheaves but have different Galois symmetries. We get an injection

$\displaystyle H^1_{\mathrm{et}}( \mathbb{P}^1_{\overline{\mathbb{Q}}}, \mathbb{G}_m) \otimes \mathbb{Z}/n \hookrightarrow H^2_{\mathrm{et}}(\mathbb{P}^1_{\overline{\mathbb{Q}}}, \mu_n).$

By étale descent theory, we have that ${H^1_{\mathrm{et}}( \mathbb{P}^1_{\overline{\mathbb{Q}}}, \mathbb{G}_m) \simeq\mathrm{Pic}( \mathbb{P}^1_{\overline{\mathbb{Q}}})}$ is ${\mathbb{Z}}$ (generated e.g. by ${\mathcal{O}(1)}$ ), and we get an injection

$\displaystyle \mathbb{Z}/n \hookrightarrow H^2_{\mathrm{et}}(\mathbb{P}^1_{\overline{\mathbb{Q}}}, \mu_n).$

This is in fact an isomorphism: one knows this from topology (and can prove it by ${H^2_{\mathrm{et}}(\mathbb{P}^1_{\overline{\mathbb{Q}}}, \mathbb{G}_m) = 0}$ by Tsen’s theorem and some additional analysis). It follows that one has an isomorphism, ${\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$ -equivariant,

$\displaystyle H^2_{\mathrm{et}}(\mathbb{P}^1_{\overline{\mathbb{Q}}}, \mu_n ) \simeq \mathbb{Z}/n.$

Thus we have a ${\mathrm{Gal}( \overline{\mathbb{Q}}/\mathbb{Q})}$ -equivariant isomorphism

$\displaystyle H^2_{\mathrm{et}}(\mathbb{P}^1_{\overline{\mathbb{Q}}}, \mathbb{Z}/n ) \simeq \mu_n^{-1}.$

Concretely, this means the following: there is a cyclotomic character

$\displaystyle \mathrm{cyc}: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \rightarrow \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})_{\mathrm{ab}} \simeq \widehat{\mathbb{Z}}^{\times}$

given by restriction to the maximal cyclotomic extension. The action of ${\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$ on ${H^2_{\mathrm{et}}(\mathbb{P}^1_{\overline{\mathbb{Q}}}, \mathbb{Z}/n )}$ is given by multiplication by ${\mathrm{cyc}(\sigma)^{-1}}$ .

We find:

Proposition 3 The action of ${\sigma \in \mathrm{Gal}( \overline{\mathbb{Q}}/\mathbb{Q})}$ on ${\mathbb{P}^1}$ induces multiplication by ${\mathrm{cyc}(\sigma)^{-1}}$ on ${H^2}$ (with finite or profinite coefficients).

4. More examples of Galois symmetry

Many other natural examples of spaces come from varieties over ${\mathbb{Q}}$ (or ${\mathrm{ind}}$ -varieties). They consequently exhibit a Galois symmetry on their profinite completions.

Example 1 ${\mathbb{CP}^n}$ for any ${n}$ comes from a variety over ${\mathbb{Q}}$ (even ${\mathbb{Z}}$ ). In particular, the colimit ${\widehat{\mathbb{CP}^\infty} = K(\widehat{\mathbb{Z}}, 2)}$ has a Galois symmetry: the action on ${\pi_2}$ of an element ${\sigma \in \mathrm{Gal}( \overline{\mathbb{Q}}/\mathbb{Q})}$ is given by multiplication by ${\mathrm{cyc}(\sigma)}$ . This follows from the computation we did earlier for ${\mathbb{CP}^1}$ and naturality thanks to the inclusion

$\displaystyle \mathbb{CP}^1 \rightarrow \mathbb{CP}^\infty$

For ${\mathbb{CP}^\infty}$ , we did not need étale homotopy theory to produce the Galois symmetry in the profinite completion ${K(\widehat{\mathbb{Z}}, 2)}$ : we could have used the fact that ${G \mapsto K(G, 2)}$ is a functor. For ${\mathbb{CP}^n}$ , and for the following example, the Galois symmetry is not at all evident.

Example 2 Each Grassmannian ${\mathrm{Gr}(k, n)}$ of ${k}$ -planes in ${n}$ -space (in fact, these exist as schemes over ${\mathbb{Z}}$ ). Taking the colimit in ${n}$ , we get a Galois symmetry on

$\displaystyle \varinjlim \widehat{\mathrm{Gr}(k, n)} = \widehat{BU(k)}.$

Using the splitting principle for the ( ${\mathrm{ind}}$ -algebraic!) map

$\displaystyle (\mathbb{CP}^\infty)^k \rightarrow BU(k),$

we find that the Galois action on ${\widehat{BU(k)}}$ is such that ${\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$ acts via

$\displaystyle \sigma( c_i) = \mathrm{cyc}(\sigma)^{-i} c_i,$

for ${c_i}$ (for ${i \leq k}$ ) the ${i}$ th Chern class.

Since the maps ${BU(n-1) \rightarrow BU(n)}$ come from maps of ${\mathrm{ind}}$ -varieties over ${\mathbb{Q}}$ , the Galois action on the profinite completions is compatible with these inclusion maps.

5. The profinite Adams operations

Our goal in this section is to use the Galois symmetry in the profinite completions of ${BU(n)}$ to produce the promised unstable versions of the Adams operations. These will be maps

$\displaystyle \psi^k: \widehat{BU(n)}_p \rightarrow \widehat{BU(n)}_p$

(for ${p \nmid k}$ ) which are self-homotopy equivalences and which are compatible with the usual (profinitely completed) Adams operations ${\widehat{BU} \rightarrow \widehat{BU}}$ .

To do this, choose ${\sigma \in \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}$ whose cyclotomic character ${\mathrm{cyc}(\sigma) \in \widehat{\mathbb{Z}}^{\times}}$ projects to ${k^{-1} \in \mathbb{Z}_p^{\times}}$ . As in the last section, we get a map

$\displaystyle \sigma: \widehat{BU(n)}_p \rightarrow \widehat{BU(n)}_p.$

These maps are compatible in ${n}$ for varying ${n}$ , and ${\sigma}$ has the action in ${\mathbb{Z}_p}$ -cohomology

$\displaystyle c_i \mapsto k^i c_i;$

in particular, it is an equivalence (since we are completed at ${p}$ ).

Proposition 4 The map ${\sigma}$ is an unstable version of the Adams operations.

Proof: We need to show that the diagram

is homotopy commutative, where ${\psi^k_{\mathrm{ord}}}$ is the ordinary Adams operation. To do this, we have to show that two elements in

$\displaystyle \widehat{K}_p( \widehat{BU(n)}_p) \simeq \widehat{K}_p( BU(n))$

are equal, where ${\widehat{K}_p}$ is ${p}$ -completed ${K}$ -theory. For this in turn it suffices (by the splitting principle for ${K}$ -theory) to pull back along the map ${(\mathbb{CP}^\infty)^n \rightarrow BU(n)}$ . In other words, we need to show that the following diagram is homotopy commutative:

But we already know that ${\sigma}$ is given by “multiplication by ${k}$ ” on profinitely completed ${\mathbb{CP}^\infty}$ from our earlier analysis (and naturality): in particular, the diagram thus homotopy commutes from the definition of the Adams operations as ${\psi^k(\mathcal{L}) = \mathcal{L}^{\otimes k}}$ for a line bundle ${\mathcal{L}}$ .

5. The Adams conjecture

The Adams conjecture is now a corollary of the above analysis. Namely, as above, we were reduced to showing that the two maps

$\displaystyle \widehat{BU(n)}_p \stackrel{f}{\rightarrow} \widehat{BU} \stackrel{J}{\rightarrow} B \mathrm{gl}_1(\widehat{S}_p), \quad f = \iota, \psi^k$

are homotopic, when ${f}$ is taken either to be the inclusion ${\iota}$ or ${\psi^k \circ \iota}$ . However, we have seen that ${\psi^k}$ (for ${p \nmid k}$ ) can be defined at the level of ${\widehat{BU(n)}_p}$ as an “unstable” Adams operation constructed through étale homotopy.

Now the first map (for ${f = \iota}$ ) classifies the tautological spherical fibration

$\displaystyle \widehat{BU(n-1)}_p \rightarrow \widehat{BU(n)}_p$

(which is algebraically defined!). The second map, when we twist by ${\psi^k}$ , classifies the pull-back of this fibration along ${\psi^k}$ . But the homotopy commutative (and cartesian!) square

shows that “twisting” by ${\psi^k}$ doesn’t change the ( ${p}$ -completed) spherical fibration. This is the Adams conjecture.

Climbing Mount Bourbaki