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
is nullhomotopic. (The map was defined in the previous post.)
Since the homotopy groups of are finite, it will follow (by the Milnor exact sequence) that we can let
and conclude that the map
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 , we can get a splitting
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 , the map
is nullhomotopic. Here is the space that classifies families of
-adically completed copies
of the sphere spectrum; it has the property that
but the higher homotopy groups are that of . In any event, from all this, we get a map
classifying a “completed” stable spherical fibration over . Since
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 are homotopic. But we can interpret these in terms of (completed) spherical fibrations. The map
classifies the completed spherical fibration over associated to fibration
given by -completing the inclusion map
. The map
classifies the pull-back of the completed spherical fibration over
under
.
The key observation of Sullivan that leads to a proof of the Adams conjecture is:
Theorem 1 Suppose
. Then the map
(canonically) factors through
, giving an unstable Adams operation
(which is a homotopy equivalence). These Adams operations are compatible with one another as
varies.
Note that, integrally, the maps are endomorphisms of
, not of
. 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
classified by
is exactly the pull-back of
along
.
- The above diagram (and the fact that the horizontal maps are homotopy equivalences!) shows that the two
-completed spherical fibrations over
(defined by
and
) are homotopy equivalent.
In particular, it follows that .
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
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
be the functor that sends a pro-object to its homotopy limit. Then, given a variety
, we have a homotopy equivalence
between the profinite completion of the space
of complex points and the (homotopy limit of the) étale homotopy type of
.
This is essentially all we need to know. The important is the following:
Theorem 2 Let
be a variety defined over
. Then there is an action of
on the profinite completion
.
This is a remarkable fact: the elements of the Galois group do not in any sense induce continuous automorphisms of
. However, étale homotopy theory implies that they do act on the profinite completion.
Let’s consider a basic example: . In this case, we find that we have a
action on
. What does it look like? We can determine the action in cohomology: the étale cohomology of
with
-coefficients can be computed from the exact sequence (of étale sheaves)
To work with (the sheaf of
th roots of unity) is equivalent to working with
: they are the same as étale sheaves but have different Galois symmetries. We get an injection
By étale descent theory, we have that is
(generated e.g. by
), and we get an injection
This is in fact an isomorphism: one knows this from topology (and can prove it by by Tsen’s theorem and some additional analysis). It follows that one has an isomorphism,
-equivariant,
Thus we have a -equivariant isomorphism
Concretely, this means the following: there is a cyclotomic character
given by restriction to the maximal cyclotomic extension. The action of on
is given by multiplication by
.
We find:
Proposition 3 The action of
on
induces multiplication by
on
(with finite or profinite coefficients).
4. More examples of Galois symmetry
Many other natural examples of spaces come from varieties over (or
-varieties). They consequently exhibit a Galois symmetry on their profinite completions.
Example 1 for any
comes from a variety over
(even
). In particular, the colimit
has a Galois symmetry: the action on
of an element
is given by multiplication by
. This follows from the computation we did earlier for
and naturality thanks to the inclusion
For , we did not need étale homotopy theory to produce the Galois symmetry in the profinite completion
: we could have used the fact that
is a functor. For
, and for the following example, the Galois symmetry is not at all evident.
Example 2 Each Grassmannian of
-planes in
-space (in fact, these exist as schemes over
). Taking the colimit in
, we get a Galois symmetry on
Using the splitting principle for the (-algebraic!) map
we find that the Galois action on is such that
acts via
for (for
) the
th Chern class.
Since the maps come from maps of
-varieties over
, 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 to produce the promised unstable versions of the Adams operations. These will be maps
(for ) which are self-homotopy equivalences and which are compatible with the usual (profinitely completed) Adams operations
.
To do this, choose whose cyclotomic character
projects to
. As in the last section, we get a map
These maps are compatible in for varying
, and
has the action in
-cohomology
in particular, it is an equivalence (since we are completed at ).
Proposition 4 The map
is an unstable version of the Adams operations.
Proof: We need to show that the diagram
is homotopy commutative, where is the ordinary Adams operation. To do this, we have to show that two elements in
are equal, where is
-completed
-theory. For this in turn it suffices (by the splitting principle for
-theory) to pull back along the map
. In other words, we need to show that the following diagram is homotopy commutative:
But we already know that is given by “multiplication by
” on profinitely completed
from our earlier analysis (and naturality): in particular, the diagram thus homotopy commutes from the definition of the Adams operations as
for a line bundle
.
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
are homotopic, when is taken either to be the inclusion
or
. However, we have seen that
(for
) can be defined at the level of
as an “unstable” Adams operation constructed through étale homotopy.
Now the first map (for ) classifies the tautological spherical fibration
(which is algebraically defined!). The second map, when we twist by , classifies the pull-back of this fibration along
. But the homotopy commutative (and cartesian!) square
shows that “twisting” by doesn’t change the (
-completed) spherical fibration. This is the Adams conjecture.
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