Motivic Homotopy

Oct. 21, 2004

The Hurewicz theorem in 𝔸^{1} homotopy theory.

Part of today’s discussion uses algebraic geometry and part of it is more formal (in the sense of being abstract model categorical/homotopy theoretic). For today we will concentrate on the formal part, assuming the following facts. Marc Levine will go over these next week.

• If a chain complex C_{∗} of Nisnevich sheaves of abelian groups is (n − 1)- connected, then L^{𝔸^{1}}C_{∗} is (n − 1)-connected.

• C_{∗} is 𝔸^{1}-local if and only if H^{k}C_{∗} is strictly 𝔸^{1}-invariant for all k.

• A connected space X is 𝔸^{1}-local if and only if the homotopy sheaves π_{∗}(X) are strictly 𝔸^{1}-invariant. (This is a consequence of the previous fact, but no proof will be given here.)

• If C_{∗} is 0-connected, then it is 𝔸^{1}-local if and only if K(C_{∗}) is 𝔸^{1}-local. Similarly, a map C_{∗} → D_{∗} of 0-connected chain complexes is an 𝔸^{1}- equivalence if and only if K(C_{∗}) → K(D_{∗}) is an 𝔸^{1}-equivalence.

Definition. A model category is left proper if weak equivalences are preserved by pushouts along cofibrations. In other words, if we have a pushout diagram

\xymatrix{X \ar@{^{(}->}[d] \ar[r]^{∼} & Y \ar[d] \\ X̃ \ar[r] & Ỹ}

(meaning that the top map is a weak equivalence and the lefthand map is a cofibration), then the bottom map is also a weak equivalence.

This gives us the following consequence, which is left as an (entertaining) exercise: Suppose we have a diagram as below.

\xymatrix{A_{1} \ar[d]^{∼} & B_{1} \ar[l] \ar[r] \ar[d]^{∼} & C_{1} \ar[d]^{∼} \\ A_{2} & B_{2} \ar[l] \ar[r] & C_{2}}

If one of the maps in each row is a cofibration, then the induced map of pushouts A_{1} ∪_{B_{1}} C_{1} → A_{2} ∪_{B_{2}} C_{2} is a weak equivalence.

The 𝔸^{1} homotopy category is left proper.

As an application, consider the following pushout diagram. (We use the notation 𝔾_{m} = 𝔸^{1} \ {0}.)

\xymatrix{𝔾_{m} × 𝔾_{m} \ar@{^{(}->}[d] \ar@{^{(}->}[r] & 𝔸^{1} × 𝔾_{m} \ar[d] \\ 𝔾_{m} × 𝔸^{1} \ar[r] & 𝔸^{2} \ \{0\}}

We can understand the 𝔸^{1}-homotopy type by considering the following maps of pushout diagrams. Here C denotes the (simplicial) cone on a space.

\xymatrix{𝔾_{m} × 𝔸^{1} & 𝔾_{m} × 𝔾_{m} \ar@{^{(}->}[l] \ar@{^{(}->}[r] & 𝔸^{1} × 𝔾_{m} \\ 𝔾_{m} × (C𝔾_{m} × 𝔸^{1}) \ar[u] \ar[d] & 𝔾_{m} × 𝔾_{m} \ar@{^{(}->}[l] \ar@{^{(}->}[r] \ar[u] \ar[d] & (C𝔾_{m} × 𝔸^{1}) × 𝔾_{m} \ar[u] \ar[d] \\ 𝔾_{m} × C𝔾_{m} & 𝔾_{m} × 𝔾_{m} \ar@{^{(}->}[l] \ar@{^{(}->}[r] & C𝔾_{m} × 𝔾_{m}}

The vertical maps are all weak equivalences, so this establishes an 𝔸^{1} homotopy equivalence between the pushout of the top row (which is 𝔸^{2} \ {0}) and the pushout of the bottom row. The pushout of the bottom row is more commonly known as the join 𝔾_{m} ∗ 𝔾_{m}.

The join has another expression in 𝔸^{1} homotopy theory. Given pointed spaces (X, x_{0}) and (Y, y_{0}), there is an 𝔸^{1} homotopy equivalence from the join X ∗ Y to the space Σ(X ∧ Y); it is given by collapsing the subspace (X ∗ y_{0}) ∪ (x_{0} ∗ Y) to a point. More explicitly, there is a map of diagrams as follows.

\xymatrix{X∗y_{0} \ar[d] & x_{0}∗y_{0} \ar@{^{(}->}[l] \ar@{^{(}->}[r] \ar[d] & x_{0}∗Y \ar[d] \\ ∗ & ∗ \ar[l] \ar[r] & ∗}

The vertical maps are all equivalences, so the pushout X ∗ y_{0} ∪ x_{0} ∗ Y is 𝔸^{1} equivalent to a point. The pushout diagram

\xymatrix{X∗y_{0} ∪ x_{0}∗Y \ar[d]^{∼} \ar@{^{(}->}[r] & X∗Y \ar[d] \\ ∗ \ar[r] & Σ(X ∧ Y)}

then shows that that the right-hand arrow is an 𝔸^{1} equivalence.

𝔾_{m} can be naturally pointed by the identity element, and so we get an 𝔸^{1} equivalence 𝔸^{2} \ {0} ≃ Σ(𝔾_{m} ∧ 𝔾_{m}).

Applying these arguments inductively shows that 𝔸^{n} \ {0} is weakly equivalent to the n-fold join 𝔾_{m} ∗ … ∗ 𝔾_{m} or to the space Σ^{n−1}(𝔾_{m} ∧ … ∧ 𝔾_{m}). As such, it is (n − 2)-connected.

We have a locally trivial 𝔾_{m}-bundle (𝔸^{n+1} \ {0}) → ℙ^{n}, which gives (up to homotopy) a fibration even in our 𝔸^{1}-model category. (This requires proof.) As a result, there is a long exact sequence of homotopy groups.

Suppose n ≥ 2. Then the connectivity of 𝔸^{n+1} \ {0} shows that π_{1}(ℙ^{n}) ≅ π_{0}(𝔾_{m}) = 𝔾_{m}. (𝔾_{m} is a constant simplicial sheaf. To compute π_{∗}, we usually need to replace it by a fibrant simplicial sheaf, but this is unnecessary for π_{0}.)

The 𝔸^{1} Hurewicz theorem then tells us that π_{1}^{𝔸^{1}}ℙ^{n} = π_{1}L^{𝔸^{1}}ℙ^{n}, which is the free strictly 𝔸^{1}-invariant sheaf generated by 𝔾_{m}. However, 𝔾_{m} is already strictly 𝔸^{1}-invariant, so π_{1}^{𝔸^{1}}ℙ^{n} = 𝔾_{m}.

We note that the space ℙ^{n}, which in our model category is the constant simplicial sheaf represented by ℙ^{n}, is not fibrant in our model category structure; otherwise π_{1} would vanish.

Recall that this proof was only valid for n ≥ 2; it is not true for ℙ^{1}. If n = 1, there is still a fibration sequence 𝔾_{m} → (𝔸^{2} \ {0}) → ℙ^{1}. It turns out that π_{1}(𝔸^{2} \ {0}) = K_{2}^{M}, the second Milnor K-group of the underlying field.

(This proved by somehow establishing an equivalence between 𝔸^{2}\{0} and SL_{2}.)

This shows π_{1}(𝔸^{2} \ {0}) is abelian, and we get an extension

0 → K_{2}^{M} → π_{1}ℙ^{1} → 𝔾_{m} → 0

of groups. This extension was known to Merkurjev and Suslin, and comes from the multiplication in Milnor K-theory.

The Hurewicz theorem. Let X be a space which is (n−1)-connected, n ≥ 3. (This theorem is in fact true for n ≥ 2.)

Then π_{i}^{𝔸^{1}}(X) = π_{i}(L^{𝔸^{1}}X) = 0 for i ≤ n − 1, and π_{n}^{𝔸^{1}}(X) is the free strictly 𝔸^{1}- invariant sheaf generated by π_{n}(X). Also, π_{n}^{𝔸^{1}}(X) = H_{n}^{𝔸^{1}}(X) = π_{n}(L^{𝔸^{1}}ℤX).

Our agenda is to prove this and then do some calculations using it.

There is also an analogue of the H_{1} = (π_{1})_{ab} theorem from ordinary topology for connected spaces. We note that almost all the “facts” we have assumed say something about sheaves of abelian groups, and so we need a mechanism for passing from these to simplicial spaces. As usual, this method is the Postnikov tower. The Postnikov tower.

First, let’s review the Postnikov tower for topological spaces. Given a (connected) space X, there is a tower of maps

\xymatrix{ & ⋮ \ar[d] \\  & P^{3}X \ar[d] \\  & P^{2}X \ar[d] \\ X \ar[uuur] \ar[uur] \ar[ur] \ar[r] & P^{1}X}

such that the map X → P^{n}X is an isomorphism on π_{i} for i ≤ n and π_{i}P^{n}X = 0 for i > n. This tower can be formed by making use of iterated cell attachments.

This implies that K(π_{n+1}X, n+1) → P^{n+1}X → P^{n}(X) is a fiber sequence, but this is not really a principal bundle. Instead, we have a fiber square

\xymatrix{P^{n+1}X \ar[r] \ar[d] & P^{1}X \ar[d] \\ P^{n}X \ar[r] & BK_{n+1},}

where K_{n+1} is an abelian group object over P^{1}X.

Why do we only encounter troubles with π_{1}? This happens because for n > 1, the homotopy automorphism group hAutK(π, n) ≅ Aut(π). Roughly speaking (in other words, the following statement is false), π_{k}hAut(K(π, n)) = [Σ^{k}K(π, n), K(π, n)] = 0. The spaces Σ^{k}K(π, n) have no lower cohomology groups. The problem with n = 1 is that Σ^{1}K(π, 1) can have non-abelian one-dimensional cohomology groups.

(For the record, the actual statement would be that elements of π_{k}hAut(K(π, n), ϕ) based at an automorphism ϕ are determined by maps S^{k}×K(π, n) → K(π, n), or equivalently elements of H^{n}(S^{k} ×K(π, n), π), which restrict to ϕ on ∗×K(π, n) and respect the basepoint. This is enough to show that there is only one homotopy class of them.)

This story, including the part about π_{1}, generalizes. We’re going to skip the details. However, there is one part of the story that does not immediately generalize.

For topological spaces, there is a weak equivalence X → holimP^{n}X. (One often takes the maps P^{n+1}X → P^{n}X to be fibrations, and in that case we simply take the inverse limit rather than the homotopy inverse limit.) The fact that this is a weak equivalence is obvious for spaces because weak equivalences are defined in terms of homotopy groups.

Let’s carry out this construction in the category of simplicial Nisnevich sheaves. The same thing works, but objects now have different meanings. We can still construct a tower X → {P^{N}X} with the desired properties on homotopy sheaves, and such that we have a pullback diagram as before. However, the “obvious” fact that X ≃ lim P^{N}X is an equivalence is no longer obvious.

We need to show that π_{n}X → π_{n} lim P^{N}X is an isomorphism. This is a map of sheaves, and so it is equivalent to prove it on stalks; we need to show (π_{n}X)_{x} → π_{n}(lim P^{N}X)_{x} is an isomorphism for all points x in the site. (We should note that since sheafification preserves stalks, doing this means that we do not have to sheafify the presheaf π_{n} to check that this is an isomorphism.)

Recall that by definition (π_{n}X)_{x} = colim_{x∈U} π_{n}X(U), where π_{n}X(U) = [Σ^{n}U, X]. x here is going to range over all points in the Nisnevich site; x is not a point of X. U is allowed to range over smooth varieties of finite type.

On the other hand,

π_{n}(lim P^{N}X)_{x} = colim_{x∈U} π_{n}(lim_{N} P^{N}X(U))
= colim_{x∈U} lim_{N} π_{n}(P^{N}X(U))

We know that lim_{N} colim_{x∈U} π_{n}(P^{N}X(U)) = lim_{N}(π_{n}P^{N}X)_{x} = lim_{N} π_{n}(P^{N}X(x)), because taking stalks commutes with sheafification. The sheaf π_{n}X was the sheaf associated to the presheaf U ↦ π_{n}X(U). By definition, the map π_{n}(X) → π_{n}(P^{N}X) is an isomorphism of sheaves for N large; this is the same as saying that it is an isomorphism on stalks, so π_{n}(X(x)) → π_{n}(P^{N}X(x)) is an isomorphism for N large. This expresses the problem in another way; proving that the map π_{n}X → π_{n} lim P^{N}X is an isomorphism is equivalent to this limit interchange problem.

The points on the Nisnevich site are supposed to correspond to Henselian local rings; in particular, they have a specific dimension attached to them. Every point in the site is a colimit of points living in objects of a bounded dimension. For a fixed point x, we can restrict the colimit to a colimit over x ∈ U, where dimU < M. As a result, H^{k}(U, π) = 0 for k ≥ M.

Take the fibration sequence K(π_{N}X, N) → P^{N}X → P^{N−1}X and apply the functor [Σ^{n}U, −]. We get a long exact sequence with the following portion.

… → H^{N−n}(U, π_{N}X) → π_{n}P^{N}X(U) → π_{n}P^{N−1}X(U) → H^{N−n−1}(U, π_{N}X) → …

For N > M + n, the cohomology groups on both sides vanish, and so the limit is actually achieved at a finite stage independent of U. Using the Postnikov setup, we now have an approach to the Hurewicz theorem. We will construct L^{𝔸^{1}}(X) in a controlled way by building it out of L^{𝔸^{1}}K(π_{n}X, n). This is similar to Sullivan’s method for rationalizing or plocalizing a space.

Let X be a fibrant simplicial Nisnevich sheaf. We’ll assume π_{0}X = π_{1}X = ∗. We construct its Postnikov tower.

\xymatrix{ & ⋮ \ar[d] \\  & P^{4} \ar[r] \ar[d] & K(π_{5}X, 6) \\  & P^{3} \ar[r] \ar[d] & K(π_{4}X, 5) \\ X \ar[r] \ar[ur] \ar[uur] & K(π_{2}X, 2) \ar[r] & K(π_{3}X, 4)}

We just finished proving X ≃ lim P^{N}X.

Now we will make a new tower by localizing these Eilenberg-MacLane objects. We have inductively defined a tower P̃^{N} and a map P^{N} → P̃^{N} which is 𝔸^{1} localization. Now define P̃^{N+1} so that the following map is a map of fiber sequences:

\xymatrix{P^{N+1} \ar[r] \ar[d] & P̃^{N+1} \ar[d] \\ P^{N} \ar[r] \ar[d] & P̃^{N} \ar[d] \\ K(π_{N+1}X, N + 2) \ar[r] & L^{𝔸^{1}}K(π_{N+1}X, N + 2)}

We have to show that P̃^{N+1} is 𝔸^{1}-local and the top map is an 𝔸^{1}-equivalence. The first of these statements is straightforward using the long exact sequences for [X, −] and [X × 𝔸^{1}, −].

However, the second part is harder. The definition of 𝔸^{1}-equivalence is that [−, Z] is an isomorphism for all 𝔸^{1}-local Z, and we don’t yet have much control maps out of a space defined by a fibration sequence.

Fix an 𝔸^{1}-local object Z. We will prove this statement by using the Postnikov tower for Z. By definition, the homotopy sheaves of P^{k}Z are the same as those of Z, and being 𝔸^{1}-local is characterized by strict 𝔸^{1}-invariance of homotopy sheaves, so the spaces P^{k}Z are all 𝔸^{1}-local. Therefore, it suffices to check it on P^{k}Z. Using the long exact sequence in homotopy, it again suffices to check on a space of the form K(π_{k}Z, k + 1).

Therefore, if Y and Y′ are simply connected, then Y → Y′ is a weak equivalence if and only if for all strictly 𝔸^{1}-invariant sheaves π of abelian groups, H^{∗}(Y, π) ← H^{∗}(Y′, π). (The simple connectivity means that we do not need to worry about π_{1}.)

In the case of the map P^{N+1}X → P̃^{N+1}X, this can be checked using the Serre spectral sequence. For any point x in the site, there two Serre spectral sequences:

H^{∗}(P^{N}X, H^{∗}(K(π_{N+1}X, N + 1), π)) ⇒ H^{∗}(P^{N+1}X)
H^{∗} (P̃^{N}X, H^{∗} (L^{𝔸^{1}}K(π_{N+1}X, N + 1), π)) ⇒ H^{∗}(P̃^{N+1}X)

The isomorphism of the left-hand term yields an isomorphism of abutments.

((yes, I know this actually doesn’t say anything about the stalk we’re looking at. there should be some discussion of how the colimit on neighborhoods passes to an inverse limit on cohomology and how the Serre sequence deals with that.))

We now claim that the map X → lim P̃^{N}X is an 𝔸^{1} equivalence, so L^{𝔸^{1}}X = lim P̃^{N}X.

Construct a homotopy fibration diagram F → lim P̃^{N} → P̃^{N}. Then π_{i}F = 0 for i ≤ N, because π_{i}L^{𝔸^{1}}(K(π, M)) = 0 for M > i using Morel’s connectivity theorem.

Using the Serre spectral sequence, H^{i}(lim P̃^{N}, π) = H^{i}(P̃^{N}, π) = H^{i}(X, π) for i ≤ N. (Recall H^{i}(X, π) = [X, K(π, i)].)

Now, these results can all be done without the use of the Serre spectral sequence, but time is running low. So, out result is that the map X → lim P̃^{N}X is an 𝔸^{1} equivalence. By comstruction of P̃^{N}X, and by the facts from the beginning of the lecture, if X is (n − 1)-connected we can take P^{N} = 0 for N < n, and π_{n}L^{𝔸^{1}}X = π_{n}L^{𝔸^{1}}K(π_{n}X, n).

As we’re running out of time, we have to skip proving that this is the free strictly 𝔸^{1}-invariant sheaf generated by X. The rest of the theorem is really deduced from the Hurewicz theorem for simplicial sets by using stalks. To identify this with homology:

By working with the Moore-Postnikov factorization of a map X →^{f} Y, we would have concluded that in the case where X and Y are simply connected, π_{i}f being an isomorphism in dimensions less than n implies π_{i}L^{𝔸^{1}}f is an isomorphism in dimension n. We can then apply this to the map X → ℤX. By the classical Hurewicz theorem applied stalkwise, this is an isomorphism in homotopy up to 1 plus the connectivity of X. Therefore, π_{i}L^{𝔸^{1}}X → π_{i}L^{𝔸^{1}}ℤX is an isomorphism for i ≤ n, and specifically π_{n}L^{𝔸^{1}}X ≅ H_{n}^{𝔸^{1}}X.