Motivic Homotopy

Dec. 2, 2004

We’re going to start off with some background from ordinary stable homotopy theory.

Let E be a (reduced) multiplicative cohomology theory. Then E is complex orientable if there is an element x ∈ E^{2}(ℂℙ^{∞}) such that the restriction E^{2}(ℂℙ^{∞}) → E^{2}(ℂℙ^{1}) ≅ E^{0} sends x to 1 ∈ E^{2}(ℂℙ^{1}) ≅ E^{0}(S^{0}). x is called a complex orientation of E.

If E has a chosen a complex orientation, then:

• The Thom isomorphism holds for all complex vector bundles, and is compatible with Whitney sums.

• If Z ⊂ M is a holomorphic embedding of smooth complex manifolds of codimension d, then there is an isomorphism E^{∗+2d}(M, M \ Z) ≅ E^{∗}(Z).

In the language of spectra, a complex orientation is a particular choice of a Thom class MU → E. (MU is the universal complex vector bundle of virtual dimension 0.)

Using the Thom isomorphism, it is straightforward to compute:

• E^{∗}(∏^{n} ℂℙ^{∞}) ≅ E^{∗}[[x_{1}, …, x_{n}]].

• E_{∗}(ℂℙ^{∞}) ≅ E_{∗}{b_{0}, b_{1}, …}, where the b_{i} are dual to x^{i}.

(These are usually proved using the Atiyah-Hirzebruch spectral sequence, which arises using either the Postnikov tower for E or a cellular decomposition for ℂℙ^{∞}. Usually we prefer the latter method because ℂℙ^{∞} is such a nice space.)

• E^{∗}(BU(n)) ≅ E^{∗}(BT^{n})^{Σ_{n}} = E^{∗}[[c_{1}, …, c_{n}]], where |c_{i}| = 2i. Here T^{n} is the maximal torus of U(n), acted on by the symmetric group Σ_{n}.

(Again, this computation can be derived from the cell decomposition of BU(n).)

• E_{∗}(BU) = Sym_{E_{∗}}(E_{∗}ℂℙ^{∞})/(b_{0} − 1).

• By the Thom isomorphism, E_{∗}(MU) = E_{∗}[x_{1}, x_{2}, …].

A formal group law collects all this information together. There is an associative and commutative map ℂℙ^{∞} × ℂℙ^{∞} → ℂℙ^{∞} that classifies the tensor product of line bundles. Applying E^{∗} we get a map E^{∗}[[x]] → E^{∗}[[x_{1}, x_{2}]], which we write as x ↦ F^{E}(x_{1}, x_{2}). F^{E} is a formal group law over E_{∗}, meaning that it is commutative, associative, and unital. (For this reason, we also sometimes use the notation F^{E}(x_{1}, x_{2}) = x_{1} +_{F^{E}} x_{2}.)

Suppose x and y are different complex orientations of E. We know that y = g(x) for some invertible power series g. Using these orientations, we get two different formal group laws F_{x} and F_{y}. Using an appropriate commutative diagram, we find that F_{y}(y_{1}, y_{2}) = g(F_{x}(g^{−1}y_{1}, g^{−1}y_{2})). In other words, g gives an isomorphism of formal group laws.

The formal group is the formal spectrum SpfE^{∗}ℂℙ^{∞}.

There is a universal formal group law F^{L} over a ring L, the Lazard ring. This is universal in the sense that there is a natural bijection

{formal groups over R} → Hom(L, R).

It is straightforward to give a definition of the ring R. Write F^{L} = ∑ a_{ij}x^{i}y^{j}. F^{L} defines a formal group law if and only if it satisfies the following conditions:

1. F^{L} is unital: a_{i0} = 0 for i > 1, a_{i1} = 1

2. F^{L} is commutative: a_{ij} = a_{ji}.

3. F^{L} is associative: some complicated relation holds.

Let L be the quotient of ℤ[a_{ij}] by these three relations.

The structure of L can be determined as follows.

Theorem. (Lazard) L is a polynomial algebra ℤ[a_{1}, a_{2}, …].

It is useful to have a better handle on where these polynomial generators come from. Let I be the kernel of the natural map L → ℤ which sends all a_{ij} to zero. (This map classifies the additive formal group over ℤ.)

Them F^{L}(x, y) = x + y + ⋯ + a_{n}c_{n}(x, y) + … mod I^{2}. Here

c_{n}(x, y) = \frac{(x + y)^{n} − x^{n} − y^{n}}{d_{n}} ,

where d_{n} = p if n = p^{k} for a prime p and d_{n} = 1 if n is not a prime power. (In fact, any choice of a_{n} for which the above holds gives an expression of L as a polynomial algebra.)

The relevance of this to topology is the following:

Theorem. (Quillen) The map L → MU_{∗} classifying the formal group law of F^{MU} is an isomorphism. Question: Given a formal group law F over a ring R, is there a cohomology theory with E_{∗} = R, F = F^{E}?

As a first answer: We have a map F : L → R. Then consider the functor

MU_{∗} ⊗_{L} R.  (1)

Theorem. (Landweber) Equation 1 defines a generalized homology theory if for all primes p, the sequence (p, a_{p^{0}}, a_{p^{1}}, …) maps to a regular sequence in R.

Example. R = ℚ, with the additive formal group law F. All of the a_{i} go to 0, and the sequence (p, 0, 0, …) is a regular sequence in ℚ. (Note that this does not work for the additive formal group law on ℤ.) Using this method, we can construct rational homology ℍℚ, but not ℍℤ.

Example. F (x, y) = x+y−xy, the formal group law of complex K-theory. The map L → ℤ[u, u^{−1}] sends a_{1} to u and a_{i} to 0 for i > 1. This satisfies Landwebers theorem, and hence constructs a spectrum (which happens to coincide with complex K-theory).

Again, this approach has faults, because we cannot get some formal group laws that we know can be realized by spectra.

As a second answer: Suppose we want to make ℍℤ. Note that the formal group law gives an isomorphism ℤ ≅ L/(a_{1}, a_{2}, …).

Start with a_{1} ∈ π_{2}MU, i.e. a_{1} : S^{2} → MU. Smash this map with the identity map of MU, giving a composite

S^{2} ∧ MU →^{a_{1}} MU ∧ MU →^{μ} MU.

(μ is the multiplication map of MU.) This map has a cofiber

MU →^{a_{1}} MU → MU/(a_{1}).

This is still multiplicative if you “do this correctly.” Then take a_{2} ∈ π_{4}(MU/(a_{1})) and repeat. Taking the colimit of these quotients and denote it my MU/(a_{1}, a_{2}, …). This has ℤ as its coefficient group, with the additive formal group law.

What about if we want to do this for K-theory instead? Instead, form the quotient ku = MU/(a_{2}, a_{3}, …). This has a formal group law that looks multiplicative, but this spectrum is not the K-theory spectrum because there is no negative homotopy. In fact,π_{∗}(ku) = ℤ[a_{1}] with |a_{1}| = 2, whereas π_{∗}(K) = ℤ[u, u^{−1}].

In order to form ordinary K-theory, we form the localization a_{1}^{−1}ku, which has the correct homotopy groups and formal group law. Explicitly, we take the (homotopy) colimit of the sequence of maps

ku ⟶^{Ω^{2}a_{1}} Ω^{2}ku ⟶^{Ω^{4}a_{1}} Ω^{4}ku ⟶ ⋯  (2)

and denote it by a_{1}^{−1}ku, or K.

The colimit in equation 2 has layers equal to desuspensions of ℍℤ. In fact, from this we get a construction of K-theory as a filtered spectrum with associated graded

⋁_{m∈ℤ} Σ^{2m}ℍℤ.

From this filtration, we get the Atiyah-Hirzebruch spectral sequence for Ktheory. Now let’s move on to motivic stable homotopy.

In topology, a spectrum E is (or used to be) a collection of spaces E_{n} with structure maps s_{n} : ΣE_{n} → E_{n+1}. The spectrum homotopy groups are given by π_{n}E = lim π_{n+k}E_{k}, and similarly E^{n}X = lim[Σ^{k}X, E_{n+k}] if X is a finite cell complex.

As motivating examples, we have the Thom spectra. The standard inclusions O(n_{1}) → O(n) give maps BO(n − 1) → BO(n) → ⋯ → BO. Let MO(n) be the Thom complex of the universal n-plane bundle ν_{v} on BO(n). Then for any n, there is a pullback diagram of bundles:

\xymatrix{ν_{n−1} ⊕ 1 \ar[r] \ar[d] & ν_{n} \ar[d] \\ BO(n − 1) \ar[r] & BO(n)}

Taking Thom complexes, we get a map Σ^{2}MO(n − 1) → MO(n).

Moving now to the motivic setting, we can do the same thing for the Grassmannians in the algebraic setting. Consider the following pullback diagram of vector bundles as N grows large:

\xymatrix{ν_{n−1} ⊕ 1 \ar[r] \ar[d] & ν_{n} \ar[d] \\ Gr_{n−1}(𝔸^{N}) \ar[r] & Gr_{n}(𝔸^{N})}

Taking Thom complexes and limits in N, we get a map

\frac{MGL_{n−1} ∧ 𝔸_{+}^{1}}{MGL_{n−1} ∧(𝔸^{1} \ \{0\})_{+}} → MGL_{n},

which is 𝔸^{1}-equivalent to a map ℙ^{1} ∧ X_{n−1} → X_{n}.

In motivic stable homotopy, we work with ℙ^{1}-spectra

{X_{n}, t_{n} : ℙ^{1} ∧ X_{n} → X_{n+1}}.

We are “inverting both spheres” to form the stable category.

The above Thom complex example gives us a ℙ^{1}-spectrum which we call the motivic complex cobordism spectrum and denoted by MGL.

Now in motivic homotopy, we have two kinds of spheres, Σ = S^{1}(0) = S^{1,0} and 𝔾_{m} = S^{0}(1) = S^{1,1}, and these are both being inverted in the stable category. As a consequence, for E a ℙ^{1}-spectrum we have a bigraded cohomology theory E^{p,q}(X), defined as follows:

E^{p,q}(X) = [X, S^{p,q} ∧ E]_{Spectra}

In ℙ^{1}-spectra, we have S^{p,q} for p, q ∈ ℤ. In particular, we have bigraded coefficient groups π_{p,q}E.

As in the topological case, a complex orientation of a spectrum E is an element x ∈ Ẽ^{2,1}(ℙ^{∞}) whose restriction to ℙ^{1} is a free generator of Ẽ^{2,1}(ℙ^{1}) ≅ E_{0,0}.

Now ℙ^{∞}×⋯×ℙ^{∞} and MGL_{n} all have algebraic cell decompositions (see Dugger-Isaksen), and these cell decompositions only have cells of the form S^{2n,n} = S^{n}(n). Therefore, we get the same Atiyah-Hirzebruch type spectral sequence results: if E is complex oriented, then there are isomorphisms

E^{∗}BGL = E^{∗}[[c_{1}, c_{2}, …]]
E^{∗}ℙ^{∞} × … × ℙ^{∞} = E^{∗}[[x_{1}, …, x_{n}]].

Here BGL is the colimit lim_{n} lim_{N} Gr_{n}(𝔸^{N}), which is not necessarily the same as the simplicial bar construction.

As a consequence, associated to any complex oriented ℙ^{1}-spectrum there is a formal group law.

Quillen’s theorem still holds. We’ll leave it open for now. If L = ℤ[a_{1}, a_{2}, …], where a_{i} has bidegree (2i, i), then the map L → π_{2∗,∗}MGL is an isomorphism “in some sense”.

Landweber’s theorem is still true. Given L → R satisfying Landweber’s criterion, then MGL_{∗}(−) ⊗_{L} R is a homology theory. (This is purely formal.)

Using algebraic cell ceompositions, we find that the homology theory MGL_{∗}(−)⊗_{L} ℤ[u, u^{−1}] classifying the multiplicative formal group 𝔾̂_{m} is algebraic K-theory.

We can just use topological results to show Snaith’s result K = u^{−1}(Σ^{∞}(B𝔾_{m})_{+}).

What about the second version of the construction? We can still define ℍℤ′ = MGL/(a_{1}, a_{2}, …), k = MGL/(a_{2}, a_{3}, …). These lead to homology theories with the right kinds of properties. A topological argument implies that a_{1}^{−1}k = K, the algebraic K-theory spectrum. As a consequence, we get a filtration of K with associated graded

⋁_{n∈ℤ} Σ^{2n,n}ℍℤ′,

and a spectral sequence starting with (ℍℤ′)^{∗}(−) and converging to K_{∗}^{alg}. If I = (a_{1}, a_{2}, …), we could also consider the sequence of spectra

⋯ → MGL/I^{2} → MGL/I,

which has associated graded ℍℤ′[a_{1}, a_{2}, …]. Associated to this filtration is a spectral sequence π_{∗}(ℍℤ) ⊗ L converging to MGL_{∗}. (There are apparently no differentials in this spectral sequence in bidegree (2∗, ∗), but there must be differentials somewhere by some calculations of Kriz and Hu.) But: What is ℍℤ′? Main Theorem. (Hopkins, Morel) ℍℤ′ → ℍℤ is an equivalence. (ℍℤ is Voevodsky’s motivic homology.) How do we prove this? Connectivity theorem. Suppose X is an (n − 1)-connected ℙ^{1}-spectrum. Then so is L^{𝔸^{1}}X. (Additionally there is a result about what the first nonvanishing homotopy group is.) (Recall that X being (n − 1)-connected means π_{a,b}X = 0 for a − b ≤ (n − 1).) S^{p,q} is p − q − 1-connected. Using the algebraic cell decomposition, we find

MGL_{n} = S^{0,0} ∪_{η} e^{2,1} ∪ … ∪ e^{2n,n}.

Therefore, π_{k,k}MGL = π_{k,k}(S^{0,0}∪_{η} e^{2,1}), since all the higher cells have p−q ≥ 2. Using the cofiber sequence

S^{1,1} ⟶^{η} S^{0,0} ⟶ S^{0,0} ∪_{η} e^{2,1},

we find that there is an exact sequence

π_{k−1,k−1}S^{0,0} ⟶^{η} π_{k,k}S^{0,0} → π_{k,k}(S^{0,0} ∪_{η} e^{2,1}) → 0.

In order words, π_{k,k}MGL is Milnor K-theory of the base field. What about ℍℤ′? Consider the cofiber sequence

Σ^{2,1}MGL → MGL → MGL/(a_{1}).

Applying π_{k,k}, we find π_{k,k}MGL ≅ π_{k,k}MGL/(a_{1}). Proceeding inductively, we find that π_{k,k}(ℍℤ′) = K^{M}, Milnor K-theory, and π_{a,b}(ℍℤ′) = 0 for a − b < 0. To proceed further, we need to appeal to some things that Voevodsky proves about motivic homology.

• ℍℤ is complex oriented. (This is called “purity.”)

• π_{k,k}ℍℤ is unramified Milnor K-theory.

• The formal group law of ℍℤ is additive.

Additionally, he calculates ℍℤ ∧ ℍℤ ≃ ∨_{α}Σ^{?,?}ℍℤ/n_{α}. (It is roughly like dual Steenrod algebra.)

The complex orientation and structure of the formal group law allow us to construct a map ℍℤ′ → ℍℤ.

Also, (ℍℤ ∧ MGL)_{∗} = ℍℤ[m_{1}, m_{2}, …]. You can keep track of the effect of modding out by a_{i} on these homotopy groups, which leads to a decomposition of ℍℤ ∧ ℍℤ′. In fact, we find that the map

ℍℤ ∧ ℍℤ′ → ℍℤ ∧ ℍℤ

is an equivalence.

We now know that the map ℍℤ′ → ℍℤ satisfies the following properties:

• It is an isomorphism on π_{a,b} for a − b ≤ 0.

• ℍℤ ∧ ℍℤ′ → ℍℤ ∧ ℍℤ is an equivalence.

We want to conclude that the original map is an equivalence. Let X be the fiber of this map. Then ℍℤ ∧ X ≃ ∗. We claim that ℍℤ ∧ Y ≃ ∗ implies that [Y, ℍℤ′] = 0. (This implies the result.)

We use the Postnikov tower for ℍℤ′. Let P^{n} = P^{n}ℍℤ′. It suffices to show [Y, P^{n}] = 0.

By induction on n, it suffices to show [Y, K^{n}] = 0, where K^{n} is the fiber of the map P^{n} → P^{n−1}. ℍℤ′ is a ring spectrum, so therefore P^{0} is a ring spectrum and each K^{n} is a P^{0}-module. It is sufficient to show P^{0} ∧ Y ≃ ∗ by using the factorization below.

\xymatrix{Y \ar[r] \ar[d]_{unit} & K^{n} \\ P^{0} ∧ Y \ar[r] & P^{0} ∧ K^{n} \ar[u]_{μ}}

However, P^{0}ℍℤ′ = P^{0}ℍℤ. Therefore, if ℍℤ ∧ Y ≃ ∗, we can use the factorization

\xymatrix{P^{0} ∧ Y \ar[r] \ar[dr]_{id} & ℍℤ ∧ P^{0} ∧ Y ≃ ∗ \ar[d] \\  & P^{0} ∧ Y}

to show P^{0} ∧ Y ≃ ∗.