Notes for talks on EO_{2} resolutions

Mark Mahowald Mathematics Department Northwestern University Evanston IL 60208

December 29, 1998

Abstract These are my notes for a series of talks on EO_{2} resolutions. The new mathematics presented here is joint with Charles Rezk and has profited from conversations with Paul Goerss and Mike Hopkins

1 Introduction

I would like to discuss an understanding of the Morava stabilizer group at the prime 2 for height 2 formal groups. I will need to introduce this topic by considering some easier examples. Most topologists have some understanding of the image of J and the J spectrum. From Adams and others we know that there is a map ψ^{3} − 1 : KO → KO whose fiber is the J spectrum at 2. This is a spectrum which contains the image of the J homomorphism and some additional classes which are always a part of this homotopy.

We want to understand this case from the formal group point of view. The Morava stabilizer group is the group of automorphisms of a formal group. For the image of J case we want to look at the multiplicative formal group which is a formal group of height 1 and we want to concentrate on the prime 2. Let R be a local ring with a maximal ideal, m, such that R/m = 𝔽_{2}. The Lubin-Tate theory asserts that the lifts of the multiplicative formal group to a formal group over R is co-representable and the co-representing ring is just ℤ_{2}. The Morava stabilizer group is just the group of automorphisms of this ring. This group is the multiplicative group of ℤ_{2} which is ℤ/2⊕ℤ_{2}. Algebraic topologists insist on a grading and so the Lubin-Tate ring is ℤ_{2}[u, u^{−1}] where |u| = 2. This graded ring is the homotopy of a spectrum, K. The Morava stabilizer group acts on the homotopy. A theorem of Adams gives an action on the spectrum K, itself.

Theorem 1.1 K^{hℤ/2} = KO.

There is a spectral sequence converging to the homotopy of K^{hℤ/2} whose E_{2} term is H^{∗}(ℤ/2; K_{∗}). The non-zero element of ℤ/2 acts on u by sending it to −u. The result is the well known homotopy of KO. It is important to note that the class which corresponds to the generator ξ of H^{∗}(ℤ/2, ℤ) has Adams filtration (2,0) and so is in the minus two stem. The class u^{2}ξ = η^{2}. We should note that in this setting, the connected cover of the spectrum is not so easy to describe.

2 Formal groups of height 2

Now let’s look at the case of a formal group of height 2 over 𝔽_{4}. The Lubin-Tate theory is similar. Let R be a local ring so that R/m = 𝔽_{4}. The ring which co-represents the deformation space of lifts is now 𝕎_{𝔽_{4}}[[u_{1}]][u, u^{−1}]. Again, this is the graded answer where |u_{1}| = 0 and |u| = 2. The Morava stabilizer group is the group of automorphisms of the formal group over 𝔽_{4}. This is a much more complicated non-commutative group that the previous case.

Let’s call the group S_{2}. It is possible to give a description of the group in a more conventional fashion. We begin by considering the integral lattice in ℍ generated by ±1, ±i, ±j, ±k. We can extend the lattice by including the 16 elements (1/2)(±1 ± i ± j ± k). In the resulting ring we complete at the ideal generated by 2. The Morava stabilizer group, S_{2}, is the group of units in this ring. There is a finite group of order 24, G_{24}, the group of units before completing. There is a class S which represents the square root of 2, The torsion free part is a free module over W_{𝔽_{4}} on 1 and S.

There is a spectrum, E_{2}, whose homotopy is the co-representing ring, 𝕎_{𝔽_{4}}[[u_{1}]][u, u^{−1}]. The group S_{2} acts on this ring. A fundamental result of Hopkins and Miller is the following theorem.

Theorem 2.1 S_{2} acts on the spectrum E_{2} as a group of E_{∞} operations. E_{2}^{hG_{24}} = EO_{2}.

The homotopy of EO_{2} is quite interesting. Our story here is less specific. The question which we ask is: Is there a finite resolution of EO_{2}-modules which calculates L_{K(2)}(S^{0}). The E_{2} term of the Adams-Novikov spectral sequence is H^{∗}(S_{2}; E_{2∗}). The steps in doing this would seem to be the following.

Theorem 2.2 (Ravenel)

H^{∗}(S_{2}; ℤ/2[u^{±1}]) = (K(2)_{∗}[h_{1,0}, h_{1,1}, g]/(h_{1,0}h_{1,1}, v_{2}h_{1,0}^{3} − h_{1,1}^{3}) ⊗ Λ(β)
⊕K(2)_{∗}[ζ]⟨ξ, ξ^{2}⟩) ⊗ Λ(ρ)

The filtration of |g| = (4, 0), |ζ| = (1, 0), |ξ| = (1, 0).

This next theorem gives the connection between G_{24} and this calculation.

Theorem 2.3 H ∗(G_{24}; ℤ/2[u^{±1}]) = K(2)_{∗}[h_{1,0}, h_{1,1}, g]/(h_{1,0}h_{1,1}v_{2}h_{1,0}^{3}−h_{1,1}^{3})

This suggests that the cohomology of S_{2} is given by Λ(ρ) (chain complex of length 4).

Λ(ρ)(EO_{2∗}(V (1)) → ℤ/2[ζ, v_{2}^{±1}] → ξℤ/2[ζ, v_{2}^{±1}] → βEO_{2∗}(V (1)))

We have two questions about this. First, what spectrum has ℤ/2[ζ, v_{2}^{±1}] as its homotopy? Second, where do the maps come from?

If we consider the ψ^{3} − 1 map of the image of J spectrum case as the right source for the maps we could look at the following. Consider the class i + j + k = \sqrt{−3}. This conjugates G_{24} to another finite group of order 24, G′_{24}. The intersection of these two groups is a cyclic group of order 6. We can think of this in terms of the following diagram.

[figure: E_{2}^{hG_{24}}; E_{2}^{hG24}; E_{2}^{hG′_{24}}; E_{2}^{hℤ/6}; 1; [x]]

We would expect that this diagram does not commute and the difference should define a map 1 − [x] : E_{2}^{hG_{24}} → E_{2}^{hℤ/6}. This is reminiscent to what happened in the K-theory situation. This suggests that E_{2}^{hℤ/6} should be interesting. We note the following: The ℤ/3 leaves u^{3} = v_{2} as an invariant. The ℤ/2 acts in the usual fashion and the result is H^{∗}(ℤ/2; E_{2∗}(V (1))) = K(2)_{∗}[ζ] which looks right.

I do not know if the map above 1 − [x] is the correct map. Let us go back to the image of J picture again to get some hints.

The image of J spectrum is the fiber of the map ψ^{3}−1 : KO → KO. When restricted to connected covers, this map can be lifted to a map g : bo → Σ^{4}bsp. Here bo is the connected spectrum constructed from Bott peridocity and bsp is the connected spectrum constructed from Bott peridocity using BSP, the symplectic group. The homotopy groups are:

π_{j}(bo) = \{ \begin{matrix}ℤ & j ≡ 0(4), j ≥ 0 \\ ℤ/2 & j ≡ 1, 2(8), j > 0 \\ 0 & otherwise \end{matrix} π_{j}(bsp) = \{ \begin{matrix}ℤ & j ≡ 0(4), j ≥ 0 \\ ℤ/2 & j ≡ 5, 6(8), j > 0 \\ 0 & otherwise \end{matrix}

There is another way to construct a map like g. Consider a bo-resolution.

bo ⇉ bo ∧ bo ⋯

Theorem 2.4 bo ∧ bo = bo ∨ Σ^{4}bsp ∨ X where X is some other spectrum.

This allows one to define a map f : bo → Σ^{4}bsp.

Theorem 2.5 Any map such as f which induces an isomorphism in cohomology in dimension 4 induces the same map in homotopy as does g above.

This gives a formally alternative understanding of the J-spectrum and a version of its connective cover.

Corollary 2.6 J = L_{K(1)}S^{0} = v_{1}^{−1}(fib(bo → Σ^{4}bsp)).

This suggests that we should look at a connected version of EO_{2} which we will call eo_{2}. Hopkins and I have constructed it and Hopkins and others have produced versions with good properties. We need only a minimal version. Here is a table of conparsions between eo_{2} and bo.

bo  eo_{2}
H^{∗}  A ⊗_{A(1)} ℤ/2 A ⊗_{A(2)} ℤ/2
π_{∗}  easy  computed but complicated
periodicity  v_{1}^{4}  v_{2}^{32}
Hurwitz image ℤ/2 ∈ π_{8k+1,2} very large

3 Conjectural part

Now we come to the conjectural part of the talk. I will try to make clear what is a theorem and what is conjectural.

We can form an eo_{2}-resolution.

eo_{2} ⇉ eo_{2} ∧ eo_{2} ⋯

We do not have a splitting theorem for eo_{2} ∧ eo_{2}. We do have an algebraic version.

Theorem 3.1 As an A-module, H^{∗}(eo_{2} ∧ eo_{2}) = ⊕_{i≥0}A ⊗_{A(2)} M_{i} where M_{i} = H^{∗}(Σ^{8i}bo_{i}) and bo_{i} is the ith bo-Brown-Gitler spectrum. (H^{∗}(bo_{i}) = A/A{Sq^{1}, Sq^{2}, χSq^{4j}, j > i})

We know that eo_{2} ∧ eo_{2} does not split as ∨Σ^{8i}eo_{2} ∧ bo_{i}. This follows from some particular calculations in π_{∗}(S^{0}) in the 38 and 39 stems.

Conjecture 3.2 eo_{2} ∧ eo_{2} = eo_{2} ∨ eo_{2} ∧ (Σ^{8}bo_{1} ∪ Σ^{16}bo_{2}) ∨ Y where Y is some unspecified spectrum. There is some map connecting the second and third parts.

There is an interesting map between Σ^{15}bo_{2} → Σ^{8}bo_{1} which induces the known differential in the Adams spectral sequence. Let D be the cofiber. Then eo_{2} ∧ D has an extra homotopy class in dimension 32. If we cone this off with a single copy of eo_{2} we get a spectrum C whose Adams spectral sequence can be completly computed. If we invert v_{2} in C we get a spectrum whose Adams-Novikov spectral sequence looks very much like a candidate for E_{2}^{hℤ/6}. In particular, they have the same E_{1}-term.

If we suppose that this all will work, then we need to compute the map between eo_{2} and C. Shimomura has calculated something which we can use as describing this map at least if we concentrate on just the Moore space. If we use this picture, we can read off all of the Adams-Novikov differentials from the known eo_{2} structrue and the map defining C.

Next we should come back to the short chain complex discussed earlier. Let F be the fiber of the map v_{2}^{−1}eo_{2} → v_{2}^{−1}C. Let CF be Brown-Commentez dual of the above map. Then L_{K(2)}(V (0) should fit into an exact sequence

CF → L_{K(2)}(V (0) → F.

This is almost right. We have to take into account the two copies of this construction given by Λ(ρ).

This understanding gives a conceptual approach to understanding Shimomura’s computations and also suggests what the differentials should be in the Adams-Novikov spectral sequence.

The short complex gives a complex like:

eo_{2} → C → C → eo_{2}

If we invert v_{1} we have

KO[v_{2}^{4}/v_{1}^{12}] → (v_{2}/v_{1}^{3})KO[v_{2}/v_{1}^{3}] → (v_{2}/v_{1}^{3})KO[v_{2}/v_{1}^{3}] → KO[v_{2}^{4}/v_{1}^{12}]

It is useful to describe these maps by what happens to various powers of v_{2}. These results are a consequence of Shimomura’s calculations. In each case we assume n > 1.

\begin{aligned}2^{n}(2t + 1) & → & 2^{n−1}(4t + 1) \\ & & 2^{n−2}(4t + 3) & → & 2^{n−1}(2t + 1) \\ & & 2^{n}(2t + 1) + 1 & → & 2^{n−1}(4t + 1) + 1 \\ & & & & 2^{n−1}(4t + 3) + 1 & → & 2^{n}(2t + 1) \end{aligned}

This leaves 4 free copies. The first two represent the image of J and the second two is a kind of dual image of J. This pattern is typical of n = 2 phenomena.