Slides for talks on eo_{2}-resolutions

Mark Mahowald Mathematics Department Northwestern University

Evanston IL 60208

December 28, 1998

1 Introduction

My goals in this talk are:

• To discuss the Morava stablizer group, S_{2} at the prime 2.

• To understand the calculation of H^{∗}(S_{2}) with trivial coefficients and for E_{2}(V (0)) as coeficients. This is related to Shimomura’s calculations.

• We want to understand both of these items in terms of the Hopkins-Miller spectrum EO_{2}.

• We begin by discussing the case of S_{1} which is quite well understood.

First let’s look at S_{1} at 2. The image of J spectrum is given by the fiber of a map

ψ^{3} − 1 : KO → KO

The formal group approach

• We begin with the multiplicative formal group over 𝔽_{2} F (x, y) = x + y + xy

• The group of automorphisms is ℤ_{2}^{+} = ℤ/2 ⊕ ℤ_{2}

• The Lubin-Tate ring is ℤ_{2}. As a graded ring it is ℤ_{2}[u^{±1}]

• The Lubin-Tate ring is the homotopy of a spectrum K

• S_{1} acts on the Lubin-Tate ring and on the spectrum K

• K^{hℤ/2} = KO; ψ^{3} is a topological generator of ℤ_{2}

• J = K^{hS_{1}} = (K^{hℤ/2})^{hℤ_{2}}.

Now let’s consider the case n = 2

The formal group is not so familiar. Over 𝔽_{4} one can take the group with 2 series [2] = x^{4}.

Another choice which is interesting is the formal completion of the group structure in the elliptic curve over 𝔽_{4} defined by x^{3} = y^{2}+y In this case, the 2 series is [2] = z^{4}(Σ_{i≥0}z^{12(2^{i}−1)}.

The Lubin-Tate ring is 𝕎_{𝔽_{4}}[[u_{1}]]

The graded Lubin-Tate ring is the homotopy of a spectrum E_{2}

S_{2} is the automorphism group of the formal group over 𝔽_{4}

S_{2} is the group of units in the ring defined by the quaterions. It has a finite subgroup of order 24 given by ±1, ±i, ±j, ±k and (1/2)(±1 ± i ± j ± k)

Theorem 1.1 (Hopkins and Miller) 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 first step in calculating this is the following Theorem 1.2 (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 1.3

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.

• What spectrum has ℤ/2[ζ, v_{2}^{±1}] as its homotopy?

• Where do the maps come from?

Following the image of J case we consider the diagram where x = \sqrt{−3} and G′_{24} is another finite group obtained by conjugation with x.

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

This suggests that [x] − 1 : E_{2}^{hG_{24}} → E_{2}^{hℤ/6} should be an interesting map.

Can we understand the correct map E_{2}^{hG_{24}} → E_{2}^{hℤ/6} which occurs in the resolution from some other point of view?

The map

ψ^{3} − 1 : KO → KO

when restricted to connected covers can be lifted to a map

g : bo → Σ^{4}bsp.

Consider a bo-resolution.

bo ⇉ bo ∧ bo ⋯

Theorem 1.4 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 1.5 Any map such as f which induces an isomorphism in cohomology in dimension 4 induces the same map in homotopy as does g above.

Corollary 1.6 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}.

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 1.7 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})

Conjecture 1.8 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.

D → Σ^{15}bo_{2} → Σ^{8}bo_{1}

Σ^{32}eo_{2} → D → C

Conjecture 1.9 v_{2}^{−1}C = E_{2}^{hℤ/6} and the map in the resolution above is obtained from the eo_{2} resolution.

• Next we should come back to the short chain complex discussed earlier.

• Let F be the fiber of the map L_{2}eo_{2} → L_{2}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 not quite the end since there is the exterior generator ρ

Shimomura’s calculations of the E_{2} term for L_{K(2)}(V (0)) fit this resolution. The resolution conjectures all of the differentials in the Adams-Novikov spectral sequence.