The Rational Homotopy of the K(2)-Local Sphere

and the Chromatic Splitting Conjecture

for the Prime 3 and Level 2

Paul G. Goerss, Hans-Werner Henn, and Mark Mahowald^{1}

Received: March 7, 2014

Communicated by Stefan Schwede

Abstract. We calculate the rational homotopy of the K(2)-local sphere L_{K(2)}S^{0} at the prime 3 and confirm Hopkins’ chromatic splitting conjecture for p = 3 and n = 2.

2010 Mathematics Subject Classification: 55P42

Keywords and Phrases: Chromatic stable homotopy theory

1. Introduction

Let K(n) be the n-th Morava K-theory at a fixed prime p. The Adams-Novikov Spectral Sequence for computing the homotopy groups of the K(n)-local sphere L_{K(n)}S^{0} can be identified by [2] with a descent spectral sequence

(1)  E_{2}^{s,t} ≅ H^{s}(𝔾_{n}, (E_{n})_{t}) ⟹ π_{t−s}(L_{K(n)}S^{0}) .

1272 Paul G. Goerss, Hans-Werner Henn, Mark Mahowald

Here 𝔾_{n} denotes the automorphism group of the pair (𝔽_{p^{n}}, Γ_{n}), where Γ_{n} is the Honda formal group law; the group 𝔾_{n} is a profinite group and cohomology is continuous cohomology. The spectrum E_{n} is the 2-periodic Landweber exact ring spectrum so that the complete local ring (E_{n})_{0} classifies deformations of Γ_{n}. In this paper we focus on the case p = 3 and n = 2. In [4], we constructed a resolution of the K(2)-local sphere at the prime 3 using homotopy fixed point spectra of the form E_{2}^{hF} where F ⊆ 𝔾_{2} is a finite subgroup. These fixed point spectra are well-understood. In particular, their homotopy groups have all been calculated (see [4]) and they are closely related to the Hopkins-Miller spectrum of topological modular forms. The resolution was used in [6] to redo and refine the earlier calculation of the homotopy of the K(2)-localization of the mod-3 Moore spectrum [14]. In this paper we show how the results of [6] imply the calculation of the rational homotopy of the K(2)-local sphere. Let ℚ_{p} be the field of fractions of the p-adic integers and Λ the exterior algebra functor.

1The first author was partially supported by the National Science Foundation (USA). The second author was partially supported by ANR “HGRT”. The first two authors dedicate this work to the memory of their friend, teacher and coauthor, Mark Mahowald.

Theorem 1.1. There are classes ζ ∈ π_{−1}(L_{K(2)}S^{0}) and e ∈ π_{−3}(L_{K(2)}S^{0}) ⊗ ℚ that induce an isomorphism of algebras

Λ_{ℚ_{3}}(ζ, e) ≅ π_{∗}(L_{K(2)}S^{0}) ⊗ ℚ .

Our result is in agreement with the result predicted by Hopkins’ chromatic splitting conjecture [8], and in fact, we will establish this splitting conjecture for n = 2 and p = 3. We will prove a more general result which will be useful for calculations with the Picard group of Hopkins [17]. Before stating that, let us give some notation. If X is a spectrum, then we define

(E_{n})_{∗}X =^{def} π_{∗}L_{K(n)}(E_{n} ∧ X) .

Despite the notation, (E_{n})_{∗}(−) is not quite a homology theory, because it doesn’t take wedges to sums; however, it is a sensitive and tested algebraic invariant for the K(n)-local category. The (E_{n})_{∗}-module (E_{n})_{∗}X is equipped with the 𝔪-adic topology where 𝔪 is the maximal ideal in (E_{n})_{0}. With respect to this topology, the group 𝔾_{n} acts through continuous maps and the action is twisted because it is compatible with the action of 𝔾_{n} on the coefficient ring (E_{n})_{∗}. This topology is always topologically complete but need not be separated. See [4] §2 for some precise assumptions which guarantee that (E_{n})_{∗}X is complete and separated. All modules which will be used in this paper will in fact satisfy these assumptions. Let E(n) denote the nth Johnson-Wilson spectrum and L_{n} localization with respect to E(n). Note that E(0)_{∗} is rational homology and E(1) is the Adams summand of p-local complex K-theory. Let S_{p}^{n} denote the p-adic completion of the sphere.

Theorem 1.2. Let p = 3 and let X be any K(2)-local spectrum so that (E_{2})_{∗}X ≅ (E_{2})_{∗} ≅ (E_{2})_{∗}S^{0} as a twisted 𝔾_{2}-module. Then there is a weak equivalence of E(1)-local spectra

L_{1}X ≅ L_{1}(S_{3}^{0} ∨ S_{3}^{−1}) ∨ L_{0}(S_{3}^{−3} ∨ S_{3}^{−4}) .

We will use Theorem 1.1 to prove Theorem 1.2, but we note that Theorem 1.1 is subsumed into Theorem 1.2. Indeed, π_{∗}X ⊗ ℚ ≅ π_{∗}L_{1}X ⊗ ℚ and

π_{∗}L_{1}S_{3}^{0} ⊗ ℚ ≅ π_{∗}L_{0}S_{3}^{0} ≅ ℚ_{3}

all concentrated in degree zero. The generality of the statement of Theorem 1.2 is not vacuous; there are such X which are not weakly equivalent to L_{K(2)}S^{0} – “exotic” elements in the K(2)-local Picard group. See [5] and [10]. We remark that Theorem 1.1 disagrees with the calculation by Shimomura and Wang in [15]. In particular, Shimomura and Wang find the exterior algebra on ζ only.

The Rational Homotopy of the K(2)-Local Sphere … 1273

An interesting feature of our proof of Theorem 1.1 is that it does not require a preliminary calculation of all of π_{∗}(L_{K(2)}S^{0}). In fact, we get away with much less, namely with only a (partial) understanding of the E_{2}-term of the Adams-Novikov Spectral Sequence converging to π_{∗}L_{K(2)}(S/3) where S/3 denotes the mod-3 Moore spectrum (see Corollary 3.4). Our method of proof can also be used to recover the rational homotopy of L_{K(2)}S^{0} as well as the chromatic splitting conjecture at primes p > 3 [16]; we only need to use the analog of Corollary 3.4 for the E_{2}-term of the Adams-Novikov spectral sequence of the K(2)-localization of the mod-p Moore spectrum for p > 3. In section 2 we give some general background on the automorphism group 𝔾_{2} and we review the main results of [4]. In section 3 we recall those results of [6] which are relevant for the purpose of this paper. Section 4 gives the calculation of the rational homotopy groups of L_{K(2)}S^{0} and in the final section 5 we prove Theorem 1.2 and the chromatic splitting conjecture for n = 2 and p = 3. See Corollary 5.11.

2. Background

Let Γ_{2} be the Honda formal group law of height 2; this is the unique 3-typical formal group law over 𝔽_{9} with 3-series [3](x) = x^{9}. We begin with a short analysis of the Morava stabilizer group 𝔾_{2}, the group of automorphisms of the pair (𝔽_{9}, Γ_{2}). Let 𝕎 = 𝕎(𝔽_{9}) denote the Witt vectors of 𝔽_{9} and let (−)^{σ} : 𝕎 → 𝕎 be the lift of the Frobenius. Define

O_{2} = 𝕎⟨S⟩/(S^{2} = 3, wS = Sw^{σ}) .

Then O_{2} is isomorphic to the ring of endomorphisms of Γ_{2} over 𝔽_{9}; hence O_{2}^{×} is isomorphic to the group 𝕊_{2} of automorphisms of Γ_{2} over 𝔽_{9}. This is a subgroup of the group 𝔾_{2} of automorphisms of the pair (𝔽_{9}, Γ_{2}), which is the group of pairs (f, ϕ) with ϕ : 𝔽_{9} → 𝔽_{9} a field isomorphism and f : ϕ_{∗}Γ_{2} → Γ_{2} an isomorphism of formal group laws over 𝔽_{9}. Since Γ_{2} is defined over 𝔽_{3}, there is a splitting

𝔾_{2} ≅ 𝕊_{2} ⋊ Gal(𝔽_{9}/𝔽_{3})

with Galois action given by ϕ(x + yS) = x^{σ} + y^{σ}S. The 3-adic analytic group 𝕊_{2} ⊆ 𝔾_{2} contains elements of order 3; indeed, an explicit such element is given by

a = −\frac{1}{2}(1 + ωS)

where ω is a fixed primitive 8-th root of unity in 𝕎. If C_{3} is the cyclic group of order 3, the map H^{∗}(𝕊_{2}, 𝔽_{3}) → H^{∗}(C_{3}, 𝔽_{3}) defined by a is surjective and, hence, 𝕊_{2} and 𝔾_{2} cannot have finite cohomological dimension. As a consequence, the trivial module ℤ_{3} cannot admit a projective resolution of finite length. Nonetheless, 𝔾_{2} has virtual finite cohomological dimension, and admits a finite length resolution by permutation modules obtained from finite subgroups. Such a resolution was constructed in [4] using the following two finite subgroups of 𝔾_{2}. The notation ⟨−⟩ indicates the subgroup generated by the listed elements.

(1) Let G_{24} = ⟨a, ω^{2}, ωϕ⟩ ≅ C_{3} ⋊ Q_{8}. Here Q_{8} is the quaternion group of order 8. Note ω^{2} acts non-trivially and ωϕ acts trivially on C_{3}.

(2) SD_{16} = ⟨ω, ϕ⟩. This subgroup is isomorphic to the semidihedral group of order 16.

Remark 2.1. The group 𝔾_{2} splits as a product 𝔾_{2} ≅ 𝔾_{2}^{1} × ℤ_{3}. To be specific, the center of 𝔾_{2} is isomorphic to ℤ_{3}^{×} and there is an isomorphism from the additive group ℤ_{3} onto the multiplicative subgroup 1 + 3ℤ_{3} ⊆ ℤ_{3}^{×} sending 1 to 4. There is also a reduced determinant map 𝔾_{2} → ℤ_{3}. (See [4].) The composition ℤ_{3} → 𝔾_{2} → ℤ_{3} is multiplication by 2, giving the splitting. All finite subgroups of 𝔾_{2} are automatically finite subgroups of 𝔾_{2}^{1}.

Because of this splitting, any resolution of the trivial 𝔾_{2}^{1}-module ℤ_{3} can be promoted to a resolution of the trivial 𝔾_{2}-module. See Remark 2.4 below. Thus we begin with 𝔾_{2}^{1}. If X = limX_{α} is a profinite set, define ℤ_{3}[[X]] = lim ℤ/3^{n}[X_{α}]. The following is the main algebraic result of [4].

Theorem 2.2. There is an exact complex of ℤ_{3}[[𝔾_{2}^{1}]]-modules of the form

0 → C_{3} → C_{2} → C_{1} → C_{0} → ℤ_{3}

with

C_{0} = C_{3} ≅ ℤ_{3}[[𝔾_{2}^{1}/G_{24}]]

and

C_{1} = C_{2} ≅ ℤ_{3}[[𝔾_{2}^{1}]] ⊗_{ℤ_{3}[SD_{16}]} ℤ_{3}(χ)

where ℤ_{3}(χ) is the SD_{16} module which is free of rank 1 over ℤ_{3} and with ω and ϕ both acting by multiplication by −1.

We recall that a continuous ℤ_{3}[[𝔾_{2}]]-module M is profinite if there is an isomorphism M ≅ lim_{α}M_{α} where each M_{α} is a finite ℤ_{3}[[𝔾_{2}]] module.

Corollary 2.3. Let M be a profinite ℤ_{3}[[𝔾_{2}^{1}]]-module. Then there is a first quadrant cohomology spectral sequence

E_{1}^{p,q}(M) ≅ Ext_{ℤ_{3}[[𝔾_{2}^{1}]]}^{q}(C_{p}, M) ⟹ H^{p+q}(𝔾_{2}^{1}, M)

with

E_{1}^{0,q}(M) = E_{1}^{3,q}(M) ≅ H^{q}(G_{24}, M)

and

E_{1}^{1,q}(M) = E_{1}^{2,q}(M) ≅ \begin{cases}Hom_{ℤ_{3}[SD_{16}]}(ℤ_{3}(χ), M) & q = 0 \\ 0 & q > 0 . \end{cases}

Remark 2.4. These ideas and techniques can easily be extended to the full group 𝔾_{2} using the splitting 𝔾_{2} ≅ 𝔾_{2}^{1} × ℤ_{3}. Let ψ ∈ ℤ_{3} be a topological generator; then there is a resolution

\xymatrix{0 \ar[r] & ℤ_{3}[[ℤ_{3}]] \ar[r]^{ψ−1} & ℤ[[ℤ_{3}]] \ar[r] & ℤ_{3} \ar[r] & 0 .}

Write C_{•} → ℤ_{3} for the resolution of Theorem 2.2. Then the total complex of the double complex

C_{•} ˆ⊗\{ \xymatrix{ℤ_{3}[[ℤ_{3}]] \ar[r]^{ψ−1} & ℤ[[ℤ_{3}]]} \}

defines an exact complex D_{•} → ℤ_{3} of ℤ_{3}[[𝔾_{2}]]-modules. The symbol ˆ⊗ indicates the completion of the tensor product. From this we get a spectral sequence analogous to that of Corollary 2.3.

Remark 2.5. In our arguments below, we will use the functors on profinite ℤ_{3}[[𝔾_{2}^{1}]]-modules to profinite abelian groups given by

M ↦ E_{2}^{p,0}(M) = H^{p}(Hom_{ℤ_{3}[[𝔾_{2}^{1}]]}(C_{•}, M)) .

Here C_{•} is the resolution of Theorem 2.2; thus, we are using the q = 0 line of the E_{2}-page of the spectral sequence of Corollary 2.3. We would like some information on the exactness of these functors; for this we need a hypothesis. If M is a profinite ℤ_{3}[[𝔾_{2}]]-module then

Hom_{ℤ_{3}[[𝔾_{2}^{1}]]}(C_{•}, M) = lim_{α}Hom_{ℤ_{3}[[𝔾_{2}^{1}]]}(C_{•}, M_{α})

is also necessarily profinite as a ℤ_{3}-module. Since profinite ℤ_{3}-modules are closed under kernels and cokernels, the groups E_{2}^{p,0}(M) are also profinite. We will use later that if M is a finitely generated profinite ℤ_{3}-module and M/3M = 0, then M = 0.

Lemma 2.6. Suppose 0 → M_{1} → M_{2} → M_{3} → 0 is an exact sequence of profinite ℤ_{3}[[𝔾_{2}^{1}]]-modules such that H^{1}(G_{24}, M_{1}) = 0. Then there is a long exact sequence of profinite ℤ_{3}-modules

0 → E_{2}^{0,0}(M_{1}) → E_{2}^{0,0}(M_{2}) → E_{2}^{0,0}(M_{3}) → E_{2}^{1,0}(M_{1}) → …

⋯ → E_{2}^{3,0}(M_{2}) → E_{2}^{3,0}(M_{3}) → 0 .

Proof. In general the sequence of complexes

0 → Hom_{ℤ_{3}[[𝔾_{2}^{1}]]}(C_{•}, M_{1}) → Hom_{ℤ_{3}[[𝔾_{2}^{1}]]}(C_{•}, M_{2}) → Hom_{ℤ_{3}[[𝔾_{2}^{1}]]}(C_{•}, M_{3}) → 0

of profinite ℤ_{3}-modules need not be exact; however, by Corollary 2.3, the failure of exactness is exactly measured by H^{1}(G_{24}, M_{1}). Therefore, if that group is zero, then we do get an exact sequence of complexes, and the result follows. □

Remark 2.7. By [4], the resolution C_{•} → ℤ_{3} of Theorem 2.2 can be promoted to a resolution of (E_{2})_{∗}E_{2}^{h𝔾_{2}^{1}} by twisted 𝔾_{2}-modules

(2) \begin{aligned}(E_{2})_{∗}E_{2}^{h𝔾_{2}^{1}} → (E_{2})_{∗}E_{2}^{hG_{24}} → & (E_{2})_{∗}Σ^{8}E_{2}^{hSD_{16}} \\ & → (E_{2})_{∗}Σ^{8}E_{2}^{hSD_{16}} → (E_{2})_{∗}E_{2}^{hG_{24}} → 0. \end{aligned}

We have Σ^{8}E_{2}^{hSD_{16}} because C_{1} is twisted by a character. From §5 of [4] we get the following topological refinement: there is a sequence of maps between spectra

(3)  E_{2}^{h𝔾_{2}^{1}} → E_{2}^{hG_{24}} → Σ^{8}E_{2}^{hSD_{16}} → Σ^{40}E_{2}^{hSD_{16}} → Σ^{48}E_{2}^{hG_{24}}

realizing the resolution (2) and with the property that any two successive maps are null-homotopic and all possible Toda brackets are zero modulo indeterminacy. Note that there is an equivalence Σ^{8}E_{2}^{hSD_{16}} ≃ Σ^{40}E_{2}^{hSD_{16}}, so that suspension is for symmetry only; however,

Σ^{48}E_{2}^{hG_{24}} ≄ E_{2}^{hG_{24}}

even though

(E_{2})_{∗}Σ^{48}E_{2}^{hG_{24}} ≅ (E_{2})_{∗}E_{2}^{hG_{24}}.

This suspension is needed to make the Toda brackets vanish. Because these Toda brackets vanish, the sequence of maps in the topological complex (3) further refines to a tower of fibrations

(4)  \xymatrix{E_{2}^{h𝔾_{2}^{1}} \ar[r] & X_{2} \ar[r] & X_{1} \ar[r] & E^{hG_{24}} \\ Σ^{45}E_{2}^{hG_{24}} \ar[u] & Σ^{38}E_{2}^{hSD_{16}} \ar[u] & Σ^{7}E_{2}^{hSD_{16}} \ar[u]}

There is a similar tower for the sphere itself, using the resolution of Remark 2.4.

Remark 2.8. Let Σ^{−p}F_{p} denote the successive fibers in the tower (4); thus, for example, F_{3} = Σ^{48}E_{2}^{hG_{24}}. Then combining the descent spectral sequences for the groups G_{24}, SD_{16} and 𝔾_{2}^{1} with Corollary 2.3 and the spectral sequence of the tower, we get a square of spectral sequences

(5)  [figure: E_{1}^{p,q}((E_{2})_{t}X); 1; 2; H^{p+q}(𝔾_{2}^{1}, (E_{2})_{t}X); 3; π_{t−q}L_{K(2)}(F_{p} ∧ X); 4; π_{t−(p+q)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) .]

We will use information about spectral sequences (1) and (2) to deduce information about spectral sequences (3) and (4). See Lemmas 4.4 and 5.3. There is a similar square of spectral sequences where the lower right corner becomes π_{∗}L_{K(2)}S^{0}. This uses the resolution of Remark 2.4 and the subsequent tower for the sphere.

3. The algebraic spectral sequences in the case of (E_{2})_{∗}/3

Let S/3 denote the mod-3 Moore spectrum. Then, in the case of (E_{2})_{∗}/3 = (E_{2})_{∗}(S/3) the spectral sequence of Corollary 2.3 was completely worked out in [6]. We begin with some of the details. First note that this is a spectral sequence of modules over H^{∗}(𝔾_{2}; (E_{2})_{∗}/3). We will describe the E_{1}-term as a module over the subalgebra

𝔽_{3}[β, v_{1}] ⊆ H^{∗}(𝔾_{2}; (E_{2})_{∗}/3)

where β ∈ H^{2}(𝔾_{2}, (E_{2})_{12}/3) detects the image of the homotopy element β_{1} ∈ π_{10}S^{0} in π_{10}(L_{K(2)}(S/3)) and v_{1} := u_{1}u^{−2} detects the image of the homotopy element in π_{4}(S/3)

\xymatrix{S^{4} \ar[r] & Σ^{4}(S/3) \ar[r]^{A} & S/3}

of the inclusion of the bottom cell composed with the v_{1}-periodic map constructed by Adams. In the next result, the element α of bidegree (1, 4) detects the image of the homotopy element α_{1} ∈ π_{3}S^{0} and the element α̃ of bidegree (1, 12) detects an element in π_{11}(L_{K(2)}(S/3)) which maps to the image of β_{1} in π_{10}(L_{K(2)}S^{0}) under the pinch map S/3 → S^{1} to the top cell. For more details on these elements, as well as for the proof of the following theorem we refer to [6]. We write

E_{r}^{p,q,t} = E_{r}^{p,q}((E_{2})_{t}/3)

for the E_{r}-term of the spectral sequence of Corollary 2.3. For example, if p = 0 or p = 3, then

E_{1}^{p,∗,t} = H^{∗}(G_{24}, (E_{2})_{t}/3) .

By the calculations of [4] §3, there is an invertible class Δ ∈ H^{0}(G_{24}, (E_{2})_{24}). We also write Δ for its image in H^{0}(G_{24}, (E_{2})_{24}/3).

Theorem 3.1. There are isomorphisms of 𝔽_{3}[β, v_{1}]-modules, with β acting trivially on E_{1}^{p,∗,∗} if p = 1, 2:

E_{1}^{p,∗,∗} ≅ \begin{cases}𝔽_{3}[[v_{1}^{6}Δ^{−1}]][Δ^{±1}, v_{1}, β, α, α̃]/(α^{2}, α^{2}, v_{1}α, v_{1}α̃, αα̃ + v_{1}β)e_{p} & p = 0, 3 \\ ω^{2}u^{4}𝔽_{3}[[u_{1}^{4}]][v_{1}, u^{±8}]e_{p} & p = 1, 2 . \end{cases}

Remark 3.2. The module generators e_{p} are of tridegree (p, 0, 0). If p = 0 or p = 3, then E_{1}^{p,0,∗} is isomorphic to a completion of the ring of mod-3 modular forms for smooth elliptic curves. Indeed, by Deligne’s calculations [1] §6, the ring of modular forms is 𝔽_{3}[b_{2}, Δ^{±1}] where b_{2} is the Hasse invariant and Δ is the discriminant. The Hasse invariant of an elliptic curve can be computed as v_{1} of the associated formal group, so we can write b_{2} = v_{1}. If p = 1 or p = 2, we have written E_{1}^{p,0,∗} as a submodule of (E_{2})_{∗}/3 = 𝔽_{9}[[u_{1}]][u^{±1}]. Recall that there is a 3-typical choice for the universal deformation of the Honda formal group Γ_{2} with v_{1} = u_{1}u^{±2} and v_{2} = u^{−8}.

All differentials in the spectral sequence of Corollary 2.3 with M = (E_{2})_{∗}/3 are v_{1}-linear. This follows from the fact that v_{1} is an element in the homotopy groups of the spectrum S/3. In particular, d_{1} is determined by continuity and the following formulae established in [6].

Theorem 3.3. There are elements

Δ_{k} ∈ E_{1}^{0,0,24k}, b_{2k+1} ∈ E_{1}^{1,0,16k+8}, \overline{b}_{2k+1} ∈ E_{1}^{2,0,16k+8}, \overline{Δ}_{k} ∈ E_{1}^{3,0,24k}

for each k ∈ ℤ satisfying

Δ_{k} ≡ Δ^{k}e_{0}, b_{2k+1} ≡ ω^{2}u^{−4(2k+1)}e_{1}, \overline{b}_{2k+1} ≡ ω^{2}u^{−4(2k+1)}e_{2}, \overline{Δ}_{k} ≡ Δ^{k}e_{3}

(where the congruences are modulo the ideal (v_{1}^{6}Δ^{−1}) resp. (v_{1}^{4}u^{8}) and in the case of Δ_{0} we even have equality Δ_{0} = Δ^{0}e_{0} = e_{0}) such that

d_{1}(Δ_{k}) = \begin{cases}(−1)^{m+1}b_{2.(3m+1)+1} & k = 2m + 1, m ∈ ℤ \\ (−1)^{m+1}mv_{1}^{4.3^{n}−2}b_{2.3^{n}(3m−1)+1} & k = 2m.3^{n}, m ∈ ℤ, \\ & m ≢ 0 mod (3), n ≥ 0 \\ 0 & k = 0 \end{cases}

d_{1}(b_{2k+1}) = \begin{cases}(−1)^{n}v_{1}^{6.3^{n}+2}\overline{b}_{3^{n+1}(6m+1)} & k = 3^{n+1}(3m + 1), \\ & m ∈ ℤ, n ≥ 0 \\ (−1)^{n}v_{1}^{10.3^{n}+2}\overline{b}_{3^{n}(18m+11)} & k = 3^{n}(9m + 8), \\ & m ∈ ℤ, n ≥ 0 \\ 0 & else \end{cases}

d_{1}(\overline{b}_{2k+1}) = \begin{cases}(−1)^{m+1}v_{1}^{2}\overline{Δ}_{2m} & 2k + 1 = 6m + 1, m ∈ ℤ \\ (−1)^{m+n}v_{1}^{4.3^{n}} \overline{Δ}_{3^{n}(6m+5)} & 2k + 1 = 3^{n}(18m + 17), \\ & m ∈ ℤ, n ≥ 0 \\ (−1)^{m+n+1}v_{1}^{4.3^{n}} \overline{Δ}_{3^{n}(6m+1)} & 2k + 1 = 3^{n}(18m + 5), \\ & m ∈ ℤ, n ≥ 0 \\ 0 & else . \end{cases}

We will actually only need the following consequence of these results, which follows after a little bookkeeping.

Corollary 3.4. There is an isomorphism

E_{2}^{p,0}((E_{2})_{0}/3) ≅ \begin{cases}𝔽_{3} & p = 0, 3 \\ 0 & p = 1, 2 . \end{cases}

Remark 3.5. We also record here the integral calculation H^{∗}(G_{24}, (E_{2})_{∗}) from [4]; we will use this in Proposition 5.5. There are elements c_{4}, c_{6} and Δ in H^{0}(G_{24}, (E_{2})_{∗}) of internal degrees 8, 12 and 24 respectively. The element Δ is invertible and there is a relation

c_{4}^{3} − c_{6}^{2} = (12)^{3}Δ .

Define j = c_{4}^{3}/Δ and let M_{∗} be the graded ring

M_{∗} = ℤ_{3}[[j]][c_{4}, c_{6}, Δ^{±1}]/(c_{4}^{3} − c_{6}^{2} = (12)^{3}Δ, Δj = c_{4}^{3}).

There are also elements α ∈ H^{1}(G_{24}, (E_{2})_{4}) and β ∈ H^{2}(G_{24}, (E_{2})_{12}) which reduce to the restriction (from 𝔾_{2} to G_{24}) of the elements of the same name in Theorem 3.1. There are relations

3α = 3β = α^{2} = 0
(6)  c_{4}α = c_{4}β = 0
c_{6}α = c_{6}β = 0.

Finally

H^{∗}(G_{24}, (E_{2})_{∗}) = M_{∗}[α, β]/R

where R is the ideal of relations given by (6). The element Δ has already appeared in Theorem 3.1. Modulo 3, c_{4} ≡ v_{1}^{2} and c_{6} ≡ v_{1}^{3} up to a unit in H^{0}(G_{24}, (E_{2})_{0}/3) = 𝔽_{3}[[j]]. Compare [3], Proposition 7.

4. The rational calculation

The purpose of this section is to give enough qualitative information about the integral calculation of H^{∗}(𝔾_{2}, (E_{2})_{∗}) in order to prove Theorem 1.1. Much of this is more refined than we actually need, but of interest in its own right. The following result implies that the rational homotopy will all arise from H^{∗}(𝔾_{2}, (E_{2})_{0}).

Proposition 4.1. a) Suppose t = 4.3^{k}m with m ≢ 0 mod (3). Then 3^{k+1}H^{∗}(𝔾_{2}, (E_{2})_{t}) = 0.

b) Suppose t is not divisible by 4. Then H^{∗}(𝔾_{2}, (E_{2})_{t}) = 0.

Proof. Part (b) is the usual sparseness for the Adams-Novikov Spectral Sequence. We can prove this here by considering the spectral sequence

H^{p}(𝔾_{2}/{±1}, H^{q}({±1}, (E_{2})_{t}) ⟹ H^{p+q}(𝔾_{2}, (E_{2})_{t})

given by the inclusion of the central subgroup {±1} ⊂ ℤ_{3}^{×} ⊂ 𝔾_{2}. The central subgroup ℤ_{3}^{×} acts trivially on (E_{2})_{0} and by multiplication on u; that is, if g ∈ ℤ_{3}^{×} then g_{∗}(u) = gu. In particular we find

H^{q}({±1}, (E_{2})_{t}) = 0

unless t is a non-zero multiple of 4 and q = 0. From this (b) follows. For (a) we use the spectral sequence

H^{p}(𝔾_{2}^{1}, H^{q}(ℤ_{3}, (E_{2})_{t})) ⟹ H^{p+q}(𝔾_{2}, (E_{2})_{t})

If ψ ∈ ℤ_{3} is a topological generator, then ψ ≡ 4 modulo 9. In particular,

ψ(u^{t/2}) = (1 + 2.3^{k+1}m)u^{t/2} mod (3^{k+2})

and we have that H^{q}(ℤ_{3}, (E_{2})_{t}) = 0 unless q = 1 and

3^{k+1}H^{1}(ℤ_{3}^{×}, (E_{2})_{t}) = 0 .

Then (a) follows. □

It’s not possible to be quite so precise in the case of 𝔾_{2}^{1}. However, we do have the following result.

Proposition 4.2. Suppose s > 3 or t is not divisible by 4. Then

H^{s}(𝔾_{2}^{1}, (E_{2})_{t}) ⊗ ℚ = 0 .

Proof. This follows from tensoring the spectral sequence of Corollary 2.3 with ℚ and noting that

H^{s}(G_{24}, (E_{2})_{t}) ⊗ ℚ = H^{s}(SD_{16}, (E_{2})_{t}) ⊗ ℚ = 0

if s > 0 or t is not divisible by 4. □

To isolate the torsion-free part of the cohomology of either 𝔾_{2} or 𝔾_{2}^{1} we use the spectral sequences of Corollary 2.3. From Remark 3.5 we have an inclusion which is an isomorphism in positive cohomological degrees

ℤ_{3}[β^{2}Δ^{−1}]/(3β^{2}Δ^{−1}) ⊆ H^{∗}(G_{24}, (E_{2})_{0}).

In the notation of the spectral sequences of Corollary 2.3 and Remark 2.4 we then have inclusions

ℤ_{3}[β^{2}Δ^{−1}]/(3β^{2}Δ^{−1})e_{p} ⊆ E_{1}^{p,∗}(𝔾_{2}^{1}, (E_{2})_{0}), p = 0, 3.

Here is the main algebraic result.

Theorem 4.3. a) There is an element e ∈ H^{3}(𝔾_{2}^{1}, (E_{2})_{0}) of infinite order so that

H^{∗}(𝔾_{2}^{1}, (E_{2})_{0}) ≅ Λ(e) ⊗ ℤ_{3}[β^{2}Δ^{−1}]/(3β^{2}Δ^{−1}) .

b) There is an element ζ ∈ H^{1}(𝔾_{2}, (E_{2})_{0}) of infinite order so that

H^{∗}(𝔾_{2}, (E_{2})_{0}) ≅ Λ(ζ) ⊗ H^{∗}(𝔾_{2}^{1}, (E_{2})_{0}) .

Proof. For the proof of part (a) we consider the functors from the category of profinite ℤ_{3}[[𝔾_{2}^{1}]]-modules to 3-profinite abelian groups introduced in Remark 2.5 and given by

M ↦ E_{2}^{p,0}(M) = H^{p}(Hom_{ℤ_{3}[[𝔾_{2}^{1}]]}(C_{•}, M)) .

Here C_{•} is the resolution of Theorem 2.2. From Remark 3.5 we know that the hypothesis of Lemma 2.6 is satisfied for the short exact sequence

0 → (E_{2})_{0} ⟶^{×3} (E_{2})_{0} → (E_{2})_{0}/3 → 0 .

Then Corollary 3.4, the long exact sequence of Lemma 2.6, and the fact that the groups E_{2}^{p,0}(𝔾_{2}^{1}, (E_{2})_{0}) are profinite ℤ_{3}- modules give

E_{2}^{p,0}(𝔾_{2}^{1}, (E_{2})_{0}) ≅ \begin{cases}ℤ_{3}, & p = 0, 3; \\ 0, & p = 1, 2 . \end{cases}

See Remark 2.5. This implies that the E_{2}-term of the spectral sequence of Corollary 2.3 is isomorphic to

Λ(e_{3}) ⊗ ℤ_{3}[β^{2}Δ^{−1}]/(3β^{2}Δ^{−1}) .

Since there can be no further differentials, part (a) follows. Since the central ℤ_{3} acts trivially on (E_{2})_{0}, we have a Künneth isomorphism

H^{∗}(ℤ_{3}, ℤ_{3}) ⊗ H^{∗}(𝔾_{2}^{1}, (E_{2})_{0}) ≅ H^{∗}(𝔾_{2}, (E_{2})_{0}) .

Part (b) follows. □

We are now ready to state and prove the main result on rational homotopy. Note that Theorem 1.1 of the introduction is an immediate consequence of Proposition 4.1, of Theorem 4.3, of the spectral sequence

H^{s}(𝔾_{2}, (E_{2})_{t}) ⊗ ℚ ⟹ π_{t−s}L_{K(2)}S^{0} ⊗ ℚ . and part (b) of the following Lemma. Let κ_{2} be the set of isomorphism classes of K(2)-local spectra X so that (E_{2})_{∗}X ≅ (E_{2})_{∗} = (E_{2})_{∗}S^{0} as twisted 𝔾_{2}-modules. This is a subgroup of the K(2)-local Picard group; the group operation is given by smash product. In [5] we show that κ_{2} ≅ (ℤ/3)^{2}. For the next result, the spectra F_{p} were defined in Remark 2.8.

Lemma 4.4. (a) Let X ∈ κ_{2}. Then for p = 0, 1, 2, 3, the edge homomorphism of the localized descent spectral sequence

E_{2}^{p,q,t} = Ext_{ℤ_{3}[[𝔾_{2}^{1}]]}^{q}(C_{p}, (E_{2})_{t}X) ⊗ ℚ ⟹ π_{t−q}L_{K(2)}(F_{p} ∧ X) ⊗ ℚ

induces an isomorphism

π_{∗}L_{K(2)}(F_{p} ∧ X) ⊗ ℚ ≅ Hom_{ℤ_{3}[[𝔾_{2}^{1}]]}(C_{p}, (E_{2})_{∗}X) ⊗ ℚ .

(b) Let F = 𝔾_{2}^{1} or 𝔾_{2}. Then the localized spectral sequence

H^{s}(F, (E_{2})_{t}X) ⊗ ℚ ⟹ π_{t−s}L_{K(2)}(E_{2}^{hF} ∧ X) ⊗ ℚ

converges and collapses.

Proof. For (a), the spectral sequence

H^{s}(F, (E_{2})_{t}X) ⟹ π_{t−s}L_{K(2)}(E_{2}^{hF} ∧ X)

has a horizontal vanishing line at E_{∞} by the calculations of §3 of [4]. Thus the rationalized spectral sequence

H^{s}(F, (E_{2})_{t}X) ⊗ ℚ ⟹ π_{t−s}L_{K(2)}(E_{2}^{hF} ∧ X) ⊗ ℚ

converges. The result follows in this case. For (b) we first do the case of 𝔾_{2}^{1}. We localize the square of spectral sequences of (5) to get a new square of spectral sequences

(7)  [figure: E_{1}^{p,q}((E_{2})_{t}X) ⊗ ℚ; 1; 2; H^{p+q}(𝔾_{2}^{1}, (E_{2})_{t}X) ⊗ ℚ; 3; π_{t−q}L_{K(2)}(F_{p} ∧ X) ⊗ ℚ; 4; π_{t−(p+q)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) ⊗ ℚ .]

We will show that spectral sequence (3) converges and the result will follow. First note that spectral sequences (2) and (4) are the localizations of finite and convergent spectral sequences, so must converge. We have noted in the proof of part (a) that the spectral sequences of (1) converge. Now we note that spectral sequence (2) with q = 0 and the spectral sequence of (4) have the same d_{1}, by the construction of the tower. From this we conclude that the E_{2}-term of the spectral sequence (4) is

E_{2}^{p,t} ≅ H^{p}(𝔾_{2}^{1}, (E_{2})_{t}X) ⊗ ℚ .

Proposition 4.2 implies that the spectral sequence (4) collapses and that, in fact, if

π_{n}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) ⊗ ℚ ≠ 0

there are unique integers p and t with t − p = n and

π_{n}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) ⊗ ℚ ≅ H^{p}(𝔾_{2}^{1}, (E_{2})_{t}X) ⊗ ℚ.

It follows immediately that spectral sequence (3) converges and collapses. There is an analogous argument for 𝔾_{2}, using the expanded square of spectral sequences for this group. See Remark 2.8. The needed properties of H^{p}(𝔾_{2}, (E_{2})_{t}X) ⊗ ℚ are obtained by combining Proposition 4.1 with Theorem 4.3.b. □

Theorem 4.3 and Lemma 4.4 immediately imply the following results. Let S_{p}^{n} denote the p-complete sphere.

Theorem 4.5. Let X ∈ κ_{2}. Then the rational Hurewicz homomorphism

π_{0}L_{0}X⟶ π_{0}L_{0}L_{K(2)}(E_{2} ∧ X) ≅ (E_{2})_{0}X ⊗ ℚ

is injective. Given a choice of isomorphism f : (E_{2})_{∗} → (E_{2})_{∗}X of twisted 𝔾_{2}-modules the image of the multiplicative unit 1 under the isomorphism

ℚ_{3} ≅ ℚ ⊗ H^{0}(𝔾_{2}, (E_{2})_{0}) ≅ π_{0}L_{0}X

extends to a weak equivalence of L_{0}L_{K(2)}S^{0}-modules

L_{0}L_{K(2)}S^{0} ≃ L_{0}X .

Theorem 4.6. The localized spectral sequence of Lemma 4.4

ℚ ⊗ H^{s}(𝔾_{2}, (E_{2})_{t}) ⟹ π_{t−s}L_{0}L_{K(2)}S^{0}

determines an isomorphism

Λ_{ℚ_{3}}(ζ, e) ≅ π_{∗}L_{0}L_{K(2)}S^{0}.

Furthermore, there is a weak equivalence

L_{0}(S_{3}^{0} ∨ S_{3}^{−1} ∨ S_{3}^{−3} ∨ S_{3}^{−4}) ≃ L_{0}L_{K(2)}S^{0}.

5. The chromatic splitting conjecture

In this section we prove a refinement of Theorem 1.2 of the introduction. Our main result, Theorem 5.10, analyzes L_{1}X for X ∈ κ_{2}. For this we will use the chromatic fracture square

(8)  \xymatrix{L_{1}X \ar[r] \ar[d] & L_{K(1)}X \ar[d] \\ L_{0}X \ar[r] & L_{0}L_{K(1)}X .}

We made an analysis of L_{0}X in Theorem 4.5. The calculation of L_{K(1)}X has a number of interesting features, so we dwell on it. In particular, we will produce weak equivalences

L_{K(1)}S^{0} → L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X)

which will be the key to the entire calculation. We begin with the following general result. Let S/p^{n} denote the Moore spectrum.

Lemma 5.1. Let X be a spectrum. Then

L_{K(1)}X = holim_{n} v_{1}^{−1}S/p^{n} ∧ X

where v_{1}^{t} : Σ^{2t(p−1)}S/p^{n} → S/p^{n} is any choice of v_{1}-self map.

Proof. By Proposition 7.10(e) of [9] we know that

(9)  L_{K(1)}X = holim_{n} S/p^{n} ∧ L_{1}X .

Since L_{1} is smashing, we may rewrite (9) as

L_{K(1)}X = holim_{n} L_{1}(S/p^{n}) ∧ X .

Thus it is sufficient to know L_{1}(S/p^{n}) = v_{1}^{−1}S/p^{n}. This follows from the Telescope Conjecture for n = 1; see [11] and [12]. □

If R is a discrete ring, then the Laurent series over R is the ring R((x)) = R[[x]][x^{−1}].

Proposition 5.2. (a) There are isomorphisms

(10)  𝔽_{3}((v_{1}^{6}Δ^{−1}))[v_{1}^{±1}] ≅ v_{1}^{−1}H_{∗}(G_{24}, (E_{2})_{∗}/3)

and

(11)  𝔽_{3}((v_{1}^{4}v_{2}^{−1}))[v_{1}^{±1}] ≅ v_{1}^{−1}H^{∗}(SD_{16}, (E_{2})_{∗}/3) .

(b) There are isomorphisms

(12)  𝔽_{3}[v_{1}^{±1}] ⊗ Λ(v_{1}^{−1}b_{1}) ≅ v_{1}^{−1}H^{∗}(𝔾_{2}^{1}, (E_{2})_{∗}/3)

and

(13)  𝔽_{3}[v_{1}^{±1}] ⊗ Λ(v_{1}^{−1}b_{1}, ζ) ≅ v_{1}^{−1}H^{∗}(𝔾_{2}, (E_{2})_{∗}/3) .

The element b_{1} has bidegree (1, 8) and the element v^{−1}b_{1} detects the image of the homotopy class α_{1} ∈ π_{3}S^{0}/3. The element ζ has bidegree (1, 0) and is the image of the class of the same name in H^{1}(𝔾_{2}, (E_{2})_{0}) from Theorem 4.3.b.

Proof. The results in (a) are immediate consequences of Theorem 3.1. See also [4] §3. For (b), the two isomorphisms both follow from Theorem 3.3 and the algebraic spectral sequences of Corollary 2.3. That v_{1}^{−1}b_{1} detects the image of α_{1} is proved in Proposition 1.5 of [6]. □

Here is our key lemma. Compare Lemma 4.4 in the rational case.

Lemma 5.3. Let X ∈ κ_{2} and let X/3 = S/3 ∧ X.

(a) Suppose that F = G_{24} or SD_{16}. Then the edge homomorphism induces an isomorphism

\xymatrix{π_{∗}L_{K(1)}L_{K(2)}(E_{2}^{hF} ∧ X/3) \ar[r]^{≅} & v_{1}^{−1}H^{0}(F, (E_{2})_{∗}X/3) .}

(b) Let F = 𝔾_{2}^{1} or 𝔾_{2}. Then the localized spectral sequence

(v_{1}^{−1}H^{s}(F, (E_{∗}X)/3))_{t} ⟹ π_{t−s}L_{K(1)}(E_{2}^{hF} ∧ X/3)

converges and collapses.

Proof. The proof of Lemma 4.4 goes through mutatis mutandis. We need only replace the localization H^{∗}(F, M) ↦ H^{∗}(F, M) ⊗ ℚ with the localization

H^{∗}(F, M) ⟼ v_{1}^{−1}H^{∗}(F, M/3)

throughout, and use Theorem 3.3 in place of Proposition 4.1 and Theorem 4.3. □

We now have the following remarkable calculation.

Proposition 5.4. Let X ∈ κ_{2}. Then the K(1)-localized Hurewicz homomorphism

π_{0}L_{K(1)}X/3⟶ π_{0}L_{K(1)}L_{K(2)}(E_{2} ∧ X/3)

is injective. Any choice of isomorphism (E_{2})_{∗} ≅ (E_{2})_{∗}X of twisted 𝔾_{2}-modules uniquely defines a generator of

π_{0}(L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X/3)) ≅ (v_{1}^{−1}H^{0}(𝔾_{2}^{1}, (E_{2})_{∗}/3)_{0} ≅ 𝔽_{3} .

This generator extends uniquely to a weak equivalence

L_{K(1)}S^{0}/3 ≃ L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X/3) .

Proof. We use the localized spectral sequence

(v_{1}^{−1}H^{s}(𝔾_{2}^{1}, (E_{2})_{∗}/3))_{t} ⟹ π_{t−s}L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X/3).

This converges by Lemma 5.3.b. The choice of isomorphism (E_{2})_{∗} ≅ (E_{2})_{∗}X is used to identify the E_{2}-term. By the isomorphism of (12) this spectral sequence collapses. By [11], we know that there is an isomorphism

𝔽_{3}[v_{1}^{±1}] ⊗ Λ(α) ≅ π_{∗}L_{K(1)}S/3

where α is the image of α_{1} ∈ π_{3}S^{0}/3. The result now follows from Proposition 5.2. □

This result will be extended to an integral calculation in Proposition 5.7. For a complete local ring A with maximal ideal 𝔪 define

A((x)) = lim_{k} {A/𝔪^{k}((x))} .

This a completion of the ring of Laurent series. Recall that v_{1} = u_{1}u^{−2} and v_{2} = u^{−8} for the standard p-typical deformation of the Honda formal group over (E_{2})_{∗}. As a first example, Lemma 5.1 immediately gives

(14)  π_{∗}L_{K(1)}E_{2} = 𝕎((u_{1}))[u^{±1}] .

We now give a calculation of π_{∗}L_{K(1)}E_{2}^{hF} for our two important finite subgroups. The elements c_{4}, c_{6}, Δ were all introduced in Remark 3.5.

Proposition 5.5. Let X ∈ κ_{2} and fix an isomorphism (E_{2})_{∗}X ≅ (E_{2})_{∗} of twisted 𝔾_{2}-modules.

(a) The edge homomorphism of the homotopy fixed point spectral sequence induces an isomorphism

π_{∗}L_{K(1)}L_{K(2)}(E_{2}^{hG_{24}} ∧ X) ≅ lim v_{1}^{−1}H^{0}(G_{24}, (E_{2})_{∗}/3^{n}) .

Define b_{2} = c_{6}/c_{4} and j = c_{4}^{3}/Δ. Then these choices define an isomorphism

ℤ_{3}((j))[b_{2}^{±1}] ≅ lim v_{1}^{−1}H^{0}(G_{24}, (E_{2})_{∗}/3^{n}) .

(b) The edge homomorphism of the homotopy fixed point spectral sequence induces an isomorphism

π_{∗}L_{K(1)}L_{K(2)}(E_{2}^{hSD_{16}} ∧ X) ≅ lim v_{1}^{−1}H^{0}(SD_{16}, (E_{2})_{∗}/3^{n}) .

Define w = v_{1}^{4}/v_{2}. Then this determines an isomorphism

ℤ_{3}((w))[v_{1}^{±1}] ≅ lim v_{1}^{−1}H^{0}(SD_{16}, (E_{2})_{∗}/3^{n}) .

Proof. For (a), the first isomorphism follows from Proposition 5.2, Lemma 5.1, and a five lemma argument. For the second isomorphism, we know by Remark 3.5 that c_{4} ≡ v_{1}^{2} and c_{6} ≡ v_{1}^{3} modulo 3 and up to a unit. It follows that c_{4} is invertible in the inverse limit and that we have a map

ℤ_{3}((j))[b_{2}^{±1}] → lim v_{1}^{−1}H^{0}(G_{24}, (E_{2})_{∗}/3^{n}) .

By Proposition 5.2, this map induces an isomorphism modulo 3 and the result follows. Part (b) follows directly from [4] §3. □

Lemma 5.6. Let X ∈ κ_{2} and let F = G_{24} or SD_{16}. Given a choice of isomorphism (E_{2})_{∗} ≅ (E_{2})_{∗}X of twisted 𝔾_{2}-modules the image of the multiplicative unit 1 under the isomorphisms

lim(v_{1}^{−1}H^{0}(F, E_{∗}/3^{n}))_{0} ≅ lim(v_{1}^{−1}H^{0}(F, E_{∗}X/3^{n}))_{0} ≅ π_{0}L_{K(1)}L_{K(2)}(E_{2}^{hF} ∧X)

extends to a weak equivalence of L_{K(1)}E_{2}^{hF} -modules

L_{K(1)}E_{2}^{hF} ≃ L_{K(1)}L_{K(2)}(E_{2}^{hF} ∧ X) .

Proof. Let Z = L_{K(2)}(E_{2}^{hF} ∧ X). By Proposition 5.5, the given isomorphism of Morava modules determines a map g : S^{0} → L_{K(1)}Z. By induction and a five lemma argument, the induced map S^{0} → L_{K(1)}Z ∧S/3^{n} extends to a weak equivalence of L_{K(1)}E_{2}^{hF} -modules

L_{K(1)}E_{2}^{hF} /3^{n} ≃ L_{K(1)}Z ∧ S/3^{n}

and the result follows from Proposition 5.1. □

Theorem 5.7. Let X ∈ κ_{2}. Then the localized Hurewicz homomorphism

π_{0}L_{K(1)}X⟶ π_{0}L_{K(1)}L_{K(2)}(E_{2} ∧ X)

is injective. Given a choice of isomorphism (E_{2})_{∗} ≅ (E_{2})_{∗}X of twisted 𝔾_{2}- modules the image of the multiplicative unit 1 under the isomorphisms

ℤ_{3} ≅ lim(v_{1}^{−1}H^{0}(𝔾_{2}^{1}, (E_{2})_{∗}/3^{n}))_{0} ≅ π_{0}(L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X))

extends to a weak equivalence of L_{K(1)}S^{0}-modules

L_{K(1)}S^{0} ≃ L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) .

Proof. Let Y = L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X). Take the tower of 2.7 and apply the localization functor L_{K(1)}L_{K(2)}(− ∧ X) to produce a tower with homotopy inverse limit L_{K(1)}Y. By Lemma 5.6, the fibers are all of the form Σ^{8k}L_{K(1)}E_{2}^{hF} with F = G_{24} or F = SD_{16}. Using Proposition 5.5, we then see that the map

S^{0} → L_{K(1)}L_{K(2)}(E_{2}^{hG_{24}} ∧ X) ≃ L_{K(1)}E_{2}^{hG_{24}}

induced by the given isomorphism of Morava modules lifts uniquely to a map

ι : L_{K(1)}S^{0} → L_{K(1)}Y .

By Proposition 5.4 this induces a weak equivalence

L_{K(1)}S/3 ≃ L_{K(1)}Y ∧ S/3 .

Then, using the natural fiber sequence

L_{K(1)}S/3 ∧ Y → L_{K(1)}S/3^{n} ∧ Y → L_{K(1)}S/3^{n−1} ∧ Y,

induction, and Lemma 5.1, we obtain the desired weak equivalence. □

We record the following result for later use. It is an immediate consequence of Theorem 5.7.

Corollary 5.8. Let X ∈ κ_{2}. Given a choice of isomorphism (E_{2})_{∗} ≅ (E_{2})_{∗}X of twisted 𝔾_{2}-modules the image of the multiplicative unit 1 under the isomorphisms

ℤ_{3} ≅ lim(v_{1}^{−1}H^{0}(𝔾_{2}^{1}, (E_{2})_{∗}/3^{n}))_{0} ≅ π_{0}(L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X))

extends to a weak equivalence of L_{K(1)}E_{2}^{h𝔾_{2}^{1}}-modules

L_{K(1)}E_{2}^{h𝔾_{2}^{1}} ≃ L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) .

We now want to extend Theorem 5.7 to the sphere itself. Recall that there is a fiber sequence

\xymatrix{L_{K(2)}S^{0} \ar[r] & E_{2}^{h𝔾_{2}^{1}} \ar[r]^{ψ−1} & E_{2}^{h𝔾_{2}^{1}}}

where ψ is a topological generator of the central ℤ_{3} ⊆ 𝔾_{2}. For any K(2)-local X, we may apply the functor L_{K(2)}((−) ∧ X) to get a fiber sequence

(15)  \xymatrix{X \ar[r] & L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) \ar[r]^{ψ−1} & L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) .}

Theorem 5.9. a) Let X ∈ κ_{2}. Given a choice of isomorphism (E_{2})_{∗} ≅ (E_{2})_{∗}X of twisted 𝔾_{2}-modules the image of the multiplicative unit 1 under the isomorphisms

ℤ_{3} ≅ lim(v_{1}^{−1}H^{0}(𝔾_{2}, (E_{2})_{∗}/3^{n})_{∗})_{0} ≅ π_{0}L_{K(1)}X

extends to a weak equivalence of L_{K(1)}L_{K(2)}S^{0}-modules

L_{K(1)}L_{K(2)}S^{0} ≃ L_{K(1)}X .

b) The weak equivalence L_{K(1)}S^{0} ≃ L_{K(1)}L_{K(2)}E_{2}^{h𝔾_{2}^{1}} of Proposition 5.7 factors uniquely though L_{K(1)}L_{K(2)}S^{0} and extends to a weak equivalence

L_{K(1)}S^{0} ∨ L_{K(1)}S^{−1} ≃ L_{K(1)}L_{K(2)}S^{0}

where L_{K(1)}S^{−1} → L_{K(1)}L_{K(2)}S^{0} is induced by ζ ∈ π_{−1}L_{K(2)}S^{0}.

Proof. Let f : L_{K(1)}E_{2}^{h𝔾_{2}^{1}} → L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) be the equivalence of

Corollary 5.8. Since ψ : E_{2}^{h𝔾_{2}^{1}} → E_{2}^{h𝔾_{2}^{1}} is a morphism of ring spectra, we get a diagram of L_{K(1)}E_{2}^{h𝔾_{2}^{1}}-module maps

\xymatrix{L_{K(1)}E_{2}^{h𝔾_{2}^{1}} \ar[r]^{f} \ar[d]_{ψ−1} & L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) \ar[d]^{(ψ−1)∧X} \\ L_{K(1)}E_{2}^{h𝔾_{2}^{1}} \ar[r]^{f} & L_{K(1)}L_{K(2)}(E_{2}^{h𝔾_{2}^{1}} ∧ X) .}

By Theorem 5.7, there is an equivalence L_{K(1)}S^{0} ≃ L_{K(1)}E_{2}^{h𝔾_{2}^{1}}. Hence, to check that the diagram commutes, we need only verify that it commutes after applying π_{0}, and this is obvious. Part (a) follows. We now prove part (b). Let f_{0} : L_{K(1)}S^{0}⟶ L_{K(1)}E_{2}^{h𝔾_{2}^{1}} be the equivalence, as in Theorem 5.7. The composition (ψ − 1)f_{0} is zero, as ψ induces a ring map on (E_{2})_{0}. Because π_{1}L_{K(1)}S^{0} = 0, f_{0} lifts uniquely to a map f : L_{K(1)}S^{0} → L_{K(1)}L_{K(2)}S^{0} and we get a weak equivalence

f ∨ g : L_{K(1)}S^{0} ∨ L_{K(1)}S^{−1}⟶ L_{K(1)}L_{K(2)}S^{0}

where g is the desuspension of the composition

\xymatrix{L_{K(1)}S^{0} \ar[r]^{f_{0}}_{≃} & L_{K(1)}E_{2}^{h𝔾_{2}^{1}} \ar[r] & ΣL_{K(1)}L_{K(2)}S^{0} .}

As ζ is defined to be the image of unit in π_{0}E_{2}^{h𝔾_{2}^{1}} in π_{−1}L_{K(2)}S^{0}, the result follows. □

We now come to our main theorems.

Theorem 5.10. Let X ∈ κ_{2}. Then the localized Hurewicz homomorphism

π_{0}L_{1}X⟶ π_{0}L_{1}L_{K(2)}(E_{2} ∧ X)

is injective. A choice of isomorphism f : (E_{2})_{∗} → (E_{2})_{∗}X determines a generator of π_{0}L_{1}X ≅ ℤ_{3}. This generator extends uniquely to a weak equivalence of L_{1}L_{K(2)}S^{0}-modules

L_{1}L_{K(2)}S^{0} ≃ L_{1}X .

Proof. From Theorem 5.9 we have that π_{1}L_{K(1)}X = 0 for all X ∈ κ_{2}. The result then follows by the chromatic fracture square (8), Theorem 4.5 and Theorem 5.9. □

Theorem 5.11 (Chromatic Splitting). If n = 2 and p = 3, then

L_{1}L_{K(2)}S^{0} ≃ L_{1}(S_{3}^{0} ∨ S_{3}^{−1}) ∨ L_{0}(S_{3}^{−3} ∨ S_{3}^{−4})

where S_{p}^{n} denotes the p-complete sphere.

Proof. We use the chromatic square of (8). Let X = L_{K(2)}S^{0}. Theorem 5.9 implies

L_{0}L_{K(1)}X ≃ L_{0}L_{K(1)}(S^{0} ∨ S^{−1}) .

From Theorem 4.6 we have that

L_{0}X ≃ L_{0}(S_{3}^{0} ∨ S_{3}^{−1} ∨ S_{3}^{−3} ∨ S_{3}^{−4}) .

Thus we need only show that the map

L_{0}X ⟶ L_{0}L_{K(1)}X

is equivalent to the composition

L_{0}(S_{3}^{0} ∨ S_{3}^{−1} ∨ S_{3}^{−3} ∨ S_{3}^{−4})⟶ L_{0}(S_{3}^{0} ∨ S_{3}^{−1})⟶ L_{0}L_{K(1)}(S^{0} ∨ S^{−1})

where the first map is projection and the second map is the L_{0} localization of the canonical map S_{3}^{0} ∨ S_{3}^{−1} → L_{K(1)}(S^{0} ∨ S^{−1}). This follows from Theorem 4.6 and Theorem 5.9.b. □

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Paul G. Goerss Department of Mathematics Northwestern University Evanston IL 60208 U.S.A. pgoerss@math.northwestern.edu Hans-Werner Henn Institut de Recherche Mathématique Avancée C.N.R.S.

- Université de Strasbourg F-67084 Strasbourg France henn@math.unistra.fr

Mark Mahowald

Department of Mathematics Northwestern University Evanston IL 60208 U.S.A.