ALGEBRAIC K-THEORY OF THE TWO-ADIC INTEGERS

John Rognes

Abstract. We compute the two-completed algebraic K-groups K_{∗}(ℤ̂_{2})_{2}^{∧} of the twoadic integers, and determine the homotopy type of the two-completed algebraic Ktheory spectrum K(ℤ̂_{2})_{2}^{∧}. The natural map K(ℤ)_{2}^{∧} → K(ℤ̂_{2})_{2}^{∧} is shown to induce an isomorphism modulo torsion in degrees 4k + 1 with k ≥ 1.

Introduction

This paper concludes a series (see [R1], [R2], [R34]) that determines all the higher algebraic K-groups of the ring of two-adic integers ℤ̂_{2}, after two-adic completion. This is achieved in Theorem 5.5 below. Explicitly for ∗ > 0

(0.1)  K_{∗}(ℤ̂_{2})_{2}^{∧} = \{ \begin{matrix}ℤ̂_{2} ⊕ ℤ/2^{v_{2}(k)+3} & for ∗ = 4k − 1, \\ ℤ/2^{v_{2}(k)+3} & for ∗ = 4k, \\ ℤ̂_{2} ⊕ ℤ/2 & for ∗ = 4k + 1, \\ ℤ/2 & for ∗ = 4k + 2. \end{matrix}

Here v_{2}(k) denotes the two-adic valuation of k, i.e., the greatest power of 2 that divides k. We also determine the structure of the two-completed algebraic K-theory spectrum K(ℤ̂_{2}) as a connective spectrum, or infinite loop space, in Theorem 8.1.

By the localization sequence, these results also determine the algebraic K-theory groups and spectrum of the field of two-adic numbers ℚ̂_{2}. In both cases the resulting spectra agree with their K-localizations, except in low degrees, as predicted by the Lichtenbaum–Quillen conjectures.

These results extend to the case p = 2 the odd-primary theorem obtained by M. Bökstedt and I. Madsen in [BM1]. They computed the p-adically completed higher algebraic K-groups and algebraic K-theory spectrum K(ℤ̂_{p})_{p}^{∧} of the ring of p-adic integers for any odd prime p. See also [BM2], where their calculation is extended to include rings of integers in local number fields which are finite unramified extensions of the p-adic numbers ℚ̂_{p}.

The methods used in this series of papers are parallel to those of [BM1] and [BM2], but modified to account for technical and conceptual difficulties appearing in the even case. The fundamental tool is the cyclotomic trace map to topological cyclic homology (see [BHM]). The significant differences appear first in the spectral

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary: 19D50, Secondary: 12G05, 19D55, 55P42, 55Q35, 55Q50.

Key words and phrases. cyclotomic trace map, Galois reduction map, image of J, invertible spectra, Adams operations.

sequence calculations of [R34], where the algebra structure on the mod p homotopy of a ring spectrum is replaced by a module action of mod four homotopy acting upon the mod two homotopy of the ring spectrum.

Second, our method of identification of the homotopy type of K(ℤ̂_{2})_{2}^{∧} differs from the odd-primary case. At odd primes a comparison is made between algebraic Ktheory and Waldhausen’s algebraic K-theory of spaces (A-theory), which succeeds because the torsion in K_{∗}(ℤ̂_{p})_{p}^{∧} is tightly related to the Adams e-invariant, or the image of J spectrum, which is essentially the K-localization of the sphere spectrum. When p = 2 there is a distinction between the real and complex einvariant, or between the real and complex image of J spectra. It is the former which is essentially the K-localization of the sphere spectrum, while it turns out that K(ℤ̂_{2}) is more closely linked to complex topological K-theory than real topological K-theory, in the sense that it is the complex image of J spectrum which describes the torsion in K_{∗}(ℤ̂_{2})_{2}^{∧}. (This may be motivated by noting that ℤ̂_{2} does not have a real embedding.)

We therefore replace the comparison with A-theory with a Galois reduction map red: K(ℤ̂_{2}) → K(𝔽_{3}) = ImJ_{ℂ} to the complex image of J, behaving somewhat as if the prime field 𝔽_{3} were a residue field of ℤ̂_{2}. This map is developed in Section 3, and motivated by a similar odd-primary construction from [DF].

We also wish to call attention to Theorem 7.7 below, showing that the natural map K(ℤ) → K(ℤ̂_{2}) induces an isomorphism modulo torsion on homotopy in all degrees 4k + 1 with k ≥ 1. At an odd prime p the corresponding map is thought to act as multiplication by specific values of a p-adic L-function. In the case p = 2 all these values are two-adic units.

We refer to the Introduction in [R34] for an overview of the initial papers [R1],

[R2] and [R34]. In [R34] we computed the mod two homotopy of the topological cyclic homology of the integers at two, which also determines the mod two homotopy of the algebraic K-theory of the two-adic integers. Thus the task of the present paper is to erect the two-adic type upon the mod two type.

Here is an overview of the paper. In Section 1 the mod two calculations are reviewed. In Section 2 we review known relationships between algebraic and topological K-theory, including Bökstedt’s model JK(ℤ) for the analogue of étale Ktheory of the integers at two. In Section 3 we construct the Galois reduction map from K(ℤ̂_{2}) to the complex image of J infinite loop space ImJ_{ℂ}, and consider its homotopy fiber, which we call the reduced K-theory K^{red}(ℤ̂_{2}). In Section 4 we compute the K-groups of ℤ̂_{2} and the reduction map in an initial range of degrees. These considerations suffice to determine the two-completed algebraic K-groups of the two-adic integers in Section 5. (Completing at another prime p merely gives the p-completion of K(𝔽_{2}).) Then in Section 6 we construct an infinite loop map B ImJ_{ℂ} → K^{red}(ℤ̂_{2}) mapping isomorphically to the torsion in even degrees. The argument uses obstruction techniques, and a construction from [HMS] of invertible spectra in the category of K-local, two-complete spectra. In Section 7 we identify the infinite loop space cofiber of this map as BBU, carrying the torsion free part of the K-groups. We obtain the following two fiber sequences:

(0.2)

K^{red}(ℤ̂_{2}) ⟶ K(ℤ̂_{2}) ⟶^{red} ImJ_{ℂ}
B ImJ_{ℂ} ⟶ K^{red}(ℤ̂_{2}) ⟶ BBU

In Section 8 these sequences are viewed as extensions of infinite loop spaces, and classified up to homotopy equivalence. This determines the infinite loop space structure of K(ℤ̂_{2})_{2}^{∧} up to homotopy equivalence. Finally, in Section 9 we discuss the natural map K(ℤ) → K(ℤ̂_{2}), viewed as a close invariant of the algebraic Ktheory of the integers, at two. We note that \sqrt{−1} ∉ ℚ̂_{2}, so in a sense this invariant is closer to K(ℤ) than the theories known to satisfy étale descent by Thomason’s theorem [T].

Acknowledgment.

I thank Karl–Heinz Knapp for letters on the image of J, and the reference to [HMS].

1. Calculational input

In this section we review the calculational input from [R34] that will be required to determine the completed algebraic K-groups of the two-adic integers. All homotopy groups, spaces and spectra will be implicitly completed at two without further mention.

By Theorem D of Hesselholt and Madsen [HM], the cyclotomic trace map induces a two-adic equivalence from K(ℤ̂_{2}) to the connective cover of the topological cyclic homology spectrum T C(ℤ):

trc: K(ℤ̂_{2}) ⟶^{≃} T C(ℤ)[0, ∞)

Definition 1.1. The algebraic K-theory spectrum K(ℤ̂_{2}) admits a ring spectrum structure induced by the tensor product of ℤ̂_{2}-modules. Likewise the topological cyclic homology spectrum T C(ℤ) admits a ring spectrum structure related to the shuffle product ring spectrum structure on topological Hochschild homology, by Proposition 4.6 of [HM]. We do not know whether the cyclotomic trace map is multiplicative, i.e., whether the equivalence displayed above also respects these two multiplicative structures.^{1} To avoid this problem, we will work with an a priori different multiplicative structure on K(ℤ̂_{2}) than the usual one, namely the one inherited from T C(ℤ)[0, ∞) by the equivalence induced by trc. For clarity we will refer to this as the T C-multiplication on K(ℤ̂_{2}), as opposed to the K-multiplication induced by the tensor product.

In this paper we will always use the T C-multiplication on K(ℤ̂_{2}), unless the contrary is explicitly stated. All our ring spectra and ring spectrum maps are assumed to be unital.

1B. I. Dundas has recently proved [D] that the cyclotomic trace map is multiplicative, hence that the K-multiplication and T C-multiplication agree.

If the cyclotomic trace map turns out to be multiplicative, then the two ring spectrum structures agree. The choice of ring spectrum structure does not affect the additive structure of K(ℤ̂_{2}) as a spectrum, or infinite loop space. In particular, products of two classes in K_{∗}(ℤ_{2}) under the two ring spectrum structures will always agree when one of the classes is in the image of the unit map from π_{∗}Q(S^{0}). We will often use this observation, also when working with finite coefficients.

Theorem 10.9 of [R34] computes the mod two homotopy of T C(ℤ). Hence it also computes the mod two homotopy of K(ℤ̂_{2}), by restricting to nonnegative degrees. We carry over the notation from Definitions 10.3 and 10.10 of loc. cit. to name classes in K_{∗}(ℤ̂_{2}; ℤ/2), through the equivalence induced by trc:

Theorem 1.2. There are short exact sequences

0 ⟶ ℤ/2{ξ_{2r}} ⟶^{∂} K_{2r−1}(ℤ̂_{2}; ℤ/2) ⟶^{π} ℤ/2{ξ_{2r−1}(e), ξ_{2r−1}} ⟶ 0

for r ≥ 2, and

0 ⟶ ℤ/2{ξ_{2r+1}(e)} ⟶^{∂} K_{2r}(ℤ̂_{2}; ℤ/2) ⟶^{π} ℤ/2{ξ_{2r}} ⟶ 0

for r ≥ 1, determining K_{∗}(ℤ̂_{2}; ℤ/2) for ∗ ≥ 2 up to extensions. The same sequences determine K_{∗}(ℤ̂_{2}; ℤ/2) in degrees ∗ = 0, 1, when the undefined class ξ_{1}(e) is omitted. Hence

#K_{∗}(ℤ̂_{2}; ℤ/2) = \{ \begin{matrix}2 & for ∗ = 0, \\ 4 & for ∗ ≥ 2 even or ∗ = 1, \\ 8 & for ∗ ≥ 3 odd. \end{matrix}

□

Definition 1.3. Let i_{k}: π_{∗}(X) → π_{∗}(X; ℤ/2^{k}) be the mod 2^{k} coefficient reduction homomorphism, j_{k}: π_{∗}(X; ℤ/2^{k}) → π_{∗−1}(X) the mod 2^{k} Bockstein homomorphism, and ρ_{ℓ}^{k}: π_{∗}(X; ℤ/2^{k}) → π_{∗}(X; ℤ/2^{ℓ}) the coefficient reduction homomorphism. We briefly write ρ = ρ_{1}^{2}.

Let η, ν and σ denote the Hopf classes in π_{∗}Q(S^{0}), as well as their images in K_{∗}(ℤ) and K_{∗}(ℤ̂_{2}). Let μ = μ_{9} be a class in π_{9}Q(S^{0}) detected by KO-theory. (The ambiguity in μ consists of classes which vanish in K_{9}(ℤ), by [Mit] and [Wal].) Recall from Notation 0.8 of [R34] the classes ν̃_{4} ∈ K_{4}(ℤ̂_{2}; ℤ/4) and η̃_{2} ∈ K_{2}(ℤ̂_{2}; ℤ/2), together with λ ∈ K_{3}(ℤ̂_{2}) and κ ∈ K_{5}(ℤ̂_{2}). These are chosen so that j_{2}(ν̃_{4}) = ν and j_{1}(η̃_{2}) = η, while 2λ = ν and i_{1}(κ) = λη̃_{2}. (Since η̃_{2} comes from π_{2}(Q(S^{0}); ℤ/2) the ring spectrum structure is not involved in forming this product.)

The classes ξ_{2r} for r ≥ 0 are defined in K_{2r}(ℤ̂_{2}; ℤ/2) by the formulas ξ_{4i} = ρ(ν̃_{4}^{i}) and ξ_{4i+2} = ν̃_{4}^{i} · η̃_{2}. The classes ξ_{2r−1}(e) for e = [log_{2}(r) − 1] and r ≥ 2 are defined in K_{2r−1}(ℤ̂_{2}; ℤ/2) by the formulas ξ_{4i+3}(e) = ρ(ν̃_{4}^{i}) · λ and ξ_{4i+5}(e) = ρ(ν̃_{4}^{i}) · κ. (In forming products with ν̃_{4}, the ring spectrum structure is essentially involved. We are using the T C-multiplication here.)

The further classes ∂(ξ_{2r}) and ∂(ξ_{2r−1}(e)) are the images of the above classes by the composite ∂π, where ∂ and π appear in the fiber sequence defining topological cyclic homology:

ΩT F (ℤ) ⟶^{∂} T C(ℤ) ⟶^{π} T F (ℤ) ⟶^{R−1} T F (ℤ)

See [R34] for further information.

We shall see (in Proposition 4.2) that i_{1}(η) represents ξ_{1} and ηη̃_{2} represents ξ_{3} in T F_{∗}(ℤ; ℤ/2). Hence we can let ρ(ν̃_{4}^{i}) · η represent ξ_{4i+1} in K_{4i+1}(ℤ̂_{2}; ℤ/2) and let ν̃_{4}^{i} · ηη̃_{2} represent ξ_{4i+3} in K_{4i+3}(ℤ̂_{2}; ℤ/2). (This will resolve the ambiguity in Definition 10.10 of [R34].)

Next we need the action of ν̃_{4} on these generators, which is clear from Proposition 10.11 of [R34].

Proposition 1.4. T C-multiplication by ν̃_{4} on K_{∗}(ℤ̂_{2}; ℤ/2) acts injectively in all degrees, and bijectively in degrees ∗ ≥ 2. On generators it is given by

ν̃_{4} · ξ_{2r} = ξ_{2r+4}  ν̃_{4} · ξ_{2r−1}(e) = ξ_{2r+3}(f)
ν̃_{4} · ∂(ξ_{2r}) = ∂(ξ_{2r+4})  ν̃_{4} · ∂(ξ_{2r−1}(e)) = ∂(ξ_{2r+3}(f))
ν̃_{4} · ξ_{2r−1} = ξ_{2r+3}

with e = [log_{2}(r) − 1] and f = [log_{2}(r + 2) − 1]. The classes not in the image of the action are additively generated by 1, ξ_{1}, ∂(ξ_{2}), ξ_{2}, ∂(ξ_{3}(0)), ξ_{3}(0), ξ_{3}, ∂(ξ_{4}), ∂(ξ_{5}(0)) and ξ_{5}(0). □

In addition to the obvious results K_{0}(ℤ̂_{2}) = ℤ and K_{1}(ℤ̂_{2}) = ℤ̂_{2}^{×} ≅ ℤ/2 ⊕ ℤ̂_{2}, we have K_{2}(ℤ̂_{2}) = ℤ/2 generated by η^{2} by [DS], and K_{3}(ℤ̂_{2}) = ℤ̂_{2} ⊕ ℤ/8 by [Lev] and [MS]; see also [K]. Here λ generates the torsion subgroup in K_{3}(ℤ̂_{2}). We shall not assume the K_{3}-result for our proofs.

For units a, b ∈ ℤ̂_{2} we write {a} and {a, b} for their symbols in K_{1}(ℤ̂_{2}) and K_{2}(ℤ̂_{2}), respectively. Recall that K_{2}(ℤ̂_{2}) is detected by the Hilbert symbol {a, b} ↦ (a, b)_{2} ∈ μ_{2} = {±1} (see [Mil]), given by (a, b)_{2} = +1 if and only if ax^{2} + by^{2} = 1 has a solution with x, y ∈ ℚ_{2}, which is equivalent to the equation having a solution in ℤ/8. The pairing K_{1}(ℤ̂_{2}) ⊗ K_{1}(ℤ̂_{2}) → K_{2}(ℤ̂_{2}) taking {a} ⊗ {b} to {a, b} is induced by the K-multiplication on K(ℤ̂_{2}). It therefore agrees with the (default) T C-multiplication when a = −1 or b = −1, since the symbol {−1} is the image of η ∈ π_{1}Q(S^{0}).

K_{∗}(ℤ̂_{2}) is degreewise finitely generated as a ℤ̂_{2}-module by [Wag], and

(1.5)  K_{∗}(ℤ̂_{2})/(torsion) ≅ \{ \begin{matrix}ℤ̂_{2} & for ∗ = 0 or ∗ ≥ 1 odd, \\ 0 & otherwise. \end{matrix}

Combining (1.2) and (1.5) we find

(1.6)  K_{∗}(ℤ̂_{2}) ≅ \{ \begin{matrix}ℤ̂_{2} & for ∗ = 0, \\ ℤ̂_{2} ⊕ (cyclic) & for ∗ ≥ 1 odd, \\ (cyclic) & for ∗ ≥ 2 even. \end{matrix}

by a universal coefficient sequence argument. Each group named “cyclic” is a nontrivial cyclic group of order a power of two.

Remark 1.7. The localization sequence

Hℤ̂_{2} ≃ K(𝔽_{2}) ⟶ K(ℤ̂_{2}) ⟶ K(ℚ̂_{2})

and the mod two calculations for K(ℤ̂_{2}) give a mod two calculation of K(ℚ̂_{2}). The injection K_{∗}(ℤ̂_{2}; ℤ/2) → K_{∗}(ℚ̂_{2}; ℤ/2) is an isomorphism for all ∗ ≠ 1. In degree one the classes ξ_{1} and ∂(ξ_{2}) generate K_{1}(ℤ̂_{2})/2 ≅ ℤ/2{{−1}, {5}}. We can adjoin a formal element ξ_{1}(e) ∈ K_{1}(ℚ̂_{2}; ℤ/2) with e = −1 representing the unit 2 ∈ ℚ̂_{2}^{×}, so

K_{1}(ℚ̂_{2}; ℤ/2) ≅ ℤ/2{ξ_{1}(e), ξ_{1}, ∂(ξ_{2})}.

K-multiplicatively ξ_{1}(−1)^{2} = ∂(ξ_{3}(0)), ξ_{1} ·ξ_{1}(−1) = 0 and ∂(ξ_{2})·ξ_{1}(−1) = ∂(ξ_{3}(0)) by the Hilbert symbol calculations (2, 2)_{2} = −1, (−1, 2)_{2} = +1 and (5, 2)_{2} = −1.

2. Topological models for algebraic K-theory

In this section we review a variety of results relating algebraic K-theory spectra to spectra built using real and complex topological K-theory. We will generally use infinite loop space notation, for almost all spectra that appear are connective. Recall the implicit completion at two.

Let ℤ×BO and ℤ×BU be the connective real and complex topological K-theory ring spectra. By theorems of Suslin [S] the natural maps K(ℝ) → ℤ × BO and K(ℂ) → ℤ × BU are equivalences of ring spectra, and likewise for an embedding ℚ̄_{2} → ℂ of the algebraic closure of the two-adic numbers ℚ̂_{2} the induced maps K(ℚ̄_{2}) → K(ℂ) → ℤ × BU are equivalences. We let v_{1} ∈ π_{2}(ℤ × BU) denote the periodicity element. Then π_{∗}(ℤ×BU) ≅ ℤ̂_{2}[v_{1}] and π_{∗}(ℤ×BU; ℤ/2) ≅ ℤ/2[i_{1}(v_{1})]. The unit map Q(S^{0}) → ℤ × BU takes η̃_{2} to i_{1}(v_{1}) on mod two homotopy.

The real and complex image of J-spectra ImJ_{ℝ} and ImJ_{ℂ} are ring spectra defined by the fiber sequences

ImJ_{ℝ} ⟶^{i} ℤ × BO ⟶^{ψ^{3}−1} BSpin
ImJ_{ℂ} ⟶^{i} ℤ × BU ⟶^{ψ^{3}−1} BU

respectively. Here ψ^{3} is the Adams operation, BSpin is the three-connected cover of ℤ×BO, and BU is the zero-component of ℤ×BU. Both maps labeled i are maps of ring spectra. (To see this, note that for a self-map f: X → X of a ring spectrum, the homotopy fixed point spectrum X^{hf} of the free monoid action generated by f forms a ring spectrum, which can be identified with the homotopy fiber of f − 1. The forgetful map i: X^{hf} → X is then a ring spectrum map.)

We let e_{2n−1} ∈ π_{2n−1}(ImJ_{ℂ}) denote the integral generator obtained as the image of v_{1}^{n} under the fiber map ΩBU → ImJ_{ℂ}. For n odd and k = 1, or n even and k ≤ 3, we let i_{k}(v_{1}^{n}) ∈ π_{2n}(ImJ_{ℂ}; ℤ/2^{k}) be the generator mapping under i_{∗} to the class with the same name in π_{2n}(ℤ × BU; ℤ/2^{k}). When k = 1 we shall briefly write e_{1} for i_{1}(e_{1}) ∈ π_{1}(ImJ_{ℂ}; ℤ/2), and v_{1} for the lift of i_{1}(v_{1}) to π_{2}(ImJ_{ℂ}; ℤ/2). Then π_{∗}(ImJ_{ℂ}; ℤ/2) ≅ ℤ/2[e_{1}, v_{1}]/(e_{1}^{2} = 0).

Let L_{K} denote Bousfield’s K-localization functor [Bou]. The localization map Q(S^{0}) → L_{K}Q(S^{0}) has a factorization

Q(S^{0}) ⟶^{e_{ℝ}} ImJ_{ℝ} ⟶ L_{K}Q(S^{0})

where the right map is an equivalence above degree one by a theorem of Mahowald (see [Bou]), and the left map induces (the negative of) Adams’ real e-invariant e′_{ℝ} on homotopy groups (see [A] and [Q2]). Hence we refer to the unit map e_{ℝ} as the real e-invariant.

In the complex case there is an equivalence of ring spectra q: ImJ_{ℂ} → K(𝔽_{3}), by Quillen’s theorem [Q1]. We identify ImJ_{ℂ} with K(𝔽_{3}) through this equivalence. By the complex version of Quillen’s argument from [Q2] the unit map Q(S^{0}) → ImJ_{ℂ} induces (the negative of) Adams’ complex e-invariant e_{ℂ} on homotopy groups (see [A] and [Q2]). Hence we refer to the unit map e_{ℂ} as the complex e-invariant.

The complexification map c: ℤ × BO → ℤ × BU is compatible with the Adams operations, and induces a ring spectrum map c: ImJ_{ℝ} → ImJ_{ℂ}. By the (confirmed) Adams conjecture e_{ℝ} admits a space level section, which we denote α: ImJ_{ℝ} → Q(S^{0}). Thus e_{ℝ} ∘ α ≃ id.

The mod three reduction homomorphism π̂: ℤ̂_{3} → 𝔽_{3} induces an equivalence π̂: K(ℤ̂_{3}) → K(𝔽_{3}) by a rigidity theorem of Gabber [G]. Following Friedlander [F] we choose an embedding of fields ℚ̂_{3} → ℂ such that the composite map

K(ℤ̂_{3}) ⟶ K(ℚ̂_{3}) ⟶ K(ℂ) ⟶^{≃} ℤ × BU

agrees with Quillen’s Brauer lifting b: K(𝔽_{3}) → ℤ × BU from [Q1] under Gabber’s equivalence, i.e., makes the following diagram homotopy commutative:

[figure: K(ℤ̂_{3}); π̂; ≃; K(𝔽_{3}); q; ≃; b; ImJ_{ℂ}; i; K(ℚ̂_{3}); K(ℂ); ≃; ℤ × BU]

Here i is the fiber map of ψ^{3} − 1, as before.

We next review Bökstedt’s model JK(ℤ) for the étale K-theory of the integers. The ring spectrum JK(ℤ) is defined in [B] as the homotopy fiber of the composite map

c ∘ (ψ^{3} − 1): ℤ × BO ⟶^{ψ^{3}−1} BSpin ⟶^{c} BSU.

Hence there is a square of fiber sequences

[figure: BSO; ∗; BBSO; η; ImJ_{ℝ}; i; j; ℤ × BO; ψ^{3}−1; BSpin; c; JK(ℤ); ℤ × BO; c(ψ^{3}−1); BSU.]

The unit map of JK(ℤ) factors as

(2.1)  Q(S^{0}) ⟶^{e_{ℝ}} ImJ_{ℝ} ⟶^{j} JK(ℤ)

because the homotopy fiber CokJ of e_{ℝ} is five-connected and K-acyclic, and JK(ℤ) agrees with its K-localization above degree one.

The alternative factorization c ∘ (ψ^{3} − 1) = (ψ^{3} − 1) ∘ c gives rise to the following square of fiber sequences

(2.2)  [figure: BBO; B(ℤ × BO); S^{1}; JK(ℤ); ℤ × BO; (ψ^{3}−1)c; c; BSU; ImJ_{ℂ}; i; ℤ × BU; ψ^{3}−1; BU.]

Hence JK(ℤ) fits into the following three fiber sequences:

(2.3)  \begin{matrix}ImJ_{ℝ} & ⟶^{j} JK(ℤ) ⟶ BBSO \\ BBO & ⟶ JK(ℤ) ⟶ ImJ_{ℂ} \\ SU & ⟶ JK(ℤ) ⟶ ℤ × BO \end{matrix}

There is a commutative square of ring spectra

[figure: K(ℤ); K(ℝ); c; K(ℤ̂_{3}); K(ℂ)]

induced by a corresponding commutative square of rings. By the equivalences K(ℤ̂_{3}) ≃ ImJ_{ℂ}, K(ℝ) ≃ ℤ × BO and K(ℂ) ≃ ℤ × BU, this square induces a map from K(ℤ) to the pullback P B in the lower left square of (2.2). The latter square is nearly homotopy Cartesian, since its iterated homotopy fiber ΩS^{1} ≃ ℤ has trivial homotopy away from degree zero. By a check on π_{1} it follows that the induced map K(ℤ) → P B has a lift

Φ: K(ℤ) ⟶ JK(ℤ)

which is at least four-connected. Φ a map of ring spectra.

The Lichtenbaum–Quillen conjecture for ℤ extended to the prime two (as in [DF]) then asserts that the map Φ is an equivalence.^{2} In [B] Bökstedt constructed a space level section

ϕ: ΩJK(ℤ) ⟶ ΩK(ℤ)

to the map ΩΦ: ΩK(ℤ) → ΩJK(ℤ). Thus ΩΦ ∘ ϕ ≃ id. Hence Φ induces split surjections on all homotopy groups. The space level section ϕ is constructed so as to make the following diagram of spaces homotopy commutative:

(2.4)  [figure: Ω ImJ_{ℝ}; Ωα; Ωj; ΩQ(S^{0}); Ωι; ΩJK(ℤ); ϕ; ΩK(ℤ)]

2After this paper was written, this conjecture was proved by V. Voevodsky [V]. See also [Wei].

Hence the splitting ϕ respects classes in K_{∗}(ℤ) and JK_{∗}(ℤ) coming from the (real) image of J in π_{∗}Q(S^{0}). (See [B] or p. 541 of [R] for details. Remember that α is the section to e_{ℝ} arising from a solution to the Adams conjecture.)

The homotopy groups of JK(ℤ) can easily be computed from (2.2). We transport the notation for classes in K_{∗}(ℤ) to JK_{∗}(ℤ) by Φ, i.e., we briefly write x for Φ(x) ∈ JK_{∗}(ℤ) when x ∈ K_{∗}(ℤ).

Lemma 2.5. The homotopy groups of JK(ℤ) begin

JK_{∗}(ℤ) = (ℤ{1}, ℤ/2{η}, ℤ/2{η^{2}}, ℤ/16{λ}, 0, ℤ{κ}, 0, ℤ/16{σ}, 0, ℤ{κ_{9}} ⊕ ℤ/2{μ}, …).

Clearly λη = 0, κη = 0 and ση = 0 in JK_{∗}(ℤ). There are relations λη̃_{2} = i_{1}(κ), κη̃_{2} = i_{1}(σ) and ση̃_{2} = i_{1}(κ_{9}).

For torsion classes x ∈ K_{∗}(ℤ) with ∗ ≤ 9 we have ϕ(Φ(x)) = x in K_{∗}(ℤ). (Without assuming the Lichtenbaum–Quillen conjecture we do not know if ϕ(κ) = κ in K_{5}(ℤ), or if ϕ(κ_{9}) = κ_{9} in K_{9}(ℤ).)

Hence in K_{∗}(ℤ) we have λη = 0 and λη̃_{2} = i_{1}(ϕ(κ)), ϕ(κ)η = 0 and ϕ(κ)η̃_{2} = i_{1}(σ), and ση = 0 and ση̃_{2} = i_{1}(ϕ(κ_{9})).

Proof. The group calculations are easy. Let γ_{i} ∈ π_{i}(SU) be a generator for i ≥ 3 odd. Then multiplication by η̃_{2} acts as multiplication by i_{1}(v_{1}) on the homotopy of SU, so γ_{3} · η̃_{2} = i_{1}(γ_{5}), γ_{5} · η̃_{2} = i_{1}(γ_{7}) and γ_{7} · η̃_{2} = i_{1}(γ_{9}). But the natural map SU → JK(ℤ) in (2.3) takes γ_{3}, γ_{5}, γ_{7} and γ_{9} to λ, κ, σ and κ_{9}, at least mod two.

The remaining claims follow from (2.4) and naturality with respect to the space level map ϕ. (Note that multiplication by η̃_{2} is first defined unstably on π_{3} of a space.) □

We remark that the following diagram homotopy commutes

[figure: K(ℤ); Φ; π; JK(ℤ); K(𝔽_{3}); q; ≃; ImJ_{ℂ}]

where π is induced by the ring surjection ℤ → 𝔽_{3}, and the unlabeled map is as in (2.2). Also the composite

ImJ_{ℝ} ⟶^{j} JK(ℤ) ⟶ ImJ_{ℂ}

agrees with the complexification map c.

Lemma 2.6. The natural map JK(ℤ) → ImJ_{ℂ} induces a surjection on all homotopy groups. Hence so does the composite map

π: K(ℤ) ⟶^{Φ} JK(ℤ) ⟶ ImJ_{ℂ} ≃ K(𝔽_{3})

induced by the ring surjection π: ℤ → 𝔽_{3}.

Proof. The second claim follows from the first and Bökstedt’s splitting. The first claim is clear in degree zero, so suppose i ≥ 1. There is a long exact sequence of Mayer–Vietoris type associated to the lower left square in (2.2):

⟶ JK_{i}(ℤ) ⟶ π_{i}(ImJ_{ℂ}) ⊕ π_{i}(ℤ × BO) ⟶ π_{i}(ℤ × BU) ⟶

Since π_{i}(ℤ × BU) is torsion free and π_{i}(ImJ_{ℂ}) is torsion, the latter group is in the image from JK_{i}(ℤ). □

We translate the named classes in K_{∗}(ℤ) to π_{∗}(ImJ_{ℂ}) by the map induced by π, so that

π_{∗}(ImJ_{ℂ}) = (ℤ{1}, ℤ/2{η}, 0, ℤ/8{λ}, 0, ℤ/2{κ}, 0, ℤ/16{σ}, …).

So η = e_{1}, λ = e_{3}, κ = e_{5} and σ = e_{7}. Then the complexification map c: ImJ_{ℝ} → ImJ_{ℂ} satisfies c(1) = 1, c(η) = η, c(η^{2}) = 0, c(ν) = 2λ and c(σ) = σ.

Remark 2.7. The pullback P B in (2.2) is the model JK(ℤ[\frac{1}{2}]) for the étale Ktheory of ℤ[\frac{1}{2}], and the natural map JK(ℤ) → JK(ℤ[\frac{1}{2}]) is a covering, compatible with the localization map K(ℤ) → K(ℤ[\frac{1}{2}]). The lift Φ′: K(ℤ[\frac{1}{2}]) → JK(ℤ[\frac{1}{2}]) is a map of ring spectra. By comparisons across the two homotopy Cartesian squares

[figure: K(ℤ̂_{2}); K(ℤ); Φ; JK(ℤ); K(ℚ̂_{2}); K(ℤ[\frac{1}{2}]); Φ′; JK(ℤ[\frac{1}{2}])]

the symbol {2} ∈ K_{1}(ℚ̂_{2}) represents the formally adjoined class ξ_{1}(−1) on mod two homotopy. The analogue of Lemma 2.5 shows that {2}η̃_{2} = i_{1}(λ) and ν̃_{4} · ξ_{1}(−1) = ξ_{5}(0), extending Proposition 1.4.

3. The Galois reduction map

In this section we construct a Galois reduction map K(ℤ̂_{2}) → K(𝔽_{3}), behaving somewhat as if 𝔽_{3} were a residue field of ℤ̂_{2}. We continue to identify the target with ImJ_{ℂ}. Compare Section 13 of [DM].

Let ℚ̂_{2}(μ_{2^{∞}}) be the field of two-adic numbers with the 2^{n}th roots of unity adjoined, for all n. Let ℚ̄_{2} be the algebraic closure of ℚ̂_{2}, and choose an embedding ℚ̄_{2} → ℂ. (Any field embedding will do.) Hence we have field extensions

ℚ̂_{2} ⟶ ℚ̂_{2}(μ_{2^{∞}}) ⟶ ℚ̄_{2} ⟶ ℂ.

Let θ^{3} ∈ Γ′ = Gal(ℚ̂_{2}(μ_{2^{∞}})/ℚ̂_{2}) be the Galois automorphism given by θ^{3}(ζ) = ζ^{3} for each root of unity ζ. So θ^{3} corresponds to the two-adic unit 3 under the isomorphism Γ′ ≅ Aut(μ_{2^{∞}}) ≅ ℤ̂_{2}^{×}. We will choose an extension ϕ^{3} of θ^{3} to the algebraic closure ℚ̄_{2}, with specified action on \sqrt{3}.

The following was pointed out to me by Steffen Bentzen.

Lemma 3.1. The rational integer 3 does not have a square root in ℚ̂_{2}(μ_{2^{∞}}).

Proof. An odd rational integer n has a square root in ℚ̂_{2} precisely if n ≡ 1 mod 8. Hence 3 does not have a square root in ℚ̂_{2}. So if 3 has a square root in ℚ̂_{2}(μ_{2^{∞}}) then ℚ̂_{2}(\sqrt{3}) is a quadratic subfield of ℚ̂_{2}(μ_{2^{∞}}). These correspond to index two subgroups of Γ′ ≅ ℤ̂_{2}^{×}, and are ℚ̂_{2}(\sqrt{−1}), ℚ̂_{2}(\sqrt{2}) and ℚ̂_{2}(\sqrt{−2}). But neither of these contain \sqrt{3}, since none of −1 · 3, 2 · 3 and −2 · 3 are congruent to 1 mod 8. □

Corollary 3.2. There exists an extension ϕ^{3} ∈ G = Gal(ℚ̄_{2}/ℚ̂_{2}) of θ^{3} ∈ Γ′ with ϕ^{3}(\sqrt{3}) = +\sqrt{3}.

Proof. The algebraic closure ℚ̄_{2} is a union of finite Galois extensions of ℚ̂_{2}(μ_{2^{∞}}), so G → Γ′ is surjective. Hence extensions ϕ^{3} of θ^{3} exist, and by the lemma above we may choose the sign of ϕ^{3}(\sqrt{3}) as we like. □

Definition 3.3. Hereafter we fix a choice of extension ϕ^{3} with ϕ^{3}(\sqrt{3}) = +\sqrt{3}. This will make our Galois reduction map a “plus reduction map.” An alternative extension ϕ^{3} with ϕ^{3}(\sqrt{3}) = −\sqrt{3} defines a “minus reduction map,” which has different properties, less suited to our argument.

Our choice of ϕ^{3} induces a self map of K(ℚ̄_{2}) which is compatible up to homotopy with the self map ψ^{3} of ℤ × BU under the equivalence K(ℚ̄_{2}) ≃ ℤ × BU. For by [MST] the only ring spectrum operations on ℤ×BU (up to homotopy) are the twoadic Adams operations ψ^{k}, and these are detected by their action on π_{2}. There is a natural Bockstein isomorphism β_{n}: K_{2}(ℚ̄_{2}; ℤ/2^{n}) → _{2^{n}}K_{1}(ℚ̄_{2}) = μ_{2^{n}}(ℚ̄_{2}), and ϕ^{3} multiplies the 2^{n}-torsion in K_{1} by three. Hence ϕ^{3} also acts on π_{2} of the two-adic completion of K(ℚ̄_{2}) as multiplication by three, like ψ^{3} does on ℤ × BU.

Thus we get a homotopy commutative diagram:

[figure: K(ℚ̄_{2}); ϕ^{3}−1; ≃; K(ℚ̄_{2}); ≃; ℤ × BU; ψ^{3}−1; ℤ × BU]

A choice of commuting homotopy determines an equivalence of the horizontal homotopy fibers K(ℚ̄_{2})^{hϕ^{3}} ⟶^{≃} (ℤ × BU)^{hψ^{3}}. Here K(ℚ̄_{2})^{hϕ^{3}} denotes the homotopy fixed points of the action of (the free monoid generated by) ϕ^{3} on K(ℚ̄_{2}), which we identify with the homotopy fiber of ϕ^{3} − 1, and likewise for (ℤ × BU)^{hψ^{3}}.

Since ϕ^{3} fixes the subfield ℚ̂_{2}, we have natural maps of spectra

K(ℤ̂_{2}) ⟶ K(ℚ̂_{2}) ⟶ K(ℚ̄_{2})^{ϕ^{3}} ⟶ K(ℚ̄_{2})^{hϕ^{3}} ⟶^{≃} (ℤ × BU)^{hψ^{3}}.

Here K(ℚ̄_{2})^{ϕ^{3}} denotes the fixed points of the functorially defined action of ϕ^{3} on K(ℚ̄_{2}), and the third map is the natural inclusion of fixed points into homotopy fixed points.

Definition 3.4. The composite above induces a spectrum map on connective covers

red: K(ℤ̂_{2}) ⟶ (ℤ × BU)^{hψ^{3}}[0, ∞) = ImJ_{ℂ}

which we will call the Galois reduction map. Following Dwyer and Mitchell [DM] we define the reduced K-theory K^{red}(ℤ̂_{2}) by the fiber sequence

K^{red}(ℤ̂_{2}) ⟶ K(ℤ̂_{2}) ⟶^{red} ImJ_{ℂ}.

Lemma 3.6. The reduction map is unital, i.e., the composite map

Q(S^{0}) ⟶^{ι} K(ℤ) ⟶ K(ℤ̂_{2}) ⟶^{red} ImJ_{ℂ}

is homotopic to the complex e-invariant

Q(S^{0}) ⟶^{e_{ℂ}} ImJ_{ℂ}.

Proof. This is clear, because the sequence of spectrum maps defining red respect the unit maps from Q(S^{0}). □

Let us conclude this section by discussing the Adams v_{1}^{4}-action on mod two homotopy, and Bousfield’s characterization of K-local spectra. In keeping with our infinite loop space notation, we write Q(S^{0}/q) for the mod q Moore spectrum, and similarly Q(S^{n}/q) for its nth delooping. The infinite loop space smash products satisfy Q(X) ∧ Q(Y) ≅ Q(X ∧ Y).

In [A] Adams constructed a spectrum map A: Q(S^{8}/2) → Q(S^{0}/2) inducing multiplication by v_{1}^{4} (an isomorphism) on complex topological K-theory. A satisfies j_{1}Ai_{1} = 8σ, and any such A will do, because different choices of such maps A induce the same map on topological K-theory. The degree eight induced action on π_{∗}(X; ℤ/2) for any spectrum X is called the Adams v_{1}^{4}-action, and denoted as multiplication by v_{1}^{4}. (It is typically not the fourth power of an action by some element v_{1}.)

Let us write ˜4σ_{4} for a class in π_{8}(Q(S^{0}); ℤ/4) with j_{2}(˜4σ_{4}) = 4σ. Mod four homotopy acts on mod two homotopy by a pairing m: Q(S^{0}/4)∧Q(S^{0}/2) → Q(S^{0}/2) (see [O] and [R2]). Then multiplication by ˜4σ_{4} induces a spectrum map

A′: Q(S^{8}/2) ≅ Q(S^{8}) ∧ Q(S^{0}/2) ⟶^{˜4σ_{4}∧1} Q(S^{0}/4) ∧ Q(S^{0}/2) ⟶^{m} Q(S^{0}/2)

such that j_{1}A′i_{1} = j_{1}m(˜4σ_{4} ∧ 1)i_{1} = j_{1}ρ(˜4σ_{4}) = 2j_{2}(˜4σ_{4}) = 8σ. Hence we can take A = A′ as our Adams map.

A choice of Adams v_{1}^{4}-action also determines a choice of class μ ∈ π_{9}Q(S^{0}), by i_{1}(μ) = v_{1}^{4} · i_{1}(η).

By a theorem of Bousfield [Bou], a spectrum X is K-local if and only if the v_{1}^{4}- action on π_{∗}(X; ℤ/2) is invertible, i.e., if multiplication by v_{1}^{4} is an isomorphism. Furthermore K-localization is “smashing,” so the K-localization map X → L_{K}X induces the localization homomorphism

π_{∗}(X; ℤ/2) ⟶ π_{∗}(X; ℤ/2)[(v_{1}^{4})^{−1}] ⟶^{≅} π_{∗}(L_{K}X; ℤ/2).

Hence if v_{1}^{4} acts as an isomorphism on π_{∗}(X; ℤ/2) in degrees ∗ ≥ k, then the localization map X → L_{K}X is an equivalence on (k − 1)-connected covers.

4. Calculations in low degrees

We proceed by computing the two-adic and mod two homotopy of K_{∗}(ℤ̂_{2}) in degrees zero through nine.

Let β_{k} denote the kth order (mod 2^{k}) homotopy Bockstein differential. So β_{k}(ρ_{1}^{k}x) = i_{1}j_{k}(x). Let t_{n} generate the torsion in K_{n}(ℤ̂_{2}) for n ≥ 1, and let g_{n} generate K_{n}(ℤ̂_{2}) modulo torsion for n odd. Then t_{n} is well defined up to an odd multiple, and g_{n} is well defined up to a two-adic unit, modulo multiples of t_{n}.

Definition 4.1. Recall from Lemma 1.11 of [R34] the class σ̄ defined in K_{7}(ℤ̂_{2})/4 by i_{2}(σ̄) = λν̃_{4}.

Let σ̃_{16} ∈ π_{8}(Q(S^{0}); ℤ/16) be a class satisfying j_{4}(σ̃_{16}) = σ, chosen so that ρ_{2}^{4}(σ̃_{16}) = ˜4σ_{4}. Multiplication by σ̃_{16} induces the Adams v_{1}^{4}-action on mod two homotopy. Let ρ_{3}^{4}(σ̃_{16}) = ˜2σ_{8} and ρ(˜4σ_{4}) = ˜8σ_{2}, so j_{3}(˜2σ_{8}) = 2σ and j_{1}(˜8σ_{2}) = 8σ.

Let λ̃_{8} ∈ K_{4}(ℤ̂_{2}; ℤ/8) be a class satisfying j_{3}(λ̃_{8}) = λ. Again we may assume ρ_{2}^{3}(λ̃_{8}) = ν̃_{4}.

Proposition 4.2. (a) The two-adic algebraic K-groups of ℤ̂_{2} begin as follows:

n  0  1  2  3  4
K_{n}(ℤ̂_{2}) ℤ̂_{2} ℤ/2 ⊕ ℤ̂_{2} ℤ/2 ℤ/8 ⊕ ℤ̂_{2} ℤ/8
gen. 1 η, {3} η^{2}  λ, g_{3} λ{3}
5  6  7  8  9
ℤ/2 ⊕ ℤ̂_{2} ℤ/2 ℤ/16 ⊕ ℤ̂_{2} ℤ/16 ℤ/2 ⊕ ℤ̂_{2}
t_{5}, κ κ{3}  σ, g_{7}  σ{3} μ, g_{9}

The generators satisfy η = {−1}, η{3} = η^{2}, i_{1}(g_{3}) = ∂(ρν̃_{4}), λ{3} = λ{5} = ∂(κ), i_{1}(t_{5}) = ν̃_{4} · i_{1}(η) = ξ_{5}, κ{3} = ∂(σ), i_{1}(g_{7}) = ∂(ρ(ν̃_{4}^{2})) and i_{1}(g_{9}) = ση̃_{2}.

There is a T C-multiplicative relation λ · λ = 0.

(b) The class σ̄ is represented by the image of σ ∈ π_{7}Q(S^{0}).

(c) The mod two algebraic K-groups of ℤ̂_{2} begin as follows:

n  0  1  2  3  4
K_{n}(ℤ̂_{2}; ℤ/2) ℤ/2 (ℤ/2)^{2}  ℤ/4  (ℤ/2)^{3}  (ℤ/2)^{2}
gen.  1 ξ_{1}, ∂(ξ_{2}) ξ_{2}, ∂(ξ_{3}(0)) ξ_{3}(0), ξ_{3}, ∂(ξ_{4}) ξ_{4}, ∂(ξ_{5}(0))
5  6  7  8  9
(ℤ/2)^{3}  ℤ/4  (ℤ/2)^{3}  (ℤ/2)^{2}  (ℤ/2)^{3}
ξ_{5}(0), ξ_{5}, ∂(ξ_{6}) ξ_{6}, ∂(ξ_{7}(1)) ξ_{7}(1), ξ_{7}, ∂(ξ_{8}) ξ_{8}, ∂(ξ_{9}(1)) ξ_{9}(1), ξ_{9}, ∂(ξ_{10})

The generators satisfy i_{1}(η) = i_{1}({−1}) = ξ_{1}, i_{1}({5}) = ∂(ξ_{2}), i_{1}({3}) = ξ_{1} +∂(ξ_{2}), η̃_{2} = ξ_{2}, i_{1}(η^{2}) = ∂(ξ_{3}(0)), i_{1}(λ) = ξ_{3}(0), ηη̃_{2} = ξ_{3}, i_{1}(g_{3}) = ∂(ξ_{4}), ρν̃_{4} = ξ_{4}, i_{1}(∂(κ)) = ∂(ξ_{5}(0)), i_{1}(κ) = ξ_{5}(0), ν̃_{4} · i_{1}(η) = ξ_{5}, ν̃_{4} · i_{1}({5}) = ∂(ξ_{6}), ν̃_{4} · η̃_{2} = ξ_{6}, i_{1}(∂(σ)) = ν̃_{4} · i_{1}(η^{2}) = ∂(ξ_{7}(1)), i_{1}(σ) = ξ_{7}(1), ν̃_{4} · ηη̃_{2} = ξ_{7}, i_{1}(g_{7}) = ∂(ξ_{8}), ρ(ν̃_{4}^{2}) = ξ_{8}, i_{1}(σ{3}) = ∂(ξ_{9}(1)), i_{1}(g_{9}) = ση̃_{2} = ξ_{9}(1) and i_{1}(μ) = ξ_{9}.

(d) There are nontrivial additive extensions 2·ξ_{2} = ∂(ξ_{3}(0)) and 2·ξ_{6} = ∂(ξ_{7}(1)). There are nonzero Bockstein differentials β_{1}(ξ_{2}) = ξ_{1}, β_{1}(ξ_{3}) = ∂(ξ_{3}(0)), β_{3}(ξ_{4}) = ξ_{3}(0), β_{3}(∂(ξ_{6})) = ∂(ξ_{5}(0)), β_{1}(ξ_{6}) = ξ_{5}, β_{1}(ξ_{7}) = ∂(ξ_{7}(1)), β_{4}(ξ_{8}) = ξ_{7}(1), β_{4}(∂(ξ_{10}) = ∂(ξ_{9}(1)) and β_{1}(ξ_{10}) = ξ_{9}.

Integral lifts of the classes ξ_{1}, ∂(ξ_{3}(0)), ξ_{3}(0), ∂(ξ_{5}(0)), ξ_{5}, ∂(ξ_{7}(1)), ξ_{7}(1), ∂(ξ_{9}(1)) and ξ_{9} generate the torsion in K_{∗}(ℤ̂_{2}) in these degrees.

Integral lifts of the classes 1, ξ_{1} + ∂(ξ_{2}), ∂(ξ_{4}), ξ_{5}(0), ∂(ξ_{8}) and ξ_{9}(1) generate K_{∗}(ℤ̂_{2}) modulo torsion, in these degrees.

The proof uses some notation from [R34].

Proof. For the integral calculations, it is clear that the unit 1 generates K_{0}(ℤ̂_{2}) = ℤ̂_{2}, and that K_{1}(ℤ̂_{2}) ≅ ℤ̂_{2}^{×} has generators {−1} and {3}, where the former is hit by η ∈ π_{1}Q(S^{0}). We also know by the Hilbert symbol that K_{2}(ℤ_{2}) ≅ ℤ/2 is generated by {−1, −1}, which is hit by η^{2}.

In degree three, the second and third map in the following display

e_{ℂ}: Q(S^{0}) ⟶^{ι} K(ℤ) ⟶ K(ℤ_{2}) ⟶^{red} ImJ_{ℂ}

take the generator λ ∈ K_{3}(ℤ) = ℤ/16{λ} to the generator λ ∈ π_{3}(ImJ_{ℂ}) = ℤ/8{λ}, modulo 4λ, because e_{ℂ}(ν) = 2λ and ι(ν) = 2λ. Furthermore 4ν = η^{3} ∈ π_{3}Q(S^{0}) maps to zero in K_{3}(ℤ_{2}), so the cyclic torsion subgroup of K_{3}(ℤ̂_{2}) is ℤ/8{λ}.

In degree seven the composite above takes σ ∈ π_{7}Q(S^{0}) = ℤ/16{σ} to the generator σ ∈ π_{7}(ImJ_{ℂ}) = ℤ/16{σ}, so the cyclic torsion subgroup of K_{7}(ℤ̂_{2}) is ℤ/16{σ}.

In degree nine the composite takes the order two class μ ∈ π_{9}Q(S^{0}) to the generator μ ∈ π_{9}(ImJ_{ℂ}) = ℤ/2{μ}, since the complex e-invariant agrees with the real e-invariant in degrees ∗ = 8k + 1 (see Proposition 7.19 of [A]). Hence the cyclic torsion subgroup of K_{9}(ℤ̂_{2}) is ℤ/2{μ}.

We turn to the mod two calculations. The unit 1 generates K_{0}(ℤ̂_{2}; ℤ/2) = ℤ/2. In degree one i_{1}(η) ∈ π_{1}(Q(S^{0}); ℤ/2) is detected in T F_{1}(ℤ; ℤ/2) as the class represented by ξ_{1} = te_{3}, so we may choose i_{1}(η) to represent ξ_{1} in K_{1}(ℤ̂_{2}; ℤ/2). The product ηξ_{2} is represented by te_{3} · te_{4} = t^{2}e_{3}e_{4} = d^{4}(e_{4}) in E^{∗}(S^{1}; ℤ/2), hence has filtration < −4. Thus ηξ_{2} lies in the image of π_{3}(R − 1; ℤ/2), and so η∂(ξ_{2}) = ∂(ηξ_{2}) = 0 in cok π_{3}(R − 1; ℤ/2) ⊂ K_{2}(ℤ̂_{2}; ℤ/2). The Hilbert symbol calculations (−1, 3)_{2} = −1 and (−1, 5)_{2} = +1 show that η{3} = η^{2} and η{5} = 0, so we must have ∂(ξ_{2}) = {5}.

In degree two, the universal coefficient sequence for K_{2}(ℤ̂_{2}; ℤ/2) is not split, because η times the order two element η in K_{1}(ℤ̂_{2}) equals η^{2}, which is nonzero in K_{2}(ℤ̂_{2})/2. Hence K_{2}(ℤ̂_{2}; ℤ/2) = ℤ/4{η̃_{2}}, with 2 · η̃_{2} = i_{1}(η^{2}). In the extension

0 ⟶ ℤ/2{ξ_{3}(0)} ⟶^{∂} K_{2}(ℤ̂_{2}; ℤ/2) ⟶^{π} ℤ/2{ξ_{2}} ⟶ 0

we recognize i_{1}(η^{2}) = ∂(ξ_{3}(0)) as generating the kernel of the circle trace map, and η̃_{2} = ξ_{2} by definition. Then j_{1}(η̃_{2}) = η as claimed.

In degree three the universal coefficient sequence for K_{3}(ℤ̂_{2}; ℤ/2) splits, because η times the order two element η^{2} in K_{2}(ℤ̂_{2}) equals η^{3}, which is zero in K_{3}(ℤ̂_{2})/2. In the extension

0 ⟶ ℤ/2{ξ_{4}} ⟶^{∂} K_{3}(ℤ̂_{2}; ℤ/2) ⟶^{π} ℤ/2{ξ_{3}, ξ_{3}(0)} ⟶ 0

we have ρν̃_{4} = ξ_{4} and i_{1}(λ) = ξ_{3}(0). Here λ is integral and ρν̃_{4} has a mod four lift, so both ξ_{3}(0) and ∂(ξ_{4}) are integral because K_{2}(ℤ̂_{2}) has no torsion classes of order four or greater. Since the integral lift λ of ξ_{3}(0) generates the torsion subgroup of K_{3}(ℤ̂_{2}) it also follows that an integral lift of ∂(ξ_{4}) generates K_{3}(ℤ̂_{2}) modulo torsion.

Furthermore ηη̃_{2} ∈ K_{3}(ℤ̂_{2}; ℤ/2) has j_{1}(ηη̃_{2}) = η^{2} ≠ 0, hence is nonzero, and generates K_{3}(ℤ̂_{2}; ℤ/2) modulo the integral classes. Thus its image π(ηη̃_{2}) must equal either ξ_{3} or ξ_{3}(0) + ξ_{3}. But the latter has filtration zero in E^{∗}(S^{1}; ℤ/2) while π(ηη̃_{2}) has filtration < −4 in E^{∗}(S^{1}; ℤ/2). Thus we may take ηη̃_{2} to represent ξ_{3}, as claimed.

In degree four the universal coefficient sequence for K_{4}(ℤ̂_{2}; ℤ/2) splits, because η times the order two element 2ν in K_{3}(ℤ̂_{2}) is zero in K_{4}(ℤ̂_{2})/2. The class i_{1}(κ) = ξ_{5}(0) is integral, so also ∂(ξ_{5}(0)) will be integral. Since 8λ = η^{3} = 0 in K_{3}(ℤ̂_{2}) there is a class λ̃_{8} ∈ K_{4}(ℤ̂_{2}; ℤ/8) with i_{1}j_{3}(λ̃_{8}) = i_{1}(λ) = ξ_{3}(0), and ρ_{1}^{3}λ̃_{8} = ξ_{4}, so β_{3}(ξ_{4}) = ξ_{3}(0).

There is a nonzero mod eight Bockstein from degree five to degree four. The class ∂(ξ_{6}) = ν̃_{4} · ∂(ξ_{2}) admits the mod eight lift λ̃_{8} · ∂(ξ_{2}), where ∂(ξ_{2}) = i_{1}({5}) is integral. Hence i_{1}j_{3}(∂(ξ_{6})) = i_{1}j_{3}(λ̃_{8} · ∂(ξ_{2})) = i_{1}(λ · ∂(ξ_{2})) = i_{1}∂(λ · η̃_{2}) = i_{1}(∂(κ)) = ∂(ξ_{5}(0)), which is nonzero. Thus K_{4}(ℤ̂_{2}) ≅ ℤ/8 is generated by ∂(κ) = λ{5}, with mod two reduction ∂(ξ_{5}(0)). Since ηλ = 0, we may also take λ{3} as the integral generator.

In degree five the universal coefficient sequence for K_{5}(ℤ̂_{2}; ℤ/2) splits because η times the order two element 4λ{3} in K_{4}(ℤ̂_{2}) is zero in K_{5}(ℤ̂_{2})/2. There is a nonzero mod two Bockstein from degree six to degree five. Namely, ξ_{6} = ν̃_{4} · η̃_{2} has β_{1}(ξ_{6}) = ν̃_{4} · β_{1}(η̃_{2}) = ν̃_{4} · i_{1}(η) = ξ_{5}, by regularity of the action of mod four homotopy upon mod two homotopy (see Lemma 26 of [O]). Hence ξ_{5} is integral and generates the torsion subgroup ℤ/2 in K_{5}(ℤ̂_{2}). Likewise i_{1}(κ) = ξ_{5}(0) is integral and lifts to generate K_{5}(ℤ̂_{2}) modulo torsion.

There is a nonzero mod two differential β_{1}(ξ_{7}) = ∂(ξ_{7}(1)) from degree seven to degree six. This follows from β_{1}(ξ_{3}) = ∂(ξ_{3}(0)) by multiplication by the mod four class ν̃_{4}, using regularity of the action again. Hence K_{6}(ℤ̂_{2}) = ℤ/2, generated by ∂(σ̄). We can also compute i_{1}(κ{3}) = i_{1}(κ{5}) = ξ_{5}(0) · ∂(ξ_{2}) = ∂(te_{3}e_{4} · te_{4}) = ∂(t^{2}e_{3}e_{4}^{2}) = ∂(ξ_{7}(1)) = ∂(σ̄). The first equality uses i_{1}(ηκ) = ηλη̃_{2} = 0. Thus κ{3} generates K_{6}(ℤ̂_{2}).

In degree six the universal coefficient sequence for K_{6}(ℤ̂_{2}; ℤ/2) is not split, because η times the order two element t_{5} in K_{5}(ℤ̂_{2}) reduces mod two to η · ξ_{5} = ν̃_{4} · ∂(ξ_{3}(0)) = ∂(ξ_{7}(1)), which is nonzero. Hence K_{6}(ℤ̂_{2}; ℤ/2) ≅ ℤ/4, generated by a lift of t_{5} over j_{1}, with ∂(σ̄) representing two times the generator.

In degree seven, the universal coefficient sequence for K_{7}(ℤ̂_{2}; ℤ/2) is split, because η times the order two element ∂(σ̄) = κ{3} in K_{6}(ℤ̂_{2}) reduces mod two to i_{1}(η∂(σ̄)) = ∂(i_{1}(ησ̄)) = ∂(ν̃_{4} · i_{1}(ηλ)) = 0. The classes i_{1}(σ̄) = ξ_{7}(1) and ∂(ξ_{8}) are integral. For σ̄ is integral, and β_{1}(∂(ξ_{8})) = ∂(β_{1}(ρν̃_{4}^{2})) = 0. Since the torsion in K_{6}(ℤ̂_{2}) has order two, this proves the assertion.

To proceed with the mod two analysis, we will use the construction of the previous section.

The Galois reduction map red: K(ℤ̂_{2}) → ImJ_{ℂ} takes λ to λ = e_{3} mod 4λ on π_{3}, hence it takes λ̃_{8} to i_{3}(v_{1}^{2}) mod 4i_{3}(v_{1}^{2}) in π_{4}(ImJ_{ℂ}; ℤ/8). There is a unique class ν̃_{8} ∈ π_{4}(Q(S^{0}); ℤ/8) with j_{3}(ν̃_{8}) = ν. Then ν̃_{8} maps to 2λ̃_{8} in K_{4}(ℤ̂_{2}; ℤ/8), and to 2i_{3}(v_{1}^{2}) in π_{4}(ImJ_{ℂ}; ℤ/8). We compute 2·red(λ̃_{8}^{2}) = red(ν̃_{8}·λ̃_{8}) = 2i_{3}(v_{1}^{2})·red(λ̃_{8}) = 2i_{3}(v_{1}^{4}). Hence red(λ̃_{8}^{2}) ≡ i_{3}(v_{1}^{4}) mod 4i_{3}(v_{1}^{4}). (This would be obvious in the Kmultiplication for which red is a ring spectrum map, but we are using the T Cmultiplication to form the product λ̃_{8} · λ̃_{8}.)

There is a commutative square

[figure: K_{8}(ℤ̂_{2}; ℤ/8); red; j_{3}; π_{8}(ImJ_{ℂ}; ℤ/8); ≅; j_{3}; _{8}K_{7}(ℤ̂_{2}); red; ≅; _{8}π_{7}(ImJ_{ℂ})]

where _{8}K_{7}(ℤ̂_{2}) = ℤ/8{2σ}. The calculation of red(λ̃_{8}^{2}) then implies j_{3}(λ̃_{8}^{2}) ≡ 2σ mod 8σ, since red(2σ) = j_{3}(i_{3}(v_{1}^{4})). Thus λ̃_{8}^{2} ≡ ˜2σ_{8} ∈ K_{8}(ℤ̂_{2}; ℤ/8) modulo 4 · ˜2σ_{8} and modulo classes in i_{3}(K_{8}(ℤ̂_{2}). By mod four and mod two reduction we get ν̃_{4}^{2} ≡ ˜4σ_{4} mod i_{2}(K_{8}(ℤ̂_{2})) in K_{8}(ℤ̂_{2}; ℤ/4), and ρ(ν̃_{4}^{2}) ≡ ˜8σ_{2} mod i_{1}(K_{8}(ℤ̂_{2})) in K_{8}(ℤ̂_{2}; ℤ/2).

We defined σ̄ ∈ K_{7}(ℤ̂_{2}) as a class with i_{2}(σ̄) = λν̃_{4}. By Theorem 2(a) of [O], the operation δ_{3} = i_{3}j_{3} acts as a derivation on mod eight homotopy. Thus i_{2}(2σ̄) = ρ_{2}^{3}(2λλ̃_{8}) = ρ_{2}^{3}δ_{3}(λ̃_{8}^{2}) = i_{2}(2σ) in K_{7}(ℤ̂_{2}; ℤ/4), and so σ̄ ≡ σ mod 2K_{7}(ℤ̂_{2}).

In fact λ^{2} = 0 in K_{6}(ℤ̂_{2}), with respect to the T C-multiplication. For ν̃_{4}λ = i_{2}(σ̄) ≡ i_{2}(σ) mod two, so squaring gives ν̃_{4}^{2}λ^{2} ≡ i_{2}(σ^{2}) mod two, and σ^{2} maps to zero in K_{14}(ℤ) by [Mit]. Thus ν̃_{4}^{2} · i_{1}(λ^{2}) = 0 and i_{1}(λ^{2}) = 0 by injectivity of the ν̃_{4}-action. Since K_{6}(ℤ̂_{2}) = ℤ/2 it follows that λ^{2} = 0.

Thus λλ̃_{8} ∈ K_{7}(ℤ̂_{2}; ℤ/8) is integral, and we may assume i_{3}(σ̄) = λλ̃_{8}. Then i_{3}(2σ̄) = δ_{3}(λ̃_{8}^{2}) = i_{3}(2σ) and so σ̄ ≡ σ mod 4K_{7}(ℤ̂_{2}). Since σ̄ originally was only defined modulo four, this proves that we may take σ̄ = σ.

From here on we may and will assume σ̄ ∈ K_{7}(ℤ̂_{2}) is chosen as the image of σ ∈ π_{7}Q(S^{0}). Thus i_{1}(σ) maps to ξ_{7}(1) = t^{2}e_{3}e_{4}^{2}, and σ generates the torsion in K_{7}(ℤ̂_{2}). The integral class ∂(ξ_{8}) lifts to a generator of K_{7}(ℤ̂_{2}) modulo torsion.

We can now compute the remaining Bockstein differentials.

There is a nonzero mod two Bockstein differential from degree ten to degree nine, given by β_{1}(ξ_{10}) = ξ_{9}. This follows from β_{1}(ξ_{2}) = ξ_{1} by multiplication by ν̃_{4}^{2} and regularity of the action of mod four homotopy upon mod two homotopy. Hence ξ_{9} is integral and generates the torsion in K_{9}(ℤ̂_{2}), i.e., i_{1}(μ) = ξ_{9}.

The product ση̃_{2} is also integral in K_{9}(ℤ̂_{2}; ℤ/2), since j_{1}(ση̃_{2}) = ση maps to zero in K_{8}(ℤ) by [Wal]. It is represented by ση̃_{2} = σ̄η̃_{2} = ν̃_{4}λη̃_{2} = ν̃_{4}i_{1}(κ) = ξ_{9}(1), so ξ_{9}(1) is integral, and represents a generator of K_{9}(ℤ̂_{2}) modulo torsion.

There is a nonzero mod sixteen Bockstein differential from degree nine to degree eight, given by β_{4}(∂(ξ_{10})) = ∂(ξ_{9}(1)). For the product σ{5} satisfies i_{1}(σ{5}) = σ · ∂(ξ_{2}) = ∂(σ · ξ_{2}) = ∂(ξ_{7}(1) · ξ_{2}) = ∂(ξ_{9}(1)) ≠ 0. Writing σ̃_{16} for the class with j_{4}(σ̃_{16}) = σ we get i_{1}j_{4}(σ̃_{16}{5}) = ∂(ξ_{9}(1)), and so β_{4}(y) = ∂(ξ_{9}(1)) where y = ρ_{1}^{4}(σ̃_{16}{5}) is the image of i_{1}({5}) = ∂(ξ_{2}) under the Adams v_{1}^{4}-action. We know that this agrees with multiplication by ν̃_{4}^{2}, modulo classes in the image of ∂. Thus y ≡ ν̃_{4}^{2} · ∂(ξ_{2}) = ∂(ξ_{10}) mod ∂(ξ_{10}). But y ≠ 0 since β_{4}(y) ≠ 0, so β_{4}(∂(ξ_{10}) = ∂(ξ_{9}(1)) as claimed.

Hence ∂(ξ_{9}(1)) admits the integral lift σ{3} = σ{5} which generates the torsion in K_{8}(ℤ̂_{2}).

There is a nonzero mod sixteen differential β_{4}(ξ_{8}) = ξ_{7}(1) from degree eight to degree seven. For i_{1}j_{4}(σ̃_{16}) = i_{1}(σ) = ξ_{7}(1), and z = ρ_{1}^{4}(σ̃_{16}) equals the image of the Adams v_{1}^{4}-action on i_{1}(1), hence agrees with ν̃_{4}^{2}·i_{1}(1) = ξ_{8} modulo classes in the image of ∂. But ∂(ξ_{9}(1)) is integral, so β_{4}(∂(ξ_{9}(1)) = 0, and β_{4}(z) = β_{4}(ξ_{8}) = ξ_{7}(1), as claimed.

The universal coefficient sequence for K_{8}(ℤ̂_{2}; ℤ/2) is split, because η times the order two element 8σ in K_{7}(ℤ̂_{2}) is zero. Likewise the universal coefficient sequence for K_{9}(ℤ̂_{2}; ℤ/2) is split, because η times the order two element 8σ{3} in K_{8}(ℤ̂_{2}) is zero. □

In the proof above, we have also shown most of the following proposition.

Proposition 4.3. (a) The Galois reduction map red: K(ℤ̂_{2}) → ImJ_{ℂ} takes λ̃_{8} ∈ K_{4}(ℤ̂_{2}; ℤ/8) to i_{3}(v_{1}^{2}) mod 4i_{3}(v_{1}^{2}) in π_{4}(ImJ_{ℂ}; ℤ/8), and λ̃_{8}^{2} to i_{3}(v_{1}^{4}) mod 4i_{3}(v_{1}^{4}) in π_{8}(ImJ_{ℂ}; ℤ/8).

(b) The classes ν̃_{4}^{2} and ˜4σ_{4} have the same image in K_{8}(ℤ̂_{2}; ℤ/4) mod i_{2}(K_{8}(ℤ̂_{2})), i.e., modulo integral classes. Their images in T F_{8}(ℤ; ℤ/4) agree, so the Adams v_{1}^{4}- action on T F_{∗}(ℤ; ℤ/2) equals the multiplication by ν̃_{4}^{2}. Hence these actions on T C_{∗}(ℤ; ℤ/2) and K_{∗}(ℤ̂_{2}; ℤ/2) agree modulo classes in ∂(T F_{∗+1}(ℤ; ℤ/2)). The difference of the actions is given by T C-multiplication by a multiple of σ{3}.

(c) The Adams v_{1}^{4}-actions on T C_{∗}(ℤ; ℤ/2) and K_{∗}(ℤ̂_{2}; ℤ/2) are injective in all degrees, and bijective in degrees ∗ ≥ 2. Hence T C(ℤ) and K(ℤ̂_{2}) agree with their K-localizations above degree one.

(d) The same Adams v_{1}^{4}-actions take 1 to ξ_{8} or ξ_{8} + ∂(ξ_{9}(1)), take ξ_{1} to ∂(ξ_{9}), and take ∂(ξ_{2}) to ∂(ξ_{10}).

Proof. (a) The calculation of the reduction map was part of the proof above.

(b) Since T F_{∗}(ℤ) is trivial in positive even degrees, the images in T F_{8}(ℤ; ℤ/4) of the classes ν̃_{4}^{2} and ˜4σ_{4} differ by a class from T F_{8}(ℤ) = 0, hence agree. The Adams v_{1}^{4}-action on T F_{∗}(ℤ; ℤ/2) is given by multiplication by the image of ˜4σ_{4} in the ring spectrum structure on T F (ℤ), hence this is the square of the ν̃_{4}-action.

The T C-multiplication by ν̃_{4}^{2} on K_{∗}(ℤ̂_{2}; ℤ/2) is therefore T C-multiplication by ˜4σ_{4} plus a multiple of σ{3}. The first part equals multiplication by ˜4σ_{4} using the infinite loop space structure, hence equals the Adams v_{1}^{4}-action. Thus the difference of the actions is at most a multiple of T C-multiplication by σ{3}.

(c) The Adams v_{1}^{4}-action is injective on ker π_{∗}(R − 1; ℤ/2) ⊂ T F_{∗}(ℤ; ℤ/2) and on cok π_{∗}(R − 1; ℤ/2), by Proposition 1.4, and therefore also on T C_{∗}(ℤ; ℤ/2). The bijectivity–conclusion and the corresponding result for K_{∗}(ℤ̂_{2}; ℤ/2) follow easily.

(d) By part (a) the product ˜4σ_{4} · i_{1}(1) = ˜8σ_{2} maps to ξ_{8} modulo classes in the image of ∂, i.e., modulo ∂(ξ_{9}(1)). We also saw in the proof above that the Adams v_{1}^{4}-action takes ∂(ξ_{2}) to ∂(ξ_{10}). Finally μ is chosen so that the Adams v_{1}^{4}-action takes ξ_{1} = i_{1}(η) to ξ_{9} = i_{1}(μ). The claims follow. □

We continue by computing the Galois reduction map on mod two homotopy. Recall that π_{∗}(ImJ_{ℂ}; ℤ/2) = ℤ/2[e_{1}, v_{1}]/(e_{1}^{2} = 0). The classes i_{1}(e_{2n+1}) = e_{1}v_{1}^{n}

are integral, while the classes v_{1}^{n} map to generators i_{1}(v_{1}^{n}) in π_{2n}(ℤ × BU; ℤ/2). The unit map from Q(S^{0}) takes i_{1}(η) and η̃_{2} to e_{1} and v_{1}. Multiplication by η̃_{2} on classes with a mod four lift equals formal multiplication by v_{1}, and the Adams v_{1}^{4}-action equals formal multiplication by v_{1}^{4}.

Proposition 4.4. (a) The mod two (plus) Galois reduction map

red: K_{∗}(ℤ̂_{2}; ℤ/2) ⟶ π_{∗}(ImJ_{ℂ}; ℤ/2)

is a v_{1}^{4}-module homomorphism in the sense that it commutes with the Adams v_{1}^{4}- actions on the source and target.

(b) The Galois reduction map is given on generators for the Adams v_{1}^{4}-action as follows:

deg. 0 1 1 2  2  3  3  3 4  4
x 1 ξ_{1} ∂(ξ_{2}) ξ_{2} ∂(ξ_{3}(0)) ξ_{3}(0) ξ_{3} ∂(ξ_{4}) ξ_{4} ∂(ξ_{5}(0))
red(x) 1 e_{1} e_{1} v_{1}  0  e_{1}v_{1} e_{1}v_{1} e_{1}v_{1} v_{1}^{2}  0
deg.  5  5  5 6  6  7  7  7  8  9
x ξ_{5}(0) ξ_{5} ∂(ξ_{6}) ξ_{6} ∂(ξ_{7}(1)) ξ_{7}(1) ξ_{7} ∂(ξ_{8}) ∂(ξ_{9}(1)) ξ_{9}(1)
red(x) e_{1}v_{1}^{2} e_{1}v_{1}^{2} e_{1}v_{1}^{2} v_{1}^{3}  0  e_{1}v_{1}^{3} e_{1}v_{1}^{3} e_{1}v_{1}^{3}  0  e_{1}v_{1}^{4}

Proof. (a) Any infinite loop map induces a v_{1}^{4}-module homomorphism on mod two homotopy.

(b) The listed classes clearly generate K_{∗}(ℤ̂_{2}; ℤ/2) as a v_{1}^{4}-module. The classes ∂(ξ_{3}(0)), ∂(ξ_{5}(0)), ∂(ξ_{7}(1)) and ∂(ξ_{9}(1)) are integral and in even degrees, hence map to zero.

By Lemma 3.6 the reduction map takes classes from π_{∗}(Q(S^{0}); ℤ/2) to their complex e-invariant, so red(1) = 1, red(ξ_{1}) = e_{1}, red(ξ_{2}) = v_{1}, red(ξ_{3}) = e_{1}v_{1}, red(ξ_{7}(1)) = e_{1}v_{1}^{3} and red(˜8σ_{2}) = v_{1}^{4}. Here ˜8σ_{2} = ξ_{8} or ξ_{8} + ∂(ξ_{9}(1)). By Proposition 4.3(a) the reduction map satisfies red(ξ_{3}(0)) = e_{1}v_{1} and red(ξ_{4}) = v_{1}^{2}.

In degree one we claim red(∂(ξ_{2})) = e_{1}. By construction, the integral Galois reduction map red: K_{1}(ℤ̂_{2}) → π_{1}(ImJ_{ℂ}) factors through K_{1}(ℚ̂_{2}(μ_{2^{∞}}))^{ϕ^{3}}, which can be identified with the units of ℚ̂_{2}(μ_{2^{∞}}) that are invariant under ϕ^{3}. Since ϕ^{3}(\sqrt{3}) = +\sqrt{3} by Definition 3.3, the unit symbol \{3\} is divisible by two in this group, and thus maps to zero in π_{1}(ImJ_{ℂ}) ≅ ℤ/2. Thus red({3}) = 0, and so red({5}) = red(η) = e_{1}.

Multiplying the integral class ∂(ξ_{2}) with η̃_{2} gives red(∂(ξ_{2}) · η̃_{2}) = e_{1}v_{1}, so ∂(ξ_{2}) · η̃_{2} ≠ 0 maps under π to π∂(ξ_{2}) · η̃_{2} = 0. The only possibility is ∂(ξ_{2}) · η̃_{2} = ∂(ξ_{4}), and red(∂(ξ_{4})) = e_{1}v_{1}.

Again ∂(ξ_{4}) = i_{1}(g_{3}) is integral, so we can multiply by η̃_{2} to obtain red(∂(ξ_{4}) · η̃_{2}) = e_{1}v_{1}^{2}. Thus ∂(ξ_{4}) · η̃_{2} ≠ 0 and maps to zero under π, hence ∂(ξ_{4}) · η̃_{2} = ∂(ξ_{6}) and red(∂(ξ_{6})) = e_{1}v_{1}^{2}.

Now ∂(ξ_{6}) is not integral, but we know that ∂(ξ_{8}) = i_{1}(g_{7}) is integral, and ∂(ξ_{8})·η̃_{2} = ∂(ν̃_{4}^{2})·η̃_{2} = ν̃_{4}^{2}·∂(η̃_{2}) = ∂(ξ_{10}). Also red(∂(ξ_{10})) = v_{1}^{4}·red(∂(ξ_{2})) = e_{1}v_{1}^{4}, since v_{1}^{4} · ∂(ξ_{2}) = ∂(ξ_{10}). So red(∂(ξ_{8})) · η̃_{2} = red(∂(ξ_{10})) ≠ 0, and red(∂(ξ_{8})) ≠ 0 must equal e_{1}v_{1}^{3}.

From i_{1}(λ) = ξ_{3}(0) and λη̃_{2} = i_{1}(κ) = ξ_{5}(0) we get red(ξ_{5}(0)) = red(ξ_{3}(0))η̃_{2} = e_{1}v_{1}^{2}. From ρν̃_{4} = ξ_{4} and ν̃_{4}η̃_{2} = ξ_{6} we get red(ξ_{6}) = red(ξ_{4})η̃_{2} = v_{1}^{3}. Applying the mod two Bockstein β_{1} gives red(ξ_{5}) = e_{1}v_{1}^{2}, since β_{1}(ξ_{6}) = ξ_{5} and β_{1}(v_{1}^{3}) = e_{1}v_{1}^{2}. From ρν̃_{4} = ξ_{4} and ν̃_{4}ηη̃_{2} = ξ_{7} we get red(ξ_{7}) = red(ξ_{4})ηη̃_{2} = e_{1}v_{1}^{3}. From ση̃_{2} = ξ_{9}(1) we get red(ξ_{9}(1)) = red(σ)η̃_{2} = e_{1}v_{1}^{4}. □

Proposition 4.5. The Galois reduction map induces an isomorphism from the torsion in K_{2r−1}(ℤ̂_{2}) onto π_{2r−1}(ImJ_{ℂ}), for all r ≥ 1. Hence there is a split short exact sequence

0 ⟶ K_{∗}^{red}(ℤ̂_{2}) ⟶ K_{∗}(ℤ̂_{2}) ⟶^{red} π_{∗}(ImJ_{ℂ}) ⟶ 0

in every degree, and K_{2r−1}(ℤ̂_{2}) ≅ ℤ̂_{2} ⊕ π_{2r−1}(ImJ_{ℂ}) for all r ≥ 1.

Proof. Recall that the torsion subgroup of K_{2r−1}(ℤ̂_{2}) is cyclic, by (1.6).

In degrees one, three and five, the classes η ∈ K_{1}(ℤ̂_{2}), λ ∈ K_{3}(ℤ̂_{2}) and t_{5} ∈ K_{5}(ℤ̂_{2}) generate torsion summands that map isomorphically onto π_{1}(ImJ_{ℂ}) ≅ ℤ/2, π_{3}(ImJ_{ℂ}) ≅ ℤ/8 and π_{5}(ImJ_{ℂ}) ≅ ℤ/2, respectively. There are corresponding nonzero Bockstein differentials β_{1}(ξ_{2}) = ξ_{1}, β_{3}(ξ_{4}) = ξ_{3}(0) and β_{1}(ξ_{6}) = ξ_{5}.

The Adams v_{1}^{4}-action is induced by multiplication by σ̃_{16}, for which β_{k}(σ̃_{16}) = 0 when k ≤ 3, so it follows by regularity (of the product on mod eight homotopy, and of the action of mod four homotopy on mod two homotopy) that there are nonzero Bockstein differentials β_{1}(v_{1}^{4k} · ξ_{2}) = v_{1}^{4k} · ξ_{1}, β_{3}(v_{1}^{4k} · ξ_{4}) = v_{1}^{4k} · ξ_{3}(0) and β_{1}(v_{1}^{4k} ·ξ_{6}) = v_{1}^{4k} ·ξ_{5} for all k ≥ 0. Hence there are integral lifts of v_{1}^{4k} ·ξ_{1}, v_{1}^{4k} ·ξ_{3}(0) and v_{1}^{4k} · ξ_{5} that generate torsion summands in K_{∗}(ℤ̂_{2}) which map isomorphically onto π_{∗}(ImJ_{ℂ}), in all degrees ∗ ≡ 1, 3, 5 mod 8.

The composite Q(S^{0}) → K(ℤ̂_{2}) ⟶^{red} ImJ_{ℂ} induces the complex e-invariant by Lemma 3.6, which is split surjective in degrees ∗ ≡ 7 mod 8. Hence the reduction map restricted to the cyclic torsion subgroup of K_{∗}(ℤ̂_{2}) induces a split surjection, and must therefore be an isomorphism. □

Proposition 4.6. (a) The Galois reduction map induces a surjection on mod two homotopy, in each degree.

(b) The mod two homotopy groups of K^{red}(ℤ̂_{2}) begin as follows:

K_{∗}^{red}(ℤ̂_{2}; ℤ/2) = \{ \begin{matrix}0 & for ∗ = 0, \\ ℤ/2 & for ∗ = 1, 2, 4, 6, 8, \\ (ℤ/2)^{2} & for ∗ = 3, 5, 7, 9. \end{matrix}

The generator in degree one is ξ_{1} + ∂(ξ_{2}). In degree 2r − 2 with 2 ≤ 2r − 2 ≤ 8 the generator is ∂(ξ_{2r−1}(e)). In degree 2r − 1 with 3 ≤ 2r − 1 ≤ 9 the generators are ξ_{2r−1}(e) + ∂(ξ_{2r}) and ξ_{2r−1} + ∂(ξ_{2r}).

(c) The Adams v_{1}^{4}-action on K_{∗}^{red}(ℤ̂_{2}; ℤ/2) is injective in all degrees, and bijective in degrees ∗ ≥ 2. Hence K^{red}(ℤ̂_{2}) agrees with its K-localization above degree one.

Proof. (a) The mod two homotopy of ImJ_{ℂ} is generated by the Adams v_{1}^{4}-action on the classes in degrees 0 ≤ ∗ < 8. The mod two reduction map surjects onto these classes by Proposition 4.4(b), hence onto all of π_{∗}(ImJ_{ℂ}; ℤ/2) by the v_{1}^{4}-module action.

(b) For this calculation, combine Propositions 4.2(c) and 4.4(b). In degree eight, red(ξ_{8}) = v_{1}^{4}, and in degree nine red(ξ_{9}) = red(∂(ξ_{10}) = e_{1}v_{1}^{4} by Proposition 4.3(d).

(c) The injective Adams v_{1}^{4}-action on K_{∗}(ℤ̂_{2}; ℤ/2) restricts to an injective action on the kernel K_{∗}^{red}(ℤ̂_{2}; ℤ/2) of the v_{1}^{4}-homomorphism induced by red. By counting orders it follows that the action is bijective in degrees ∗ ≥ 2. Alternatively one may note that the homotopy fiber of any map of K-local spectra is again K-local. □

Remark 4.7. By Remark 2.7 we see that K(ℚ̂_{2}) agrees with its K-localization on zero-connected covers; one degree better than K(ℤ̂_{2}).

5. The algebraic K-groups of ℤ̂_{2}

We proceed to determine the completed algebraic K-groups of ℤ̂_{2}. The odd groups were determined in Proposition 4.5.

Lemma 5.1. There are isomorphisms K_{8k+2}(ℤ̂_{2}) ≅ ℤ/2, K_{8k+4}(ℤ̂_{2}) ≅ ℤ/8 and K_{8k+6}(ℤ̂_{2}) ≅ ℤ/2 for all k ≥ 0.

Proof. In Proposition 4.2 we found nonzero Bockstein differentials β_{1}(ξ_{3}) = ∂(ξ_{3}(0)), β_{3}(∂(ξ_{6})) = ∂(ξ_{5}(0)) and β_{1}(ξ_{7}) = ∂(ξ_{7}(1)). The integral lifts of ∂(ξ_{3}(0)), ∂(ξ_{5}(0)) and ∂(ξ_{7}(1)) generate K_{∗}(ℤ̂_{2}) in degrees two, four and six, respectively.

The Adams v_{1}^{4}-action is induced by multiplication by σ̃_{16}, for which β_{k}(σ̃_{16}) = 0 when k ≤ 3, so it follows by regularity that there are nonzero Bockstein differentials β_{1}(v_{1}^{4k} · ξ_{3}) = v_{1}^{4k} · ∂(ξ_{3}(0)), β_{3}(v_{1}^{4k} · ∂(ξ_{6})) = v_{1}^{4k} · ∂(ξ_{5}(0)) and β_{1}(v_{1}^{4k} · ξ_{7}) = v_{1}^{4k} · ∂(ξ_{7}(1)).

The listed K-groups are known to be cyclic by (1.6), so this completes the proof. □

It remains to settle the degrees divisible by eight.

Definition 5.2. The symbol {3} ∈ K_{1}(ℤ̂_{2}) is represented by a unique homotopy class of infinite loop maps Q(S^{1}) → K(ℤ̂_{2}). Since red({3}) = 0 this map uniquely factors through K^{red}(ℤ̂_{2}). Since K^{red}(ℤ̂_{2}) agrees with its K-localization above degree one, a check on π_{1} shows that there is a unique extension of this map over L_{K}Q(S^{1}), and thus over B ImJ_{ℝ}. We obtain the following homotopy commutative diagram:

(5.3)  [figure: Q(S^{1}); Be_{ℝ}; {3}; B ImJ_{ℝ}; f_{ℝ}; K^{red}(ℤ̂_{2}); K(ℤ̂_{2})]

The extended map f_{ℝ}: B ImJ_{ℝ} → K^{red}(ℤ̂_{2}) then induces an isomorphism on π_{1}.

Lemma 5.4. The map f_{ℝ}: B ImJ_{ℝ} → K^{red}(ℤ̂_{2}) induces an isomorphism

K_{8k}(ℤ̂_{2}) ≅ π_{8k}(B ImJ_{ℝ})

for all k ≥ 1.

Proof. Let ι_{1} ∈ π_{1}Q(S^{0}) ≅ π_{1} ImJ_{ℝ} be the generator. Then f_{ℝ} maps i_{1}(ι_{1}) to i_{1}({3}) = ξ_{1} + ∂(ξ_{2}). Since f_{ℝ} is an infinite loop map it also maps i_{1}(σι_{1}) to i_{1}(σ{3}) = ∂(ξ_{9}(1)), and takes v_{1}^{4} ·i_{1}(ι_{1}) = σ̃_{16} ·i_{1}(ι_{1}) to v_{1}^{4} ·ξ_{1} +∂(ξ_{2}) = ξ_{9} +∂(ξ_{10}). See Propositions 4.2(c) and 4.3(d).

Thus f_{ℝ}: B ImJ_{ℝ} → K^{red}(ℤ̂_{2}) induces an injection on mod two homotopy in degrees eight and nine. The mod two homotopy of both the source and target of this map is v_{1}^{4}-periodic above degree two, so the mod two homomorphism is also injective in every degree 8k and 8k + 1 for k ≥ 1.

By a Bockstein argument, which uses that K_{8k}^{red}(ℤ̂_{2}) is finite and cyclic, it follows that π_{8k}(B ImJ_{ℝ}) maps isomorphically onto K_{8k}^{red}(ℤ̂_{2}) for all k ≥ 1. Clearly K_{8k}(ℤ̂_{2}) ≅ K_{8k}^{red}(ℤ̂_{2}). □

Combined, these lemmas prove the following result.

Theorem 5.5. The two-adically completed algebraic K-groups of the two-adic integers are given by

K_{∗}(ℤ̂_{2})_{2}^{∧} ≅ π_{∗}(ImJ_{ℂ})_{2}^{∧} ⊕ π_{∗}(B ImJ_{ℂ})_{2}^{∧} ⊕ π_{∗}(BBU)_{2}^{∧}.

Proof. It only remains to note that π_{∗}(B ImJ_{ℂ}) equals ℤ/2 for ∗ = 4k + 2, equals ℤ/8 for ∗ = 8k + 4, and equals π_{∗}(B ImJ_{ℝ}) for ∗ = 8k + 8 with k ≥ 0. □

The following results discuss the relationship of K(ℤ) and JK(ℤ), and the multiplicative structure on K_{∗}(ℤ̂_{2}).

Lemma 5.6. For any section ϕ: ΩJK(ℤ) → ΩK(ℤ) (to ΩΦ) the images of κ and ϕ(κ) in K_{5}(ℤ̂_{2}) agree. So we may take κ = ϕ(κ) in K_{5}(ℤ̂_{2}).

Proof. The classes κ and ϕ(κ) in K_{5}(ℤ) agree under Φ, hence differ by a torsion class, which maps to a torsion class in K_{5}(ℤ̂_{2}) ≅ ℤ̂_{2} ⊕ ℤ/2. The ℤ/2 is detected by the reduction map to π_{5}(ImJ_{ℂ}) = ℤ/2{κ}. We claim both κ and ϕ(κ) map to the generator under red. In the case of κ this was proven in Proposition 4.4(b). For the other case, note that η̃_{2}κ = i_{1}(σ) in JK_{7}(ℤ; ℤ/2), so η̃_{2}ϕ(κ) = i_{1}(σ) in K_{7}(ℤ; ℤ/2) by (2.4). Here red(i_{1}(σ)) ≠ 0 in π_{7}(ImJ_{ℂ}; ℤ/2), so red(ϕ(κ)) ≠ 0. Thus κ and ϕ(κ) agree when mapped to K_{5}(ℤ̂_{2}). □

Lemma 5.7. In the homotopy of K(ℤ̂_{2}) we have κη = 0, κη̃_{2} = i_{1}(σ) and λσ = 0.

Proof. The first properties follow from Lemmas 2.5 and 5.6. To see that λσ = 0 in K_{10}(ℤ̂_{2}) ≅ ℤ/2 it suffices to check that i_{1}(λσ) = 0. But ν̃_{4} · i_{1}(λσ) = i_{1}(σ^{2}) = 0 in K_{14}(ℤ̂_{2}) by Proposition 4.2(b) and [Mit]. Hence i_{1}(λσ) = 0 by Proposition 1.4. □

We also inspect the classes inducing v_{1}^{4}-periodicity a little more closely.

Lemma 5.8. The image in K_{8}(ℤ̂_{2}; ℤ/4) of ˜4σ_{4} does not depend upon the choice of lift over j_{2} of 4σ. The classes ˜4σ_{4} and ν̃_{4}^{2} differ at most by a multiple of i_{2}(σ{3}) coming from K_{8}(ℤ̂_{2})/4 = ℤ/4{σ{3}}. By altering the choice of lift ν̃_{4} of ν over j_{2} by a multiple of i_{2}(λ{3}) from K_{4}(ℤ̂_{2})/4 = ℤ/4{λ{3}} we may arrange that either ν̃_{4}^{2} = ˜4σ_{4} or ν̃_{4}^{2} = ˜4σ_{4} + i_{2}(σ{3}) (but we do not know which).

The nonpresence or presence of the term i_{2}(σ{3}) corresponds to whether the v_{1}^{4}-action on K(ℤ̂_{2}; ℤ/2) admits a square root or not, respectively.

Proof. The choice in ˜4σ_{4} stems from elements in π_{8}Q(S^{0}) = ℤ/2{ση, ν̄} in Toda’s notation, all of which map to zero in K_{8}(ℤ) by [Wal] and [Mit], respectively.

By Proposition 4.3(b) the classes ˜4σ_{4} and ν̃_{4}^{2} have the same images under j_{2}, implying the second claim.

Changing ν̃_{4} by multiple of i_{2}(λ{3}) from K_{4}(ℤ̂_{2})/4 alters ν̃_{4}^{2} by a multiple of 2λ{3}ν̃_{4} = 2i_{2}(σ{3}) since λ^{2} = 0 in K_{6}(ℤ̂_{2}). So the indeterminacy in ν̃_{4}^{2} equals even multiples of i_{2}(σ{3}). □

6. An extension argument

We now proceed towards our second aim of determining the homotopy type of K(ℤ̂_{2}) as an infinite loop space. First we extend the map f_{ℝ} from (5.3) over the (delooped) complexification map Bc: B ImJ_{ℝ} → B ImJ_{ℂ}, to obtain a map f_{ℂ}: B ImJ_{ℂ} → K^{red}(ℤ̂_{2}) that later will be seen to induce a split injection on homotopy. In Section 7 we will identify the infinite loop space cofiber of f_{ℂ} as BBU, thus determining K(ℤ̂_{2}) up to extensions. The extension will be characterized in Section 8.

In this section we apply Hopkins, Mahowald and Sadofsky’s paper [HMS] on constructions of “invertible spectra” in the K/2-local category to recognize the homotopy fiber W of Bc as such an invertible spectrum, and to show that the composite map W → B ImJ_{ℝ} → K^{red}(ℤ̂_{2}) is null homotopic. The existence of the extension then follows.

Definition 6.1. Let W be the homotopy fiber of the complexification map

Bc: B ImJ_{ℝ} → B ImJ_{ℂ}.

We use the following commutative diagram:

(6.2)  [figure: W; Q(S^{1}); Be_{ℝ}; Be_{ℂ}; B ImJ_{ℝ}; Bc; f_{ℝ}; K^{red}(ℤ̂_{2}); K(ℤ̂_{2}); B ImJ_{ℂ}; f_{ℂ}]

Theorem 6.3. The composite

W ⟶ B ImJ_{ℝ} ⟶^{f_{ℝ}} K^{red}(ℤ̂_{2})

is null-homotopic as a map of infinite loop spaces. Hence there exists an infinite loop map f_{ℂ}: B ImJ_{ℂ} → K^{red}(ℤ̂_{2}) extending f_{ℝ} over Bc and extending {3}: Q(S^{1}) → K^{red}(ℤ̂_{2}) over Be_{ℂ} = Bc ∘ Be_{ℝ}. The map f_{ℂ} induces an isomorphism on π_{1}.

Proof. We use spectrum notation in this proof.

First note that W is 2-connected, so the homotopy classes of maps W → K^{red}(ℤ̂_{2}) are in bijection with the homotopy classes of maps W → L_{K}K^{red}(ℤ̂_{2}), since K^{red}(ℤ̂_{2}) agrees with its K-localization above degree one. Furthermore such maps are clearly in bijection with maps L_{K}W → L_{K}K^{red}(ℤ̂_{2}).

Next note that L_{K}W is equivalent to the spectrum Σ^{2}X_{1/3} of [HMS]. These authors define X_{λ} with λ ∈ ℤ̂_{2} as the homotopy fiber of ψ^{3} − λ acting on the real periodic topological K-theory spectrum KO

X_{λ} ⟶ KO ⟶^{ψ^{3}−λ} KO.

Then X_{1/3} is clearly also equivalent to the homotopy fiber of 3ψ^{3} − 1. Recall the Puppe (co-)fiber sequence of spectra

(6.4)  ΣKO ⟶^{η} KO ⟶^{c} KU ⟶^{Σ^{2}(r)b^{−1}} Σ^{2}KO

where b: Σ^{2}KU → KU is the Bott periodicity equivalence, c: KO → KU and r: KU → KO denote complexification and realification, and η denotes smashing with η: S^{1} → S^{0}. The real and complex Adams operations ψ^{3} are compatibly defined on KO and KU with respect to the maps c and r, while b∘Σ^{2}(3ψ^{3}) ≃ ψ^{3}∘b. Thus we have the following diagram of horizontal and vertical fiber sequences of spectra:

[figure: Σ^{2}X_{1/3}; ΣJO; Σc; ΣJU; Σ^{3}X_{1/3}; Σ^{2}KO; Ση; Σ^{2}3ψ^{3}−1; ΣKO; Σc; Σψ^{3}−1; ΣKU; Σ^{3}rΣb^{−1}; Σψ^{3}−1; Σ^{3}KO; Σ^{3}3ψ^{3}−1; Σ^{2}KO; Ση; ΣKO; Σc; ΣKU; Σ^{3}rΣb^{−1}; Σ^{3}KO]

The fiber sequence W → B ImJ_{ℝ} → B ImJ_{ℂ} maps to the top horizontal sequence, and L_{K}B ImJ_{ℝ} ≃ ΣJO and L_{K}B ImJ_{ℂ} ≃ ΣJU, so L_{K}W ≃ Σ^{2}X_{1/3} as claimed.

Note that

π_{∗}Σ^{2}X_{1/3} ≅ \{ \begin{matrix}ℤ/2 & ∗ ≡ 1, 2, 4, 5 mod 8, \\ ℤ/4 & ∗ ≡ 3 mod 8, \\ 0 & otherwise \end{matrix}

consists of one “lightning flash” in every 8 degrees, which is one half of the v_{1}^{4}- periodic homotopy of a (two-cell) mod two Moore spectrum, namely of π_{∗}(L_{K}S^{1}/2). Each spectrum X_{λ} behaves like a one-cell spectrum in this respect.

In the proof of Theorem 5.1 of [HMS], Hopkins, Mahowald and Sadofsky construct a map S^{1}/2 → Σ^{2}X_{1/3} = X_{0}, whose natural extension f_{0} over L_{K}S^{1}/2 induces a surjection on homotopy. Inductively, for all i ≥ 0 they construct maps f_{i}: L_{K}S^{1}/2 → X_{i} which are surjective on homotopy, and then define X_{i+1} as the spectrum level cofiber of f_{i}. Then each cofiber map g_{i}: X_{i} → X_{i+1} induces the zero map on homotopy, and π_{∗}(hocolim_{i} X_{i}) = colim_{i} π_{∗}(X_{i}) = 0, so hocolim_{i} X_{i} ≃ ∗. Let h_{i}: X_{i+1} → L_{K}S^{2}/2 be the connecting maps in the Puppe sequence.

Rephrased, this construction eventually annihilates Σ^{2}X_{1/3} = X_{0} by inductively attaching cones on K-localized mod two Moore spectra until the remainder in the colimit is contractible. We obtain the following tower of spectra, with cofiber sequences mapping down, across and down again:

[figure: L_{K}S^{1}/2; f_{0}; L_{K}S^{1}/2; f_{1}; L_{K}S^{1}/2; f_{2}; Σ^{2}X_{1/3}; g_{0}; k_{0}; X_{1}; h_{0}; g_{1}; X_{2}; h_{1}; …; hocolim_{i} X_{i} ≃ ∗; L_{K}S^{2}/2; L_{K}S^{2}/2; L_{K}K^{red}(ℤ̂_{2})]

From part (iii) of the cited theorem of [HMS] it follows that each composite map

δ_{i} = h_{i−1} ∘ f_{i}: L_{K}S^{1}/2 ⟶ L_{K}S^{2}/2

fits into a diagram

[figure: L_{K}S^{1}; α_{−1}; i_{1}; L_{K}S^{2}; i_{1}; L_{K}S^{1}/2; δ_{i}; L_{K}S^{2}/2.]

Here α_{−1} generates π_{−1}L_{K}S^{0} ≅ ℤ̂_{2} and without K-localizing v_{1}^{4}α_{−1} is realized in π_{7}Q(S^{0}) as the class of the Hopf map σ.

Now we claim that the given map k_{0} : X_{0} = Σ^{2}X_{1/3} → ΣJO → L_{K}K^{red}(ℤ̂_{2}) inductively extends over each map X_{i} → X_{i+1}, and so in the limit extends over hocolim_{i} X_{i} ≃ ∗, and thus is null-homotopic.

To see this, inductively assume we have a map k_{i}: X_{i} → L_{K}K^{red}(ℤ̂_{2}) such that k_{i} ∘ f_{i} is null-homotopic. (We will check the case i = 0 after the following lemma.) Then k_{i} extends over g_{i} to a map k_{i+1} : X_{i+1} → L_{K}K^{red}(ℤ̂_{2}). Two choices of extensions differ by a map ℓ ∘ h_{i} where ℓ: L_{K}S^{2}/2 → L_{K}K^{red}(ℤ̂_{2}). We now ask whether the composite k_{i+1} ∘ f_{i+1} is null-homotopic.

This map is determined by its restriction to S^{1}/2, which amounts to a class in K_{2}^{red}(ℤ̂_{2}; ℤ/2). Likewise ℓ amounts to a class in K_{3}^{red}(ℤ̂_{2}; ℤ/2). Altering k_{i+1} by changing ℓ alters k_{i+1} ∘ f_{i+1} by the change in the composite ℓ ∘ δ_{i+1}. Hence if we show that the map induced by precomposition with δ_{i+1}:

δ_{i+1}^{∗}: K_{3}^{red}(ℤ̂_{2}; ℤ/2) ⟶ K_{2}^{red}(ℤ̂_{2}; ℤ/2)

is surjective, then it follows that k_{i+1} may be chosen so that k_{i+1} ∘ f_{i+1} is trivial, as required. Here the target group is generated by the integral class i_{1}(η^{2}), so this follows from the following inserted lemma.

Lemma 6.5. Let f_{3} ∈ K_{3}^{red}(ℤ̂_{2}) be the integral generator represented mod two by ξ_{3}(0) + ∂(ξ_{4}), and let μη = μ_{10} ∈ π_{10}Q(S^{0}) be the element detected by KO-theory. Then in K_{10}^{red}(ℤ̂_{2})

σ · f_{3} = μη

so after K-localizing we have α_{−1} · f_{3} = η^{2}. With mod two coefficients this asserts that δ_{i+1}^{∗}(i_{1}(f_{3})) = i_{1}(η^{2}).

Proof. We can also write f_{3} = λ + g_{3} in K_{3}(ℤ̂_{2}). Then σλ = 0 in K_{10}(ℤ̂_{2}) by Lemma 5.7, while σg_{3} is represented mod two by

σ · ∂(ξ_{4}) = ∂(σ · ρν̃_{4}) = ∂(ξ_{11}(1)).

We claim this represents μη. For acting by ν̃_{4}^{2} on β_{1}(ξ_{3}) = ∂(ξ_{3}(0)) yields β_{1}(ξ_{11}) = ∂(ξ_{11}(1)). So ∂(ξ_{11}(1)) is integral and generates the two-torsion in K_{10}(ℤ̂_{2}), i.e., it is i_{1}(μη). This proves the displayed formula. The other two claims follow directly from this. □

To complete the proof of Theorem 6.3 we must check the initial case of the induction argument, namely that k_{0} ∘ f_{0} : L_{K}S^{1}/2 → L_{K}K^{red}(ℤ̂_{2}) is null-homotopic. Again this amounts to a claim about a map S^{1}/2 → K^{red}(ℤ̂_{2}) or a class in K_{2}^{red}(ℤ̂_{2}; ℤ/2) ≅ ℤ/2{i_{1}(η^{2})}. But the given map factors through W → K^{red}(ℤ̂_{2}) and W is 2-connected, so the map is inessential.

Hence by induction over i each map k_{i}: X_{i} → L_{K}K^{red}(ℤ̂_{2}) extends over X_{i+1}, and so Σ^{2}X_{1/3} = X_{0} → L_{K}K^{red}(ℤ̂_{2}) extends over hocolim_{i} X_{i} ≃ ∗, and must be inessential. Thus the given map W → K^{red}(ℤ̂_{2}) is null-homotopic, and this completes the proof of the theorem. □

7. Two (co)fiber sequences

Now we know that f_{ℝ}: B ImJ_{ℝ} → K^{red}(ℤ̂_{2}) extends over Bc. We wish to recognize the infinite loop space cofiber of the extension f_{ℂ} as BBU. Let ι_{1} ∈ π_{1}B ImJ_{ℂ} be the generator.

Proposition 7.1. Any infinite loop map f: B ImJ_{ℂ} → K^{red}(ℤ̂_{2}) mapping ι_{1} to {3} on π_{1} induces an injection on mod two homotopy and a split injection on integral homotopy in all degrees. Explicitly on mod two homotopy

f_{∗}(v_{1}^{k}ι_{1}) = ξ_{2k+1} + ∂(ξ_{2k+2})
f_{∗}(e_{1}v_{1}^{k}ι_{1}) = ∂(ξ_{2k+3}(e))

for 0 ≤ k ≤ 3. Let X be the infinite loop space cofiber of such a given map, so there is a fiber sequence

B ImJ_{ℂ} ⟶ K^{red}(ℤ̂_{2}) ⟶ X.

Then π_{∗}(X; ℤ/2) ≅ π_{∗}(BBU; ℤ/2) as free v_{1}^{4}-modules of rank four. Hence X agrees with its K-localization above degree one, and has homotopy groups π_{∗}X ≅ π_{∗}BBU.

Proof. By assumption f_{∗}(i_{1}(ι_{1})) = i_{1}({3}) = ξ_{1} + ∂(ξ_{2}). Multiplying by η gives f_{∗}(e_{1}ι_{1}) = i_{1}(η{3}) = i_{1}(η^{2}) = ∂(ξ_{3}(0)) by Proposition 4.2. Here e_{1}ι_{1} admits an integral lift, so we can multiply by η̃_{2} and obtain f_{∗}(e_{1}v_{1}ι_{1}) = η̃_{2} · η^{2} = ξ_{2} · ∂(ξ_{3}(0)) = ∂(ξ_{5}(0)) = ∂(κ) = λ{3}. Again e_{1}v_{1}ι_{1} is integral, so multiplying by η̃_{2} gives f_{∗}(e_{1}v_{1}^{2}ι_{1}) = η̃_{2} · λ{3} = ξ_{2} · ∂(ξ_{5}(0)) = ∂(ξ_{7}(1)) = ∂(σ) = κ{3}. A third multiplication by η̃_{2}, or multiplying the original equation by σ, gives f_{∗}(e_{1}v_{1}^{3}ι) = ∂(ξ_{9}(1)) = σ{3}.

Multiplying f_{∗}(ι_{1}) = {3} by η̃_{2} gives f_{∗}(v_{1}ι_{1}) = η̃_{2}{3} = x. Then β_{1}(x) = ∂(ξ_{3}(0)) ≠ 0, red(x) = 0 and tr(x) = 0. (Here tr denotes the Bökstedt trace K(ℤ̂_{2}) → T (ℤ̂_{2}) ≅ T (ℤ).) The only possibility is x = ξ_{3} + ∂(ξ_{4}).

Let y = f_{∗}(v_{1}^{2}ι_{1}). Then β_{3}(y) = ∂(ξ_{5}(0)) ≠ 0 and red(y) = 0, so either y = ξ_{5}(0) + ∂(ξ_{6}) or y = ξ_{5} + ∂(ξ_{6}). Let z = f_{∗}(v_{1}^{3}ι_{1}). Then β_{1}(z) = ∂(ξ_{7}(1)) ≠ 0 and red(z) = 0, so either z = ξ_{7}(1) + ξ_{7} or z = ξ_{7} + ∂(ξ_{8}).

The class y admits a mod eight lift y′ = f_{∗}(λ̃_{8} · ι_{1}), so multiplying with η̃_{2} makes sense, and gives η̃_{2} · y′ = z. Then either y′ ≡ κ + λ̃_{8} · {5} mod two with η̃_{2} · y′ = ξ_{7}(1) + ∂(ξ_{8}), or y′ ≡ λ̃_{8} · {3} mod two with η̃_{2} · y′ = ξ_{7} + ∂(ξ_{8}). Comparing with the expressions for z, the only consistent possibility is y = ξ_{5} + ∂(ξ_{6}) and z = ξ_{7} + ∂(ξ_{8}).

Hence π_{∗}(f; ℤ/2) is an injection for all 1 ≤ ∗ ≤ 8. By v_{1}^{4}-periodicity, this holds in all degrees. Hence π_{∗}(f) induces a split injection in all degrees. □

Lemma 7.2. In K_{∗}^{red}(ℤ̂_{2}; ℤ/2) the classes ξ_{3}(0) + ∂(ξ_{4}), ξ_{5}(0) + ξ_{5}, ξ_{7}(1) + ∂(ξ_{8}) and ξ_{9}(1)+ξ_{9} admit integral lifts. Their products with η̃_{2} are ξ_{5}(0)+∂(ξ_{6}), ξ_{7}(1)+ξ_{7}, ξ_{9}(1) + ∂(ξ_{10}) and ξ_{11}(1) + ξ_{11}, respectively.

In π_{∗}(X; ℤ/2) all classes are integral, and multiplication by η̃_{2} is an isomorphism in all degrees ∗ ≥ 3.

Proof. The integral lifts are λ + g_{3}, κ + t_{5}, σ + g_{7} and g_{9} + μ, in the notation of Proposition 4.2. Their products with η̃_{2} are computed as in the proof of Proposition 7.1. The claims for X then follow, by dividing K_{∗}^{red}(ℤ̂_{2}; ℤ/2) by the image of π_{∗}(f_{ℂ}; ℤ/2) to obtain π_{∗}(X; ℤ/2) in low degrees, and using v_{1}^{4}-periodicity to obtain the general result. □

Hence the infinite loop space cofiber of f_{ℂ} satisfies the hypothesis of the following result.

Proposition 7.3. Let X be a two-complete infinite loop space with π_{∗}X ≅ π_{∗}BBU. If multiplication by η̃_{2}

π_{i}(X; ℤ/2) ⟵^{≅} π_{i}(X)/2 ⟶^{η̃_{2}} π_{i+2}(X; ℤ/2)

is injective in all degrees, then X ≃ BBU as infinite loop spaces.

Proof. Each “Postnikov segment” X[i, i + 2] of X for i ≥ 3 odd has nontrivial k-invariant, since η̃_{2} : π_{i}(X[i, i + 2]; ℤ/2) → π_{i+2}(X[i, i + 2]; ℤ/2) is assumed to be nonzero. Hence there is a fiber sequence

K(ℤ, i + 2) ⟶ X[i, i + 2] ⟶ K(ℤ, i) ⟶^{Q̄_{1}} K(ℤ, i + 3)

where Q̄_{1} lifts

Q_{1} = [Sq^{1}, Sq^{2}] = Sq^{3} + Sq^{2}Sq^{1} : K(ℤ/2, i) ⟶ K(ℤ/2, i + 3).

Then as in the proof of Proposition 2.1 of [AP], the spectrum cohomology of X satisfies H^{∗}(X; ℤ/2) ≅ A/(AQ_{0} + AQ_{1}) where A is the mod two Steenrod algebra and Q_{0} = Sq^{1}. Thus by Theorem 1.1 of loc. cit., there exists a two-adic equivalence X ≃ BBU. □

Hence we have established the following fiber sequences of infinite loop spaces

(7.4)

K^{red}(ℤ̂_{2}) ⟶ K(ℤ̂_{2}) ⟶^{red} ImJ_{ℂ}
B ImJ_{ℂ} ⟶^{f_{ℂ}} K^{red}(ℤ̂_{2}) ⟶ BBU.

We can now identify well-defined choices of integral generators for K_{∗}(ℤ̂_{2}; ℤ/2).

Proposition 7.5. Let t_{n} generate the cyclic torsion subgroup in K_{n}(ℤ̂_{2}) for n ≥ 2, and let f_{n} generate the free cyclic group K_{n}^{red}(ℤ̂_{2}) ⊂ K_{n}(ℤ̂_{2}) for n ≥ 3 odd. The mod two representatives of these classes begin as follows:

n  2  3  4  5
i_{1}(t_{n}) ∂(ξ_{3}(0))  ξ_{3}(0)  ∂(ξ_{5}(0))  ξ_{5}
i_{1}(f_{n})  ξ_{3}(0) + ∂(ξ_{4})  ξ_{5}(0) + ξ_{5}
n  6  7  8  9
i_{1}(t_{n}) ∂(ξ_{7}(1))  ξ_{7}(1)  ∂(ξ_{9}(1))  ξ_{9}
i_{1}(f_{n})  ξ_{7}(1) + ∂(ξ_{8})  ξ_{9}(1) + ξ_{9}

In all higher degrees i_{1}(t_{8k+n}) = v_{1}^{4k} · t_{n} and i_{1}(f_{8k+n}) = v_{1}^{4k} · f_{n}.

Proof. In even degrees the classes ∂(ξ_{3}(0)), ∂(ξ_{5}(0)), ∂(ξ_{7}(1)) and ∂(ξ_{9}(1)), and their multiples by v_{1}^{4k}, represent the image from π_{∗}(B ImJ_{ℂ}; ℤ/2) in K_{∗}^{red}(ℤ̂_{2}; ℤ/2) and K_{∗}(ℤ̂_{2}; ℤ/2), by Proposition 7.1. Hence these are the integral classes in even degrees of K_{∗}(ℤ̂_{2}; ℤ/2).

In degree 8k + 3 the integral classes are v_{1}^{4k} · ξ_{3}(0) and v_{1}^{4k} · ∂(ξ_{4}), since β_{3}(ξ_{4}) = ξ_{3}(0) and β_{1}(∂(ξ_{4})) = 0 while β_{1}(ξ_{3}) ≠ 0. These differentials propagate up by multiplication by v_{1}^{4k}, because multiplication by v_{1}^{4} respects β_{k} for k ≤ 3.

In degree 8k + 5 the integral classes are v_{1}^{4k} · ξ_{5}(0) and v_{1}^{4k} · ξ_{5}, since β_{1}(ξ_{4}) = ξ_{5} and β_{3}(ξ_{5}(0)) = 0 while β_{3}(∂(ξ_{6})) ≠ 0. These differentials also propagate up by multiplication by v_{1}^{4k}.

In degree 8k + 7 the class v_{1}^{4k} · ξ_{7}(1) is integral in K_{∗}(ℤ̂_{2}; ℤ/2). This is because i_{1}(σ) ∈ π_{7}(Q(S^{0}); ℤ/2) maps to ξ_{7}(1) under the unit map, and its multiples by v_{1}^{4k} are all integral in π_{8k+7}(Q(S^{0}); ℤ/2). Also v_{1}^{4k} ·∂(ξ_{8}) is integral, since β_{1}(∂(ξ_{8})) = 0 while β_{1}(ξ_{7}) ≠ 0. Again the β_{1}-differential propagates.

In degree 8k + 9 the integral classes are v_{1}^{4k} · ξ_{9}(1) and v_{1}^{4k} · ξ_{9}. For β_{1}(ξ_{10}) = ξ_{9} translates up by multiplication by v_{1}^{4k}. Also ξ_{9}(1) is the image of ση̃_{2} ∈ π_{9}(Q(S^{0}); ℤ/2), so v_{1}^{4k} · ξ_{9}(1) is the image of σx ∈ π_{8k+9}(Q(S^{0}); ℤ/2), where j_{1}(x) = μ_{8k+1} ∈ π_{8k+1}Q(S^{0}) is the Adams μ-element [A]. Then j_{1}(σx) = σμ_{8k+1}, which maps to zero in K_{∗}(ℤ), by [Wal]. Hence v_{1}^{4k} ·ξ_{9}(1) is integral in K_{∗}(ℤ̂_{2}; ℤ/2).

The given formulas now follow from Proposition 4.6. □

We can also extend Proposition 4.2(c).

Lemma 7.6. In the first two cases below assume k ≥ 1. In the remaining cases assume k ≥ 0.

In K_{8k}(ℤ̂_{2}; ℤ/2) the class v_{1}^{4(k−1)} · ∂(ξ_{9}(1)) is integral, while β_{v}(v_{1}^{4(k−1)} · ξ_{8}) = v_{1}^{4(k−1)} · ξ_{7}(1) with v = v_{2}(k) + 4.

In K_{8k+1}(ℤ̂_{2}; ℤ/2) the classes v_{1}^{4(k−1)} · ξ_{9}(1) and v_{1}^{4(k−1)} · ξ_{9} are integral, while β_{v}(v_{1}^{4(k−1)} · ∂(ξ_{10})) = v_{1}^{4(k−1)} · ∂(ξ_{9}(1)) with v = v_{2}(k) + 4. An integral lift of the class v_{1}^{4(k−1)} · (ξ_{9}(1) + ξ_{9}) generates K_{8k+1}(ℤ̂_{2}) modulo torsion.

In K_{8k+2}(ℤ̂_{2}; ℤ/2) the class v_{1}^{4k} · ∂(ξ_{3}(0)) is integral, while β_{1}(v_{1}^{4k} · ξ_{2}) = v_{1}^{4k} · ξ_{1}. In K_{8k+3}(ℤ̂_{2}; ℤ/2) the classes v_{1}^{4k} · ξ_{3}(0) and v_{1}^{4k} · (ξ_{3}(0) + ∂(ξ_{4})) are integral,

while β_{1}(v_{1}^{4k} · ξ_{3}) = v_{1}^{4k} · ∂(ξ_{3}(0)). An integral lift of the class v_{1}^{4k} · ∂(ξ_{4}) generates K_{8k+3}(ℤ̂_{2}) modulo torsion.

In K_{8k+4}(ℤ̂_{2}; ℤ/2) the class v_{1}^{4k}·∂(ξ_{5}(0)) is integral, while β_{3}(v_{1}^{4k}·ξ_{4}) = v_{1}^{4k}·ξ_{3}(0). In K_{8k+5}(ℤ̂_{2}; ℤ/2) the classes v_{1}^{4k} · ξ_{5}(0) and v_{1}^{4k} · ξ_{5} are integral, while β_{3}(v_{1}^{4k} ·

∂(ξ_{6})) = v_{1}^{4k} · ∂(ξ_{5}(0)). An integral lift of the class v_{1}^{4k} · (ξ_{5}(0) + ξ_{5}) generates K_{8k+5}(ℤ̂_{2}) modulo torsion.

In K_{8k+6}(ℤ̂_{2}; ℤ/2) the class v_{1}^{4k} · ∂(ξ_{7}(1)) is integral, while β_{1}(v_{1}^{4k} · ξ_{6}) = v_{1}^{4k} · ξ_{5}. In K_{8k+7}(ℤ̂_{2}; ℤ/2) the classes v_{1}^{4k} ·ξ_{7}(1) and v_{1}^{4k} ·∂(ξ_{8}) are integral, while β_{1}(v_{1}^{4k} ·

ξ_{7}) = v_{1}^{4k} · ∂(ξ_{7}(1)). An integral lift of the class v_{1}^{4k} · (ξ_{7}(1) + ∂(ξ_{8})) generates K_{8k+7}(ℤ̂_{2}) modulo torsion.

Proof. Combine Proposition 4.2 and Theorem 5.5 with Proposition 7.5. □

The following comparison theorem is useful.

Theorem 7.7. The natural map K_{∗}(ℤ)_{2}^{∧} → K_{∗}(ℤ̂_{2})_{2}^{∧} induces an isomorphism modulo torsion in degrees ∗ = 4i + 1 for all i ≥ 1.

Proof. Consider the maps

SU ⟶ JK(ℤ) ⟵^{Φ} K(ℤ) ⟶ K(ℤ̂_{2})

and the space level section ϕ: ΩJK(ℤ) → ΩK(ℤ). The generators γ_{4i−1} and γ_{4i+1} in π_{∗}(SU) map to classes λ_{4i−1} and κ_{4i+1} in JK_{∗}(ℤ). Here λ_{4i−1} generates the finite cyclic group JK_{4i−1}(ℤ), while κ_{4i+1} is of infinite order and generates the group JK_{4i+1}(ℤ) modulo torsion. For i even λ_{4i−1} is in the image from π_{4i−1}(ImJ_{ℝ}), while for i odd 2λ_{4i−1} is in this image. The relation γ_{4i−1}η̃_{2} = i_{1}(γ_{4i+1}) in π_{∗}(SU; ℤ/2) translates to ϕ(λ_{4i−1})η̃_{2} = i_{1}(ϕ(κ_{4i+1})) in K_{4i+1}(ℤ; ℤ/2). For i ≥ 2 this follows by naturality of multiplication by η̃_{2} with respect to ϕ, because this multiplication is unstably defined on π_{n} with n ≥ 3. The case i = 1 was treated separately in Lemma 5.6.

Now we map across to K(ℤ̂_{2}). We claim that λ_{4i−1} maps to a generator of the torsion in K_{4i−1}(ℤ̂_{2}) for all i. For i even this follows from Lemma 3.6, while for i odd the same lemma shows that 2λ_{4i−1} maps to twice a generator. Since the torsion group has order eight in these degrees, this suffices to prove the claim. Hence ϕ(λ_{8k+3}) is represented mod two by v_{1}^{4k} ·ξ_{3}(0), and ϕ(λ_{8k+7}) is represented mod two by v_{1}^{4k} · ξ_{7}(1), by Proposition 7.5. The relation η̃_{2} · ϕ(λ_{8k+3}) = i_{1}(ϕ(κ_{8k+5})) implies that ϕ(κ_{8k+5}) is represented mod two by v_{1}^{4k}·η̃_{2}ξ_{3}(0) = v_{1}^{4k}·ξ_{5}(0), and thus generates K_{8k+5}(ℤ̂_{2}) modulo torsion. Likewise the relation η̃_{2} · ϕ(λ_{8k+7}) = i_{1}(ϕ(κ_{8k+9})) implies that ϕ(κ_{8k+9}) is represented mod two by v_{1}^{4k} ·η̃_{2}ξ_{7}(1) = v_{1}^{4k} ·ξ_{9}(1), and thus generates K_{8k+9}(ℤ̂_{2}) modulo torsion.

Hence ϕ(κ_{4i+1}) generates K_{4i+1}(ℤ) modulo torsion, and maps to a class that generates K_{4i+1}(ℤ̂_{2}) modulo torsion. This proves the theorem. □

8. K(ℤ̂_{2}) as an infinite loop space

We use the multiplicative relations η^{2} ≠ 0 and σf_{3} = μη to identify the extensions in (7.4). Thus we determine the two-completed algebraic K-theory of the two-adic integers as an infinite loop space.

Theorem 8.1. The homotopy type of K(ℤ̂_{2}) as an infinite loop space is determined by the following diagram of infinite loop space fiber sequences

[figure: B ImJ_{ℂ}; f_{ℂ}; K^{red}(ℤ̂_{2}); K(ℤ̂_{2}); red; ImJ_{ℂ}; BBU]

where the connecting maps are given as follows:

The group of infinite loop maps BU → B ImJ_{ℂ} that induce the zero homomorphism on homotopy is free on one generator as a module over the two-adic integers. The vertical connecting map ∂_{1} : BU → B ImJ_{ℂ} represents a generator of this group.

The group of infinite loop maps Ω ImJ_{ℂ} → K^{red}(ℤ̂_{2}) is also free on one generator as a module over the two-adic integers, and the horizontal connecting map ∂_{2} : Ω ImJ_{ℂ} → K^{red}(ℤ̂_{2}) represents a generator of this group.

Any two generators of either group differ (multiplicatively) by a two-adic unit, hence by a two-adic automorphism of the spaces ImJ_{ℂ}, B ImJ_{ℂ} or BBU involved. Thus this theorem characterizes K(ℤ̂_{2}) as an infinite loop space, up to homotopy equivalence.

Proof. We begin with the extension B ImJ_{ℂ} → K^{red}(ℤ̂_{2}) → BBU, classified by an infinite loop map ∂_{1}. We know that ∂_{1} induces the zero map on homotopy since f_{ℂ} induces injections on homotopy. Consider the diagram

[figure: BU; ∂_{1}; ℤ × BU; ψ^{3}−1; BU; B∂; B ImJ_{ℂ}; Bi; U.]

There are no essential (infinite loop) maps BU → U, so ∂_{1} factors over an infinite loop map BU → BU. By [MST] such maps form a suitably completed group-ring ℤ̂_{2}[[ℤ̂_{2}^{×}]] and can be written as certain infinite series in the Adams operations ψ^{k} with k ∈ ℤ̂_{2}^{×}. The maps BU → B ImJ_{ℂ} are thus the quotient of this ring by the closed ideal (ψ^{3} −1) of maps factoring through ψ^{3} −1. In this quotient any ψ^{k} with k = 3^{i}, or more generally k ∈ (3) ⊂ ℤ̂_{2}^{×}, is identified with 1. Since ℤ̂_{2}^{×} ≅ ℤ̂_{2} ⊕ ℤ/2 with 3 generating the first summand and −1 generating the second, it follows that the quotient ring in question is ℤ̂_{2}[ℤ/2], which is additively free over ℤ̂_{2} on the two generators 1 = ψ^{1} and t = ψ^{−1} (complex conjugation).

We may thus write ∂_{1} = B∂ ∘ (u + vψ^{−1}) with u, v ∈ ℤ̂_{2}. Now we use that (∂_{1})_{∗} = 0. In degree 2n this map (∂_{1})_{∗} multiplies by u + (−1)^{n}v, so for n odd (when π_{2n}(B ImJ_{ℂ}) = ℤ/2) we find u − v ≡ 0 mod two, while for n even (when π_{2n}(B ImJ_{ℂ}) ≅ ℤ/2^{i} with i = v_{2}(n) + 2) we find u + v ≡ 0 mod 2^{i}. Hence twoadically u + v = 0 and ∂_{1} = u · B∂ ∘ (1 − ψ^{−1}). Thus the extension is determined by the two-adic coefficient u, and we claim that u is a unit. This will determine the extension and the infinite loop space homotopy type of K^{red}(ℤ̂_{2}), up to equivalence.

We claim that u is not divisible by two. If it were, we could factor ∂_{1} as 2y with π_{∗}(y) = 0, and get a map of fiber sequences

[figure: BU; y; B ImJ_{ℂ}; 2; Y; BBU; BU; ∂_{1}; B ImJ_{ℂ}; K^{red}(ℤ̂_{2}); BBU.]

Consider the product σ · f_{3} = μη in K_{10}^{red}(ℤ̂_{2}), from Lemma 5.5. Here f_{3} is represented on π_{3}(BBU), so σ · f_{3} is defined in π_{∗}(Y) and lifts to π_{∗}(B ImJ_{ℂ}) in the upper row. Hence μη should be divisible by two, which it is not. This contradiction shows that u is a unit. (By applying a two-adic automorphism to our identification of the infinite loop space cofiber of B ImJ_{ℂ} → K^{red}(ℤ̂_{2}) with BBU we may arrange to have ∂_{1} = B∂ ∘ (1 − ψ^{−1}).)

Next consider the extension K^{red}(ℤ̂_{2}) → K(ℤ̂_{2}) → ImJ_{ℂ}, classified by an infinite loop map ∂_{2} : Ω ImJ_{ℂ} → K^{red}(ℤ̂_{2}). Its delooping B∂_{2} fits into a diagram

[figure: ImJ_{ℂ}; B∂_{2}; b; B^{2} ImJ_{ℂ}; Bf_{ℂ}; BK^{red}(ℤ̂_{2}); B^{3}U.]

Consider maps from the defining fiber sequence

(8.2)  [figure: U; ∂; ImJ_{ℂ}; i; ℤ × BU; ψ^{3}−1; BU]

to BU. There are no essential infinite loop maps U → BU. The infinite loop maps ℤ × BU → BU are series in the ψ^{k} with k ∈ ℤ̂_{2}^{×}. Modulo those which extend over ψ^{3} − 1 these can be written as u + vψ^{−1} for some u, v ∈ ℤ̂_{2} (typically different from the u, v above). Furthermore the map to BU must be zero on π_{0}, so u + v = 0. Hence the infinite loop maps ImJ_{ℂ} → BU are all uniquely expressible as u · (1 − ψ^{−1})i with u ∈ ℤ̂_{2}. We write [ImJ_{ℂ}, BU] ≅ ℤ̂_{2}{(1 − ψ^{−1})i}.

The unit map Q(S^{0}) → ImJ_{ℂ} is 2-connected, so H_{spec}^{∗}(ImJ_{ℂ}) = 0 for ∗ = 1, 2, and so [ImJ_{ℂ}, B^{3}U] ≅ [ImJ_{ℂ}, BU] by the natural map B^{3}U ≃ BSU → BU. We also write (1 − ψ^{−1})i for the generator of [ImJ_{ℂ}, B^{3}U]. Hence b = u · (1 − ψ^{−1})i for some unique u ∈ ℤ̂_{2}.

We claim that there are no essential maps a: ImJ_{ℂ} → B^{2} ImJ_{ℂ}, so B∂_{2} is uniquely determined by b. For consider maps from ImJ_{ℂ} to the fiber sequence

B^{2}U ⟶^{B^{2}∂} B^{2} ImJ_{ℂ} ⟶^{B^{2}i} B^{2}(ℤ × BU) ⟶^{B^{2}(ψ^{3}−1)} B^{3}U.

We have seen that [ImJ_{ℂ}, B^{2}(ℤ × BU)] ≅ [ImJ_{ℂ}, BU] ≅ ℤ̂_{2}{(1 − ψ^{−1})i}, and [ImJ_{ℂ}, B^{3}U] ≅ [ImJ_{ℂ}, BSU] ≅ ℤ̂_{2}{(1 − ψ^{−1})i}. Under these identifications the map B^{2}(ψ^{3} − 1) corresponds to composition with (\frac{1}{3}ψ^{3} − 1) = −\frac{2}{3} + \frac{1}{3}(ψ^{3} − 1), which acts as multiplication by −\frac{2}{3}, and hence is injective.

Thus a factors through a map ImJ_{ℂ} → B^{2}U and we claim all such maps are inessential. Consider maps from (8.2) to B^{2}U. There are no essential maps ℤ × BU → B^{2}U ≃ SU, because all maps ℤ × BU → U are inessential and each map ℤ×BU → ℤ through the homotopy fiber of SU → U factors through the connecting map ℤ × BU ≃ ΩU → ℤ, and thus gives a trivial composite map.

Next look at maps U → B^{2}U ≃ SU and precomposition with Ω(ψ^{3} − 1). Delooping once we are led to consider the homomorphism

[BU, BSU] ⟶^{(ψ^{3}−1)^{∗}} [ℤ × BU, BSU]

which clearly is injective. Hence [ImJ_{ℂ}, B^{2}U] = 0, and thus [ImJ_{ℂ}, B^{2} ImJ_{ℂ}] = 0. We conclude that any map a: ImJ_{ℂ} → B^{2} ImJ_{ℂ} is inessential, and so B∂_{2} : ImJ_{ℂ} → BK^{red}(ℤ̂_{2}) is uniquely determined as being a lift of u·(1−ψ^{−1})i over BK^{red}(ℤ̂_{2}) → B^{3}U ≃ BSU → BU.

We claim the coefficient u is a two-adic unit. For η · η = η^{2} in K_{2}(ℤ̂_{2}) with η detected in π_{1}(ImJ_{ℂ}) and η^{2} coming from K_{2}^{red}(ℤ̂_{2}). By the same argument as above, if u were divisible by two then η^{2} would be divisible by two, which it is not. Hence [ImJ_{ℂ}, BK^{red}(ℤ̂_{2})] maps isomorphically to [ImJ_{ℂ}, B^{3}U] ≅ ℤ̂_{2}, and the fiber sequence K^{red}(ℤ̂_{2}) → K(ℤ̂_{2}) → ImJ_{ℂ} is represented by a generator of this group.

Hence the connecting map ∂_{2} is determined by the diagram

[figure: Ω ImJ_{ℂ}; u^{−1}·∂_{2}; Ωi; K^{red}(ℤ̂_{2}); Ω(ℤ × BU); 1−Ωψ^{−1}; U; BBU.]

where u is a two-adic unit, K^{red}(ℤ̂_{2}) → BBU is the infinite loop cofiber map of f_{ℂ}, and BBU ≃ SU → U is the natural covering map. □

Remark 8.3. If we pass from the ring of two-adic integers to the field of two-adic numbers, there is still a Galois reduction map K(ℚ̂_{2}) → ImJ_{ℂ} with homotopy fiber K^{red}(ℚ̂_{2}), and a fiber sequence B ImJ_{ℂ} → K^{red}(ℚ̂_{2}) → U. (We use Remark 2.7 to identify the infinite loop space cofiber as U.)

[figure: ℤ × BU; 1−ψ^{−1}; ∂_{1}; BU; B∂; B ImJ_{ℂ}; f_{ℂ}; Ω ImJ_{ℂ}; ∂_{2}; Ωi; K^{red}(ℚ̂_{2}); K(ℚ̂_{2}); red; ImJ_{ℂ}; Ω(ℤ × BU); 1−Ωψ^{−1}; U]

Up to two-adic units, the connecting map ∂_{1} : ΩU ≃ ℤ × BU → B ImJ_{ℂ} is the composite B∂ ∘ (1 − ψ^{−1}), while the connecting map ∂_{2} : Ω ImJ_{ℂ} → K^{red}(ℚ̂_{2}) uniquely lifts the composite (1 − Ωψ^{−1}) ∘ Ωi over K^{red}(ℚ̂_{2}) → U.

9. Comparison with K(ℤ)

We can now consider K(ℤ̂_{2}) as an invariant of K(ℤ), i.e., as a target for detecting classes in K(ℤ). We round off with a brief discussion of this map under the assumption that the Lichtenbaum–Quillen conjecture holds for K(ℤ) at two, that is, if Φ is an equivalence, or at least admits a spectrum-level section.

Let α_{8k−1} ∈ π_{8k−1}Q(S^{0}) be the generator of the image of J in this degree, and let μ_{8k+1} ∈ π_{8k+1}Q(S^{0}) be the order two class detected by KO-theory, also in the image of J.

Lemma 9.1. The composite homomorphism

JK_{∗}(ℤ) ⟶^{ϕ} K_{∗}(ℤ) ⟶ K_{∗}(ℤ̂_{2})

has kernel ℤ/2 in degrees 8k + 3 for k ≥ 0, and is injective otherwise. When nontrivial, the kernel is generated by μ_{8k+1}η^{2}, which is of order two in π_{8k+3}(ImJ_{ℝ}) ⊂ JK_{8k+3}(ℤ).

Proof. The homotopy of JK_{∗}(ℤ) is given by the fiber sequence

ImJ_{ℝ} ⟶^{j} JK(ℤ) ⟶ BBSO

from (2.3), where the boundary map hits the classes α_{8k−1}η and α_{8k−1}η^{2} in degrees 8k and 8k + 1 for k ≥ 1. So for ∗ ≥ 2

(9.2)  JK_{∗}(ℤ) = \{ \begin{matrix}0 & for ∗ ≡ 0, 4, 6 mod 8, \\ ℤ ⊕ ℤ/2 & for ∗ ≡ 1 mod 8, \\ ℤ/2 & for ∗ ≡ 2 mod 8, \\ ℤ/16 & for ∗ ≡ 3 mod 8, \\ ℤ & for ∗ ≡ 5 mod 8, \\ ℤ/2^{v_{2}(k)+4} & for ∗ = 8k − 1. \end{matrix}

The homotopy of K_{∗}(ℤ̂_{2}) is given by Theorem 5.5. So for ∗ ≥ 1

(9.3)  K_{∗}(ℤ̂_{2}) = \{ \begin{matrix}ℤ ⊕ ℤ/2 & for ∗ ≡ 1 mod 4, \\ ℤ/2 & for ∗ ≡ 2 mod 4, \\ ℤ ⊕ ℤ/2^{v_{2}(k)+3} & for ∗ = 4k − 1, \\ ℤ/2^{v_{2}(k)+3} & for ∗ = 4k. \end{matrix}

By Theorem 7.7 only torsion classes can be in the kernel of the given homomorphism. In degrees ∗ = 8k + 1 (with k ≥ 1) the ℤ/2 in JK_{∗}(ℤ) comes from ImJ_{ℝ} and is detected by KO-theory, hence also by the complex e-invariant. (See Proposition 7.19 of [A].)

In degrees ∗ = 8k + 2 the ℤ/2 in JK_{∗}(ℤ) is generated by μ_{8k+1}η, which maps to μ_{8k+1}ι_{1} ≠ 0 in π_{8k+2}(B ImJ_{ℂ}) ⊆ K_{8k+2}(ℤ̂_{2}). (Here ι_{1} is the generator of π_{1}(B ImJ_{ℂ}).) This is because η^{2} maps to η · {3} detected as ηι_{1}, and in general multiplication by v_{1}^{4k} takes η to μ_{8k+1}.

In degrees ∗ = 8k + 3 the homomorphism is ℤ/16 → ℤ ⊕ ℤ/8 with ℤ/8 ⊂ ℤ/16 represented by the real image of J. So twice the generator in ℤ/16 maps to twice the generator in ℤ/8, and so the kernel is the ℤ/2 generated by μ_{8k+1}η^{2}.

In degrees ∗ ≡ 4, 6, 8 mod 8 the source is trivial and in degrees ∗ ≡ 5 mod 8 there is no torsion in the source, so in all these degrees the homomorphism is injective. Finally, in degrees ∗ = 8k − 1 with k ≥ 1 the real and complex einvariants agree, so the torsion group ℤ/2^{v_{2}(k)+4} is mapped isomorphically onto the torsion in K_{8k−1}(ℤ̂_{2}). □

Now we discuss a possible normalization of the identification of the infinite loop space cofiber of f_{ℂ} with BBU, and a description of the map K(ℤ) → K(ℤ̂_{2}). Consider the following diagram of vertical fiber sequences.

(9.4)  [figure: BBO; K^{red}(ℤ); K^{red}(ℤ̂_{2}); JK(ℤ); K(ℤ); π; Φ; K(ℤ̂_{2}); red; ImJ_{ℂ}; ImJ_{ℂ}; ImJ_{ℂ}.]

The fiber sequence on the left is from (2.2). The middle sequence is induced from the map π : K(ℤ) → K(𝔽_{3}) ≃ ImJ_{ℂ}, and defines K^{red}(ℤ). The fiber sequence on the right is from Definition 3.4. We do not know if the bottom right hand square homotopy commutes.

Hypothesis 9.5. Assume that the reduction map K(ℤ̂_{2}) → ImJ_{ℂ} can be chosen to extend the natural map K(ℤ) → K(𝔽_{3}) ≃ ImJ_{ℂ}. Also assume that Φ admits an infinite loop section over the maps to ImJ_{ℂ}.

(In fact Φ is a homotopy equivalence, by [We], so the latter assumption holds.) Then there is a diagram

(9.6)  [figure: B ImJ_{ℂ}; f_{ℂ}; BBO; f; K^{red}(ℤ); K^{red}(ℤ̂_{2}); BBU.]

Proposition 9.7. Under the hypothesis above, the infinite loop space cofiber of f_{ℂ} may be identified with BBU in such a way that the composite map f above is homotopic to Bc, where c: BO → BU is the complexification map.

There are no essential infinite loop maps BBO → B ImJ_{ℂ}, so this determines the composite BBO → K^{red}(ℤ̂_{2}) up to homotopy.

Proof. By Theorem 7.7 and (9.2) the map f: BBO → BBU takes a generator in degree 4k + 1 to a generator in the same degree when k is even, and to twice a generator when k is odd, just like the delooped complexification map Bc.

Since BBO is one-connected and BBU agrees with its K-localization above degree one, there is a bijection between the infinite loop maps BBO → BBU and the spectrum maps ΣKO → ΣKU. Likewise there is a bijection of between the infinite loop maps BBO → B ImJ_{ℂ} and the spectrum maps ΣKO → ΣJU, since B ImJ_{ℂ} agrees with its K-localization above degree one. Hence we may work with the corresponding K-local spectra (and avoid encumbering the notation to account for low-dimensional corrections).

We are then considering maps from ΣKO to (the suspension of) the fiber sequence (6.4). There are no essential spectrum maps from ΣKO to Σ^{2}KO or Σ^{3}KO by Proposition 2.2 of [R], so Σc: ΣKO → ΣKU induces a bijection of maps from ΣKO. The given map f: BBO → BBU agrees with Bc on homotopy, up to two-adic units, so the corresponding spectrum map ΣKO → ΣKU factors up to homotopy as

ΣKO ⟶^{≃} ΣKO ⟶^{Σc} ΣKU

where the left map is an equivalence because it induces an isomorphism on homotopy groups. Such an equivalence is a power series in the real Adams operations ψ^{k} (with k odd), which commutes with complexification. Thus f is also homotopic to (the one-connected cover of) a composite

ΣKO ⟶^{Σc} ΣKU ⟶^{≃} ΣKU.

By adjusting the identification of the infinite loop space cofiber of f_{ℂ} with BBU by the right hand equivalence, the first conclusion follows.

For the second claim, consider maps from ΣKO into the (co)fiber sequence

KU ⟶ ΣJU ⟶ ΣKU ⟶^{Σψ^{3}−1} ΣKU

defining ΣJU. There are no essential spectrum maps ΣKO → KU by Corollary 2.3 of [R]. Since composition with Σψ^{3} − 1 acts injectively on the self maps of ΣKO, and any map ΣKO → ΣKU lifts uniquely over Σc, it follows that any composite ΣKO → ΣJU → ΣKU is null homotopic, proving the second conclusion. Hence f is uniquely determined by its composite to BBU, which we may take to be Bc. □

References

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Department of Mathematics, University of Oslo, Norway E-mail address: rognes@math.uio.no