RATIONAL ALGEBRAIC K-THEORY

OF TOPOLOGICAL K-THEORY

Christian Ausoni and John Rognes

August 14th 2007

Abstract. We show that after rationalization there is a homotopy fiber sequence BBU_{⊗} → K(ku) → K(ℤ). We interpret this as a correspondence between the virtual 2-vector bundles over a space X and their associated anomaly bundles over the free loop space LX. We also rationally compute K(KU) by using the localization sequence, and K(MU) by a method that applies to all connective S-algebras.

Introduction

We are interested in the algebraic K-theory K(ku) of the connective complex Ktheory spectrum ku. By the calculations of [AR02, Thm. 0.4] and [Au], the “mod p and v_{1}” homotopy of K(ku) is purely v_{2}-periodic, a distinctive homotopy theoretic property it shares with the spectra representing elliptic cohomology [LRS95] and topological modular forms [Ho02, §4]. The theory of 2-vector bundles from [BDR04] and [BDRR] therefore exhibits K(ku) as a geometrically defined form of elliptic cohomology. In Section 5 we outline how a 2-vector bundle with connecting data, similar to a connection in a vector bundle, is thought to specify a 1 + 1-dimensional conformal field theory. Since these 2-vector bundles are also effective cycles for the form of elliptic cohomology theory represented by K(ku), we have some justification for referring to them as elliptic objects, as proposed by Segal [Se89].

As illustrated by the authors’ calculations referred to above, the arithmetic and homotopy-theoretic information captured by algebraic K-theory becomes more accessible after the introduction of suitable finite coefficients. However, for the extraction of ℂ-valued numerical invariants from a conformal field theory, only the rational homotopy type of K(ku) will matter. We are grateful to Ib Madsen and Dennis Sullivan for insisting that for such geometric applications, we should first want to compute K(ku) rationally. To this end we can offer the following theorem. There is a unit inclusion map

w: BBU_{⊗} → BGL_{1}(ku) → BGL_{∞}(ku)^{+} → K(ku) .

Let π: K(ku) → K(ℤ) be induced by the zero-th Postnikov section ku → Hℤ. The composite π ∘ w is the constant map to the base-point of the 1-component of K(ℤ).

Typeset by AMS-TEX

Theorem 0.1. (a) After rationalization,

BBU_{⊗} ⟶^{w} K(ku) ⟶^{π} K(ℤ)

is a split homotopy fiber sequence.

(b) The Poincaré series of K(ku) is

1 + \frac{t^{3}}{1 − t^{2}} + \frac{t^{5}}{1 − t^{4}} = 1 + \frac{t^{3} + 2t^{5}}{1 − t^{4}} .

(c) There is a rational determinant map

det_{ℚ}: BGL_{∞}(ku)^{+} → BGL_{1}(ku)_{ℚ}

that, in its relative form for ku → Hℤ, rationally splits w.

By the Poincaré series of a space X of finite type, we mean the formal power series ∑_{n≥0} r_{n}t^{n} in ℤ[[t]], where r_{n} is the rank of π_{n}(X). Theorem 0.1 is proved by assembling Theorem 2.5(a) and Theorem 4.8(a). The other parts of those theorems prove similar results for K(ko) and K(ℓ), where ℓ = BP ⟨1⟩.

The splitting of K(ku)_{ℚ} shows that for a (virtual) 2-vector bundle over X, represented by a map E: X → K(ku), the rational information splits into two pieces. The less interesting piece is the decategorified information carried by the dimension bundle dim(E) = π ∘ E: X → K(ℤ). The more interesting piece is the determinant bundle

|E| = det_{ℚ} ∘ E: X → (BBU_{⊗})_{ℚ} .

To specify |E| is equivalent to specifying a rational virtual vector bundle H: LX → (BU_{⊗})_{ℚ} over the free loop space LX = Map(S^{1}, X), called the “anomaly bundle”, subject to a coherence condition relating the composition of free loops, when defined, to the tensor product of virtual vector spaces. See diagram (5.3).

The conclusion is that for rational purposes the information in a 2-vector bundle E over X is the same as that in its anomaly bundle H over LX (subject to the indicated coherence condition, which we think of as implicit, and together with the dimension bundle dim(E) over X, which we tend to ignore). In physical language, the fiber of the anomaly bundle at a free loop γ: S^{1} → X plays the role of the state space of γ viewed as a closed string in X. The advantage of 2-vector bundles over their homotopy-theoretic alternatives, such as representing maps to classifying spaces or bundles of ku-modules, is that they are geometrically modeled in terms of vector bundles, rather than virtual vector bundles. This seems to become an essential virtue when one wants to treat differential-geometric structures like connections on these bundles.

We also compute the rational algebraic K-theory K(KU) of the periodic complex K-theory spectrum KU. To this end we evaluate the (rationalized) transfer map π_{∗} in the localization sequence

K(ℤ) ⟶^{π_{∗}} K(ku) ⟶^{ρ} K(KU)

predicted by the second author and established by Blumberg and Mandell [BM].

Theorem 0.2. (a) There is a rationally split homotopy fiber sequence of infinite loop spaces

K(ku) ⟶^{ρ} K(KU) ⟶^{∂} BK(ℤ)

where ρ is induced by the connective cover map ku → KU, and BK(ℤ) denotes the first connected delooping of K(ℤ).

(b) The Poincaré series of K(KU) is

(1 + t) + \frac{t^{3} + 2t^{5} + t^{6}}{1 − t^{4}} .

See Theorem 2.12 for our proof.

As stated, Theorems 0.1 and 0.2 only concern the algebraic K-theory of topological K-theory, but we develop our proofs in the greater generality of arbitrary connective S-algebras. In Section 1 we observe how the calculation by Goodwillie [Go86] of the relative rational algebraic K-theory for a 1-connected map R → π_{0}R of simplicial rings (which generalizes earlier calculations by Hsiang and Staffeldt [HS82] for simplicial group rings), also applies to determine the relative rational algebraic K-theory for a 1-connected map A → Hπ_{0}A of connective S-algebras. The answer is given in terms of negative cyclic homology; see Theorem 1.5 and Corollary 1.6.

When π_{0}A is close to ℤ, and A → Hπ_{0}A is a “rational de Rham equivalence”, we get a very simple expression for the relative rational algebraic K-theory as the image of Connes’ B-operator on Hochschild homology; see Proposition 1.8 and Corollary 1.9. These hypotheses apply to a number of interesting examples of connective S-algebras, including the K-theory spectra ku, ko and ℓ, and the bordism spectra MU, MSO and MSp. We work these examples out in Theorem 2.5 and Theorem 3.4, respectively.

In Section 4 we consider the unit inclusion map w: BGL_{1}(A) → K(A). For commutative A, the rationalization A_{ℚ} is equivalent as a commutative Hℚ-algebra to the Eilenberg–Mac Lane spectrum HR of a commutative simplicial ℚ-algebra, so we can use the determinant GL_{n}(R) → GL_{1}(R) to define a rational determinant map

det_{ℚ}: BGL_{∞}(A)^{+} → BGL_{1}(A)_{ℚ} .

We show in Proposition 4.7 that the composite det_{ℚ} ∘w is the rationalization map, and apply this in Theorem 4.8 to show that w induces a rational equivalence from BBU_{⊗} to the homotopy fiber of π: K(ku) → K(ℤ), and similarly for ko and ℓ. This last step is a counting argument; it does not apply for MU or the other bordism spectra.

Acknowledgments. In an earlier version of this paper, we emphasized a trace map to T HH(ku) over the determinant map to (BBU_{⊗})_{ℚ}, in order to detect the image of w in K(ku). We are grateful to Bjørn Dundas for reminding us of the existence of determinants for commutative simplicial rings, which is half of the basis for the existence of the map det_{ℚ} defined in Lemma 4.6. We are also grateful to Mike Mandell and Brooke Shipley for help with some of the references concerning commutative simplicial ℚ-algebras given in Subsection 1.1.

§1. Rational algebraic K-theory of connective S-algebras

1.1. S-algebras. We work in one of the modern symmetric monoidal categories of spectra [EKMM97], [Ly99], [HSS00], [MMSS01], which we shall refer to as Smodules. The monoids (resp. commutative monoids) in this category are called S-algebras (resp. commutative S-algebras), and are equivalent to the A_{∞} ring spectra (resp. E_{∞} ring spectra) considered since the 1970’s. The Eilenberg–Mac Lane functor H: R ↦ HR maps the category of simplicial rings (resp. commutative simplicial rings) to the category of S-algebras (resp. commutative S-algebras).

Schwede proved in [Schw99, 4.5] that H is part of a Quillen equivalence from the category of simplicial rings to the category of connective Hℤ-algebras. There is a similar equivalence between the category of commutative simplicial ℚ-algebras and the category of connective commutative Hℚ-algebras.

One form of the latter equivalence appears in [KM95, II.1.3]. In a little more detail, the category of connective commutative Hℚ-algebras is “connective Quillen equivalent” [MMSS01, p. 445] to the category of connective E_{∞} Hℚ-ring spectra [EKMM97, II.4], and connective E_{∞} Hℚ-ring spectra are the E_{∞} objects in connective Hℚ-modules, which are Quillen equivalent to E_{∞} simplicial ℚ-algebras [Schw99, 4.4]. The monads defining E_{∞} algebras and commutative algebras in simplicial ℚ-modules are weakly equivalent, since for every j ≥ 0 the group homology of Σ_{j} with coefficients in any ℚ-module is concentrated in degree zero. Hence E_{∞} simplicial ℚ-algebras are Quillen equivalent to commutative simplicial ℚ-algebras [Ma03, 6.7].

The homotopy categories of commutative simplicial rings and connective commutative Hℤ-algebras are not equivalent.

1.2. Linearization. Let A be a connective S-algebra. We write π = π_{A}: A → Hπ_{0}A for its zero-th Postnikov section, and define the linearization map λ = λ_{A}: A → Hℤ ∧ A to be π ∧ id: A ≅ S ∧ A → Hℤ ∧ A. It is a π_{0}-isomorphism and a rational equivalence of connective S-algebras. For each (simplicial or topological) monoid G let S[G] = Σ^{∞}G_{+} be its unreduced suspension spectrum. For A = S[G], the linearization map λ: S[G] → Hℤ ∧ S[G] ≅ Hℤ[G] agrees with the map considered by Waldhausen [Wa78, p. 43].

In general, Hℤ ∧ A is a connective Hℤ-algebra, so by the first Quillen equivalence above there is a naturally associated simplicial ring R with Hℤ ∧ A ≃ HR. For connective commutative A, the rationalization A_{ℚ} = Hℚ ∧ A is a connective commutative Hℚ-algebra, so by the second Quillen equivalence above there is a naturally associated commutative simplicial ℚ-algebra R with A_{ℚ} ≃ HR.

1.3. Algebraic K-theory. For a general S-algebra A, the algebraic K-theory space K(A) can be defined as Ω|hS_{•}C_{A}|, where C_{A} is the category of finite cell Amodule spectra and their retracts, S_{•} denotes Waldhausen’s S_{•}-construction [Wa85, §1.3], and |h(−)| indicates the nerve of the subcategory of weak equivalences. By iterating the S_{•}-construction, we may also view K(A) as a spectrum. For connective S-algebras, K(A) can alternatively be defined in terms of Quillen’s plusconstruction as K_{0}(π_{0}A)×BGL_{∞}(A)^{+}, and then the two definitions are equivalent, see [EKMM97, VI.7.1]. We write K(R) for K(HR), and similarly for other functors.

Any map A → A′ of connective S-algebras that is a π_{0}-isomorphism and a rational equivalence induces a rational equivalence K(A) → K(A′), by [Wa78, 2.2]. The proof goes by observing that BGL_{n}(A) → BGL_{n}(A′) is a rational equivalence for each n. In particular, for R with Hℤ ∧ A ≃ HR there is a natural rational equivalence λ: K(A) → K(Hℤ ∧ A) ≃ K(R). For X ≃ BG, Waldhausen writes A(X) for K(S[G]), and λ: A(X) → K(ℤ[G]) is a rational equivalence. In this case, A(X) can also be defined as the algebraic K-theory of a category R_{f}(X) of suitably finite retractive spaces over X, see [Wa85, §2.1].

1.4. Cyclic homology. There is a natural trace map tr: K(A) → T HH(A) to the topological Hochschild homology of A, see [BHM93, §3]. The target is a cyclic object in the sense of Connes, hence carries a natural S^{1}-action. There exists a model for the trace map that factors through the fixed points of this circle action [Du04], hence it also factors through the homotopy fixed points T HH(A)^{hS^{1}}. We get a natural commutative triangle

\xymatrix{K(A) \ar[r]^{α} \ar[dr]_{tr} & T HH(A)^{hS^{1}} \ar[d]^{F} \\  & T HH(A)}

where the Frobenius map F forgets about S^{1}-homotopy invariance. For any simplicial ring R there are natural isomorphisms T HH_{∗}(HR_{ℚ}) ≅ HH_{∗}(R ⊗ ℚ) (Hochschild homology) and T HH_{∗}(HR_{ℚ})^{hS^{1}} ≅ HC_{∗}^{−}(R⊗ℚ) (negative cyclic homology). See e.g. [EKMM97, IX.1.7] and [CJ90, 1.3(3)]. With these identifications, the triangle above realizes the commutative diagram of [Go86, II.3.1]. In [Go86, II.3.4], Goodwillie proved:

Theorem 1.5. Let f: R → R′ be a map of simplicial rings, with π_{0}R → π_{0}R′ a surjection with nilpotent kernel. Then

\xymatrix{K(R)_{ℚ} \ar[r]^{α} \ar[d]_{f} & HC^{−}(R ⊗ ℚ) \ar[d]^{f} \\ K(R′)_{ℚ} \ar[r]^{α} & HC^{−}(R′ ⊗ ℚ)}

is homotopy Cartesian, i.e., the map of vertical homotopy fibers

α: K(f)_{ℚ} → HC^{−}(f ⊗ ℚ)

is an equivalence.

Here we write K(f) for the homotopy fiber of K(R) → K(R′), so that there is a long exact sequence

⋯ → K_{∗+1}(R′) → K_{∗}(f) → K_{∗}(R) → K_{∗}(R′) → …,

and similarly for other functors from (S-)algebras to spaces. (Goodwillie writes K(f) for a delooping of our K(f), but we need to emphasize fibers over cofibers.) We write K(R)_{ℚ} for the rationalization of K(R), and similarly for other spaces and S-algebras.

Corollary 1.6. Let g: A → A′ be a map of connective S-algebras, with π_{0}A → π_{0}A′ a surjection with nilpotent kernel, Then

\xymatrix{K(A)_{ℚ} \ar[r]^{α} \ar[d]_{g} & T HH(A_{ℚ})^{hS^{1}} \ar[d]^{g} \\ K(A′)_{ℚ} \ar[r]^{α} & T HH(A′_{ℚ})^{hS^{1}}}

is homotopy Cartesian, i.e., the map of vertical homotopy fibers

α: K(g)_{ℚ} → T HH(g_{ℚ})^{hS^{1}}

is an equivalence.

Proof. Given g: A → A′ we find f: R → R′ with Hℤ∧A ≃ HR and Hℤ∧A′ ≃ HR′ making the diagram

\xymatrix{A \ar[r]^{λ} \ar[d]_{g} & HR \ar[d]^{Hf} \\ A′ \ar[r]^{λ} & HR′}

homotopy commute. Then λ: K(A) → K(R) is a rational equivalence and A_{ℚ} ≃ H(R ⊗ ℚ), so the square in the corollary is equivalent to the square in Goodwillie’s theorem. □

1.7. De Rham homology. The (spectrum level) circle action on T HH(A) induces a suspension operator d: T HH_{∗}(A) → T HH_{∗+1}(A), analogous to Connes’ operator B: HH_{∗}(R) → HH_{∗+1}(R). When A_{ℚ} ≃ H(R ⊗ ℚ), these operators are compatible under the isomorphism T HH_{∗}(A_{ℚ}) ≅ HH_{∗}(R ⊗ℚ). In general dd = dη is not zero [He97, 1.4.4], where η is the stable Hopf map, but in the algebraic case BB = 0, so one can define the de Rham homology

H_{∗}^{dR}(R) = ker(B)/ im(B)

of a simplicial ring R as the homology of HH_{∗}(R) with respect to the B-operator.

For a map g: A → A′ of S-algebras, the homotopy fiber T HH(g) of T HH(A) → T HH(A′) inherits a circle action and associated suspension operator. Similarly, for a map f: R → R′ of simplicial rings there is a relative B-operator acting on the term HH_{∗}(f) in the long exact sequence

⋯ → HH_{∗+1}(R′) → HH_{∗}(f) → HH_{∗}(R) → HH_{∗}(R′) → …,

and we define H_{∗}^{dR}(f) to be the homology of HH_{∗}(f) with respect to this Boperator. We say that f: R → R′ is a de Rham equivalence if H_{∗}^{dR}(f) = 0, and that f is a rational de Rham equivalence if H_{∗}^{dR}(f ⊗ ℚ) = 0. If we assume that HH_{∗}(R) → HH_{∗}(R′) is surjective in each degree, then there is a long exact sequence

⋯ → H_{∗+1}^{dR}(R′) → H_{∗}^{dR}(f) → H_{∗}^{dR}(R) → H_{∗}^{dR}(R′) → …,

in which case f is a de Rham equivalence if and only if H_{∗}^{dR}(R) → H_{∗}^{dR}(R′) is an isomorphism in every degree.

Proposition 1.8. If f: R → R′ is a de Rham equivalence, then there is an exact sequence

0 → HC_{∗}^{−}(f) ⟶^{F} HH_{∗}(f) ⟶^{B} HH_{∗+1}(f)

that identifies HC_{∗}^{−}(f) with ker(B) ⊂ HH_{∗}(f).

Proof. By analogy with the homotopy fixed point spectral sequence for T HH(g)^{hS^{1}}, there is a second quadrant homological spectral sequence

E_{∗∗}^{2} = ℚ[t] ⊗ HH_{∗}(f) ⟹ HC_{∗}^{−}(f)

with t ∈ E_{−2,0}^{2} and d^{2}(t^{i} · x) = t^{i+1} · B(x) for all x ∈ HH_{∗}(f), i ≥ 0. So E_{∗∗}^{3} is the sum of ker(B) ⊂ HH_{∗}(f) in the zero-th column and a copy of H_{∗}^{dR}(f) in each even, negative column. By assumption the latter groups are all zero, so the spectral sequence collapses to the zero-th column at the E^{3}-term. The Frobenius F is the edge homomorphism for this spectral sequence, and the assertion follows. □

Corollary 1.9. Let A be a connective S-algebra such that π_{0}A is any localization of the integers, and let R be a simplicial ℚ-algebra with A_{ℚ} ≃ HR.

(a) The homotopy fiber sequence

K(π_{A}) → K(A) ⟶^{π_{A}} K(π_{0}A)

is rationally split, where π_{A}: A → Hπ_{0}A is the zero-th Postnikov section. (b) There are equivalences

K(π_{A})_{ℚ} ⟶_{≃}^{α} T HH(π_{Aℚ})^{hS^{1}} ≃ HC^{−}(π_{R}) ,

where π_{R}: R → π_{0}R = ℚ is the zero-th Postnikov section.

Suppose furthermore that H_{∗}^{dR}(R) ≅ ℚ is trivial in positive degrees.

(c) The map π_{R} is a de Rham equivalence, and the Frobenius map identifies HC_{∗}^{−}(π_{R}) with the positive-degree part of

ker(B) ⊂ HH_{∗}(R) ≅ T HH_{∗}(A) ⊗ ℚ .

That part is also equal to im(B) ⊂ T HH_{∗}(A) ⊗ ℚ.

(d) The trace map tr: K(A) → T HH(A) induces the composite identification of K_{∗}(π_{A}) ⊗ ℚ with the positive-degree part of ker(B) ⊂ T HH_{∗}(A) ⊗ ℚ.

Proof. (a) Write π_{0}A = ℤ_{(P)} for some (possibly empty) set of primes P. The unit map i: S → A factors through S_{(P)}, and the composite map S_{(P)} → A → Hπ_{0}A is a π_{0}-isomorphism and a rational equivalence. Hence the composite

K(S_{(P)}) → K(A) ⟶^{π_{A}} K(π_{0}A)

is a rational equivalence.

(b) The map π_{A}: A → Hπ_{0}A induces the identity on π_{0}, so α is an equivalence by Corollary 1.6. We recalled the second identification in Subsection 1.4. It is clear that π_{0}R = π_{0}A_{ℚ} = π_{0}A ⊗ ℚ = ℚ.

(c) Since HH_{∗}(ℚ) = ℚ is trivial in positive degrees, the map HH_{∗}(R) → HH_{∗}(ℚ) is surjective in each degree, so π_{R} is a de Rham equivalence if (and only if) H_{∗}^{dR}(R) ≅ H_{∗}^{dR}(ℚ) = ℚ is trivial in all positive degrees. The homotopy fiber sequence

HH(π_{R}) → HH(R) → HH(ℚ)

identifies HH_{∗}(π_{R}) with the positive-degree part of HH_{∗}(R), so ker(B) ⊂ HH_{∗}(π_{R}) is the positive-degree part of ker(B) ⊂ HH_{∗}(R). The identification im(B) = ker(B) in positive degrees is of course equivalent to the vanishing of H_{∗}^{dR}(R) in positive degrees.

(d) The trace map factors as tr = F ∘ α. □

§2. Examples from topological K-theory

2.1. Connective K-theory spectra. Let ku be the connective complex Ktheory spectrum, ko the connective real K-theory spectrum, and ℓ = BP ⟨1⟩ the Adams summand of ku_{(p)}, for p an odd prime. These are all commutative Salgebras. We write Ω^{∞}ku = BU × ℤ, Ω^{∞}ko = BO × ℤ and Ω^{∞}ℓ = W × ℤ_{(p)} for the underlying infinite loop spaces (see [Ma77, V.3-4]). The homotopy units form infinite loop spaces, namely GL_{1}(ku) = BU_{⊗} × {±1}, GL_{1}(ko) = BO_{⊗} × {±1} and GL_{1}(ℓ) = W_{⊗} ×ℤ_{(p)}^{×}. The homotopy algebras are π_{∗}ku = ℤ[u] with |u| = 2, π_{∗}ko = ℤ[η, α, β]/(2η, η^{3}, ηα, α^{2}−4β) with |η| = 1, |α| = 4, |β| = 8, and π_{∗}ℓ = ℤ_{(p)}[v_{1}] with |v_{1}| = 2p − 2. The complexification map ko → ku takes η to 0, α to 2u^{2} and β to u^{4}. The inclusion ℓ → ku_{(p)} takes v_{1} to u^{p−1}.

Proposition 2.2. (a) There are π_{0}-isomorphisms and rational equivalences

κ: S[ΩS^{3}] → S[K(ℤ, 2)] → ku

of S-algebras, so ku_{ℚ} ≃ Hℚ[ΩS^{3}] as homotopy commutative Hℚ-algebras, where ℚ[ΩS^{3}] is a simplicial ℚ-algebra, and ku_{ℚ} ≃ Hℚ[K(ℤ, 2)] as commutative Hℚalgebras, where ℚ[K(ℤ, 2)] is a commutative simplicial ℚ-algebra.

(b) There are π_{0}-isomorphisms and rational equivalences ᾱ: S[ΩS^{5}] → ko and v̄_{1} : S_{(p)}[ΩS^{2p−1}] → ℓ, so ko_{ℚ} ≃ Hℚ[ΩS^{5}] and ℓ_{ℚ} ≃ Hℚ[ΩS^{2p−1}] as Hℚ-algebras, where ℚ[ΩS^{5}] and ℚ[ΩS^{2p−1}] are simplicial ℚ-algebras.

(c) In particular, there is a rational equivalence κ: A(S^{3}) → K(ku) of Salgebras, a rational equivalence A(K(ℤ, 3)) → K(ku) of commutative S-algebras, and a rational equivalence ᾱ: A(S^{5}) → K(ko) of S-modules.

Proof. (a) Let BS^{3} → K(ℤ, 4) represent a generator of H^{4}(BS^{3}). It induces a double loop map ΩS^{3} → K(ℤ, 2), such that the composite S^{2} → ΩS^{3} → K(ℤ, 2) represents a generator of π_{2}K(ℤ, 2). The inclusions K(ℤ, 2) ≃ BU(1) → BU_{⊗} → GL_{1}(ku) are infinite loop maps, and the generator of π_{2}K(ℤ, 2) maps to a generator of π_{2}GL_{1}(ku). By adjunction we have an E_{2} ring spectrum map S[ΩS^{3}] → S[K(ℤ, 2)] and an E_{∞} ring spectrum map S[K(ℤ, 2)] → ku, with composite the E_{2} ring spectrum map κ: S[ΩS^{3}] → ku.

These are rational equivalences, because π_{∗}S[ΩS^{3}] ⊗ ℚ ≅ H_{∗}(ΩS^{3}; ℚ) ≅ ℚ[x], H_{∗}(K(ℤ, 2); ℚ) ≅ ℚ[b] and π_{∗}ku ⊗ ℚ = ℚ[u], with κ mapping x via b to u. We may take the Kan loop group of S^{3} (a simplicial group, see e.g. [Wa96]) as our model for ΩS^{3}, and rigidify κ to a map of S-algebras. Following [FV], there remains an E_{1} = A_{∞} operad action on these S-algebras and κ, which in particular implies that κ: A(S^{3}) → K(ku) is homotopy commutative.

(b) For the real case, let S^{4} → BO_{⊗} ⊂ GL_{1}(ko) represent a generator of π_{4}GL_{1}(ko). By the loop structure on the target, it extends to a loop map ΩS^{5} → GL_{1}(ko), with left adjoint an A_{∞} ring spectrum map ᾱ: S[ΩS^{5}] → ko. It is a rational equivalence, because π_{∗}S[ΩS^{5}]⊗ℚ ≅ H_{∗}(ΩS^{5}; ℚ) ≅ ℚ[y] and π_{∗}ko⊗ℚ = ℚ[α], with ᾱ mapping y to α. We interpret ΩS^{5} as the Kan loop group, and form the simplicial ℚ-algebra ℚ[ΩS^{5}] as its rational group ring. The Adams summand case is entirely similar, starting with a map S^{2p−2} → W_{⊗} ⊂ GL_{1}(ℓ).

(c) By [FV] and naturality there is an A_{∞} operad action on the induced map of spectra A(S^{3}) = K(S[ΩS^{3}]) → K(ku) (rather than of spaces), which we can rigidify to a map of S-algebras. The S-algebra multiplication A(S^{3}) ∧ A(S^{3}) → A(S^{3}) is induced by the group multiplication S^{3} × S^{3} → S^{3}. □

Lemma 2.3. (a) For any integer n ≥ 1 the simplicial ℚ-algebra R = ℚ[ΩS^{2n+1}] has Hochschild homology

HH_{∗}(R) ≅ ℚ[x] ⊗ E(dx)

with |x| = 2n, where Connes’ B-operator satisfies B(x) = dx. Here E(−) denotes the exterior algebra. (b) The de Rham homology H_{∗}^{dR}(R) ≅ ℚ is concentrated in degree zero, so π_{R}: R → ℚ is a de Rham equivalence. (c) The positive-degree part of ker(B) ⊂ HH_{∗}(R) is

ℚ[x]{dx} = ℚ{dx, x dx, x^{2} dx, … } .

Proof. (a) The Hochschild filtration on the bisimplicial ℚ-algebra HH(R) yields a spectral sequence

(2.4)  E_{∗∗}^{2} = HH_{∗}(π_{∗}(R)) ⟹ HH_{∗}(R) ,

and π_{∗}(R) = ℚ[x] with |x| = 2n. The Hochschild homology of this graded commutative ring is ℚ[x] ⊗ E(dx), where dx ∈ E_{1,2n}^{2} is the image of x under Connes’ B-operator. The spectral sequence collapses at that stage, for bidegree reasons.

(b,c) The B-operator is a derivation, hence takes x^{m} to mx^{m−1} dx for all m ≥ 0. It follows easily that the de Rham homology is trivial in positive degrees, and that ker B is as indicated. □

By combining Corollary 1.9, Proposition 2.2 and Lemma 2.3, we obtain the following result.

Theorem 2.5. (a) There is a rationally split homotopy fiber sequence

K(π_{ku}) → K(ku) ⟶^{π} K(ℤ)

and the trace map tr: K(ku) → T HH(ku) identifies

K_{∗}(π_{ku}) ⊗ ℚ ≅ ℚ[u]{du}

with its image in T HH_{∗}(ku) ⊗ ℚ ≅ ℚ[u] ⊗ E(du). Here |u| = 2 and |du| = 3, so K(π_{ku}) has Poincaré series t^{3}/(1 − t^{2}).

(b) Similarly, there are rationally split homotopy fiber sequences

K(π_{ko}) → K(ko) ⟶^{π} K(ℤ)
K(π_{ℓ}) → K(ℓ) ⟶^{π} K(ℤ_{(p)})

and the trace maps identify

K_{∗}(π_{ko}) ⊗ ℚ ≅ ℚ[α]{dα}
K_{∗}(π_{ℓ}) ⊗ ℚ ≅ ℚ[v_{1}]{dv_{1}}

with their images in T HH_{∗}(ko) ⊗ ℚ ≅ ℚ[α] ⊗ E(dα) and T HH_{∗}(ℓ) ⊗ ℚ ≅ ℚ[v_{1}] ⊗ E(dv_{1}), respectively. Hence K(π_{ko}) has Poincaré series t^{5}/(1 − t^{4}), whereas K(π_{ℓ}) has Poincaré series t^{2p−1}/(1 − t^{2p−2}).

Remark 2.6. The Poincaré series of K(ℤ) is 1 + t^{5}/(1 − t^{4}) by Borel’s calculation [Bo74]. Hence the (common) Poincaré series of K(ku) and A(S^{3}) is

1 + t^{3}/(1 − t^{2}) + t^{5}/(1 − t^{4}) = 1 + (t^{3} + 2t^{5})/(1 − t^{4}) ,

whereas the Poincaré series of K(ko) and A(S^{5}) is 1 + 2t^{5}/(1 − t^{4}). More generally, we recover the Poincaré series 1 + t^{5}/(1 − t^{4}) + t^{2n+1}/(1 − t^{2n}) of A(S^{2n+1}) for n ≥ 1, from [HS82, Cor. 1.2]. The group K_{1}(ℤ_{(p)}) is not finitely generated, so we do not discuss the Poincaré series of K(ℓ).

2.7. Periodic K-theory spectra. Let KU be the periodic complex K-theory spectrum, KO the periodic real K-theory spectrum, and L = E(1) the Adams summand of KU_{(p)}, for p an odd prime. We have maps of commutative S-algebras

Hℤ ⟵^{π} ku ⟶^{ρ} KU

with associated maps of “brave new” affine schemes [TV, §2]

(2.8)  Spec(ℤ) ⟶^{π} Spec(ku) ⟵^{ρ} Spec(KU) .

Let i: S → S[ΩS^{3}] and c: S[ΩS^{3}] → S be induced by the inclusion map ∗ → S^{3} and the collapse map S^{3} → ∗, respectively. We have a map of horizontal cofiber sequences

(2.9)  \xymatrix{Σ^{2}S[ΩS^{3}] \ar[r]^{x} \ar[d]^{Σ^{2}κ} & S[ΩS^{3}] \ar[r]^{c} \ar[d]^{κ} & S \ar[d]^{λ} \\ Σ^{2}ku \ar[r]^{u} & ku \ar[r]^{π} & Hℤ}

where the top row exhibits S as a two-cell S[ΩS^{3}]-module, and the bottom row exhibits Hℤ as a two-cell ku-module. (In each case, the two cells are in dimension zero and three.) There are algebraic K-theory transfer maps c_{∗}: A(∗) → A(S^{3}) and π_{∗}: K(ℤ) → K(ku) (with a lower star, in accordance with the variance conventions from algebraic geometry and (2.8)), that are induced by the functors that view finite cell S-modules as finite cell S[ΩS^{3}]-modules, and finite cell Hℤ-modules as finite cell ku-modules, respectively. In terms of retractive spaces, c_{∗} is induced by the exact functor R_{f}(∗) → R_{f}(S^{3}) that takes a pointed space X ⇄ ∗ to the retractive space X × S^{3} ⇄ S^{3}. The transfer maps are compatible, by (2.9), so we have a commutative diagram with vertical rational equivalences

(2.10)  \xymatrix{A(∗) \ar[r]^{c_{∗}} \ar[d]^{λ} & A(S^{3}) \ar[d]^{κ} \\ K(ℤ) \ar[r]^{π_{∗}} & K(ku) \ar[r]^{ρ} & K(KU) .}

The bottom row is a homotopy fiber sequence by the localization theorem of [BM].

Lemma 2.11. The transfer map c_{∗}: A(∗) → A(S^{3}) is null-homotopic, as a map of A(∗)-module spectra. The transfer map π_{∗}: K(ℤ) → K(ku) is rationally nullhomotopic, again as a map of A(∗)-module spectra.

Proof. The projection formula asserts that c_{∗} is an A(S^{3})-module map, where c: A(S^{3}) → A(∗) makes A(∗) an A(S^{3})-module. Restricting the module structures along i: A(∗) → A(S^{3}), we see that c_{∗} is a map of A(∗)-module spectra, and the source is a free A(∗)-module of rank one. Hence it suffices to show that c_{∗} takes a generator of π_{0}A(∗), represented say by S^{0} ⇄ ∗, to zero in π_{0}A(S^{3}) ≅ ℤ. But c_{∗} maps that generator to the class of S^{0} × S^{3} ⇄ S^{3}, which corresponds to its Euler characteristic χ(S^{3}) = 0.

The conclusion for π_{∗} follows from that for c_{∗}, via the rational equivalences λ and κ. □

Note the utility of the comparison with A-theory at this point, since we do not have an S-algebra map K(ℤ) → K(ku) that is analogous to i: A(∗) → A(S^{3}).

Theorem 2.12. There are rationally split homotopy fiber sequences

K(ku) ⟶^{ρ} K(KU) ⟶^{∂} BK(ℤ)
K(ℓ) ⟶^{ρ} K(L) ⟶^{∂} BK(ℤ_{(p)})

of infinite loop spaces. Hence the Poincaré series of K(KU) is

(1 + t) + (t^{3} + 2t^{5} + t^{6})/(1 − t^{4}) .

Proof. The claims for KU follow by combining Theorem 2.5(a) and Lemma 2.11. The proof of the claim for L is completely similar, using that Hℤ_{(p)} is a two-cell ℓ-module, with cells in dimension zero and (2p − 1). By [BM] there is a homotopy fiber sequence K(ℤ_{(p)}) → K(ℓ) → K(L). □

Remark 2.13. We do not know how to relate K(ko) with K(KO), so we do not have a rational calculation of K(KO). However, KO → KU is a ℤ/2-Galois extension of commutative S-algebras, in the sense of [Ro, §4.1], so it is plausible that K(KO) → K(KU)^{hℤ/2} is close to an equivalence. Here ℤ/2 acts on KU by complex conjugation, and

π_{∗}(K(KU)^{hℤ/2}) ⊗ ℚ ≅ [K_{∗}(KU) ⊗ ℚ]^{ℤ/2} .

The conjugation action on ku fixes K(ℤ), and acts on K_{∗}(π_{ku}) ⊗ ℚ ≅ ℚ[u]{du} by sign on u and du, hence fixes ℚ[u^{2}]{udu} ≅ ℚ[α]{dα} ≅ K_{∗}(π_{ko}) ⊗ ℚ. So K(ko) → K(ku)^{hℤ/2} is a rational equivalence. The conjugation action also fixes BK(ℤ) after rationalization, so the Poincaré series of K(KU)^{hℤ/2} is (1 + t) + (2t^{5} + t^{6})/(1 − t^{4}).

Remark 2.14. We expect that c_{∗} and π_{∗} are essential (not null-homotopic) as maps of A(S^{3})-module spectra and K(ku)-module spectra, respectively. In other words, we expect that K(ku) → K(KU) → ΣK(ℤ) is a non-split extension of K(ku)- module spectra. This expectation is to some extent justified by the fact that the cofiber T HH(ku|KU) of the T HH-transfer map π_{∗}: T HH(ℤ) → T HH(ku) sits in a non-split extension T HH(ku) → T HH(ku|KU) → ΣT HH(ℤ) of T HH(ku)- module spectra. See [Au05, 10.4], or [HM03, Lemma 2.3.3] for a similar result in an algebraic case.

§3. Examples from smooth bordism

3.1. Oriented bordism spectra. Let MU be the complex bordism spectrum, MSO the real oriented bordism spectrum, and MSp the symplectic bordism spectrum. These are all connective commutative S-algebras, given by the Thom spectra associated to infinite loop maps from BU, BSO and BSp to BSF = BSL_{1}(S), respectively. We recall that

H_{∗}(BU) ≅ ℤ[b_{k} | k ≥ 1]

with |b_{k}| = 2k, while H_{∗}(BSO; ℤ[1/2]) ≅ H_{∗}(BSp; ℤ[1/2]) ≅ ℤ[1/2][q_{k} | k ≥ 1] with |q_{k}| = 4k.

The Thom equivalence θ: MU ∧ MU → MU ∧ S[BU] induces an equivalence Hℤ ∧ MU ≃ Hℤ ∧ S[BU] = Hℤ[BU]. Combined with the Hurewicz map π: S → Hℤ we obtain a chain of maps of commutative S-algebras

MU ⟶ Hℤ ∧ MU ≃ Hℤ[BU] ⟵ S[BU] ,

that are π_{0}-isomorphisms and rational equivalences. There are similar chains MSO ⟶ Hℤ ∧ MSO ≃ Hℤ[BSO] ⟵ S[BSO] and MSp ⟶ Hℤ ∧ MSp ≃ Hℤ[BSp] ⟵ S[BSp], and all induce rational equivalences

(3.2)  \begin{matrix}K(MU) & ⟶ K(ℤ[BU]) ⟵ A(BBU) \\ K(MSO) & ⟶ K(ℤ[BSO]) ⟵ A(BBSO) \\ K(MSp) & ⟶ K(ℤ[BSp]) ⟵ A(BBSp) \end{matrix}

of commutative S-algebras. Here we view BU ≃ ΩBBU as the Kan loop group of BBU, ℤ[BU] is the associated simplicial ring, and similarly for BSO and BSp.

Lemma 3.3. (a) The simplicial ℚ-algebra R = ℚ[BU] with π_{∗}R = H_{∗}(BU; ℚ) = ℚ[b_{k} | k ≥ 1] has Hochschild homology

HH_{∗}(R) ≅ ℚ[b_{k} | k ≥ 1] ⊗ E(db_{k} | k ≥ 1) ,

with Poincaré series

h(t) = ∏_{k≥1} \frac{1 + t^{2k+1}}{1 − t^{2k}} ,

and Connes’ operator acts by B(b_{k}) = db_{k}.

(b) The de Rham homology H_{∗}^{dR}(R) ≅ ℚ is concentrated in degree zero, so π_{R}: R → ℚ is a de Rham equivalence.

(c) The Poincaré series of ker(B) ⊂ HH_{∗}(R) is

k(t) = \frac{1 + th(t)}{1 + t} .

(d) The simplicial ℚ-algebra R_{so} = ℚ[BSO] ≃ ℚ[BSp], with π_{∗}R_{so} = ℚ[q_{k} | k ≥ 1], has Hochschild homology

HH_{∗}(R_{so}) ≅ ℚ[q_{k} | k ≥ 1] ⊗ E(dq_{k} | k ≥ 1) .

Its Poincaré series is h_{so}(t) = ∏_{k≥1}(1 + t^{4k+1})/(1 − t^{4k}). The map R_{so} → ℚ is a de Rham equivalence, and ker(B) ⊂ HH_{∗}(R_{so}) has Poincaré series k_{so}(t) = (1 + th_{so}(t))/(1 + t).

Proof. (a) In this case the spectral sequence (2.4) has E_{∗∗}^{2} = HH_{∗}(ℚ[b_{k} | k ≥ 1]) ≅ ℚ[b_{k} | k ≥ 1] ⊗ E(db_{k} | k ≥ 1). The algebra generators are in filtrations 0 and 1, so E^{2} = E^{∞}. This term is free as a graded commutative ℚ-algebra, so HH_{∗}(R) is isomorphic to the E^{∞}-term.

(b) The homology of ℚ[b_{k}] ⊗ E(db_{k}) with respect to B is just ℚ, for each k ≥ 1, so by the Künneth theorem the de Rham homology of HH_{∗}(R) is also just ℚ.

(c) Write H_{n} for HH_{n}(R) and K_{n} for ker(B: H_{n} → H_{n+1}). Let h_{n} = dim_{ℚ} H_{n}, so h(t) = ∑_{n≥0} h_{n}t^{n}, and k_{n} = dim_{ℚ} K_{n}. In view of the exact sequence

0 → ℚ → H_{0} ⟶^{d} H_{1} ⟶^{d} … ⟶^{d} H_{n−1} → K_{n} → 0

we find that 1 − (−1)^{n}k_{n} = h_{0} − h_{1} + ⋯ + (−1)^{n−1}h_{n−1}, so

∑_{n≥0} t^{n}k_{n} − ∑_{n≥0}(−t)^{n} = th(t) − t^{2}h(t) + ⋯ + (−t)^{m+1}h(t) + ….

It follows that the Poincaré series k(t) = ∑_{n≥0} k_{n}t^{n} for ker(B) satisfies k(t) − 1/(1 + t) = th(t)/(1 + t).

(d) The only change from the complex to the oriented real and symplectic cases is in the grading of the algebra generators, which (as long as they remain in even degrees) plays no role for the proofs. □

Theorem 3.4. (a) There is a rationally split homotopy fiber sequence

K(π_{MU}) → K(MU) ⟶^{π} K(ℤ)

and the trace map tr: K(MU) → T HH(MU) identifies K_{∗}(π_{MU}) ⊗ ℚ with the positive-degree part of ker(B) in

T HH_{∗}(MU) ⊗ ℚ ≅ ℚ[b_{k} | k ≥ 1] ⊗ E(db_{k} | k ≥ 1) ,

where |b_{k}| = 2k and B(b_{k}) = db_{k}. Hence K(π_{MU}) has Poincaré series

k(t) − 1 = \frac{th(t) − t}{1 + t} .

(b) There are rationally split homotopy fiber sequences

K(π_{MSO}) → K(MSO) ⟶^{π} K(ℤ)
K(π_{MSp}) → K(MSp) ⟶^{π} K(ℤ)

and the trace maps identify both K_{∗}(π_{MSO})⊗ℚ and K_{∗}(π_{MSp})⊗ℚ with the positivedegree part of ker(B) in

T HH_{∗}(MSO) ⊗ ℚ ≅ T HH_{∗}(MSp) ⊗ ℚ ≅ ℚ[q_{k} | k ≥ 1] ⊗ E(dq_{k} | k ≥ 1) ,

where |q_{k}| = 4k and B(q_{k}) = dq_{k}. Hence K(π_{MSO}) and K(π_{MSp}) both have Poincaré series k_{so}(t) − 1 = (th_{so}(t) − t)/(1 + t).

Remark 3.5. Adding the Poincaré series of K(ℤ), as in Remark 2.6, we find that the Poincaré series of K(MU) and A(BBU) is

\frac{t^{5}}{1 − t^{4}} + \frac{1 + th(t)}{1 + t} ,

whereas the Poincaré series of K(MSO), A(BBSO), K(MSp) and A(BBSp) is t^{5}/(1 − t^{4}) + (1 + th_{so}(t))/(1 + t).

§4. Units, determinants and traces

4.1. Units. For each connective S-algebra A there is a natural map of spaces

w: BGL_{1}(A) → K(A)

that factors as the infinite stabilization map BGL_{1}(A) → BGL_{∞}(A), composed with the inclusion BGL_{∞}(A) → BGL_{∞}(A)^{+} into Quillen’s plus construction, and followed by the inclusion of BGL_{∞}(A)^{+} ≅ {1} × BGL_{∞}(A)^{+} into K_{0}(π_{0}A) × BGL_{∞}(A)^{+} = K(A).

Remark 4.2. This w is an E_{∞} map with respect to the multiplicative E_{∞} structure on K(A) that is induced by the smash product over A. However, we shall only work with the additive grouplike E_{∞} structure on K(A), which comes from viewing K(A) as the underlying infinite loop space of the K-theory spectrum. So when we refer to infinite loop structures below, we are thinking of the additive ones.

We write BSL_{1}(A) = BGL_{1}(π_{A}) for the homotopy fiber of the map BGL_{1}(A) → BGL_{1}(π_{0}A) induced by π_{A}: A → Hπ_{0}A. In the resulting diagram

(4.3)  BSL_{1}(A) ⟶^{w} K(A) ⟶^{π} K(π_{0}A)

the composite map has a preferred null-homotopy (to the base point of the 1component of K(π_{0}A)). The diagram is a rational homotopy fiber sequence if and only if w: BSL_{1}(A) → K(π_{A}) is a rational equivalence. Note that the natural inclusion {1} × BGL_{∞}(A)^{+} → K(A) induces a homotopy equivalence

BGL_{∞}(π_{A})^{+} ≃ K(π_{A}) ,

since K_{0}(A) ≅ K_{0}(π_{0}A).

4.4. Determinants. Suppose furthermore that A is commutative as an S-algebra. One attempt at proving that w is injective could be to construct a map det: K(A) → BGL_{1}(A) with the property that det ∘w ≃ id. However, no such determinant map exists in general, as the following adaption of [Wa82, 3.7] shows.

Example 4.5. When A = S, the map λ ∘ w: BF = BGL_{1}(S) → A(∗) → K(ℤ) factors through BGL_{1}(ℤ) ≃ K(ℤ/2, 1), and π_{2}(λ): π_{2}A(∗) → K_{2}(ℤ) ≅ ℤ/2 is an isomorphism, so π_{2}(w): π_{2}BF → π_{2}A(∗) is the zero map. But π_{2}BF ≅ π_{1}(S) ≅ ℤ/2 is not zero, so π_{2}(w) is not injective. In particular, w is not split injective up to homotopy.

However, it is possible to construct a rationalized determinant map. Recall from Subsection 1.1 that A_{ℚ} is equivalent to HR for some naturally determined commutative simplicial ℚ-algebra R.

Lemma 4.6. Let R be a commutative simplicial ring. There is a natural infinite loop map

det: BGL_{∞}(R)^{+} → BGL_{1}(R)

that agrees with the usual determinant map for discrete commutative rings, such that the composite with w: BGL_{1}(R) → BGL_{∞}(R)^{+} equals the identity.

Proof. The usual matrix determinant det: M_{n}(R) → R induces a simplicial group homomorphism GL_{n}(R) → GL_{1}(R) and a pointed map BGL_{n}(R) → BGL_{1}(R) for each n ≥ 0. These stabilize to a map BGL_{∞}(R) → BGL_{1}(R), which extends to an infinite loop map

det: BGL_{∞}(R)^{+} → BGL_{1}(R) ,

unique up to homotopy, by the multiplicative infinite loop structure on the target and the universal property of Quillen’s plus construction. To make the construction natural, we fix a choice of extension in the initial case R = ℤ, and define det_{R} for general R as the dashed pushout map in the following diagram:

\xymatrix{BGL_{∞}(ℤ) \ar[r] \ar[d] & BGL_{∞}(ℤ)^{+} \ar[d] \ar[dr]^{_{det_{ℤ}}} \\ BGL_{∞}(R) \ar[r] \ar[drr] & BGL_{∞}(R)^{+} \ar@{-->}[dr]^{_{det_{R}}} & BGL_{1}(ℤ) \ar[d] \\  &  & BGL_{1}(R)}

(Recall that Quillen’s plus construction is made functorial by demanding that the left hand square is a pushout.) □

Proposition 4.7. Let A be a connective commutative S-algebra. There is a natural infinite loop map

det_{ℚ}: BGL_{∞}(A)^{+} → BGL_{1}(A)_{ℚ}

that agrees with the rationalized determinant map for a commutative ring R when A = HR, such that the composite

BGL_{1}(A) ⟶^{w} BGL_{∞}(A)^{+} ⟶^{det_{ℚ}} BGL_{1}(A)_{ℚ}

is homotopic to the rationalization map.

Proof. We define det_{ℚ} as the dashed pullback map in the following diagram

\xymatrix{BGL_{∞}(A)^{+} \ar[r] \ar[d] \ar@{-->}[dr]^{_{det_{ℚ}}} & BGL_{∞}(A_{ℚ})^{+} \ar[dr]^{_{det{′ \atop R}}} \\ BGL_{∞}(π_{0}A)^{+} \ar[dr]_{(det_{π_{0}A})_{ℚ}} & BGL_{1}(A)_{ℚ} \ar[r] \ar[d] & BGL_{1}(A_{ℚ}) \ar[d] \\  & BGL_{1}(π_{0}A)_{ℚ} \ar[r] & BGL_{1}(π_{0}A_{ℚ})}

where the vertical maps are induced by the Postnikov section π: A → Hπ_{0}A, and the horizontal maps are induced by the rationalization q: A → A_{ℚ}. The right hand square is a homotopy pullback, since SL_{1}(A)_{ℚ} ≃ SL_{1}(A_{ℚ}).

To define the map det′_{R}, we take R to be a commutative simplicial ℚ-algebra such that A_{ℚ} ≃ HR as commutative Hℚ-algebras. A natural choice can be made for R, as discussed in Subsection 1.1, such that the identification π_{0}A_{ℚ} ≅ π_{0}R is the identity. Then det′_{R} is the composite map

BGL_{∞}(A_{ℚ})^{+} ≃ BGL_{∞}(R)^{+} ⟶^{det_{R}} BGL_{1}(R) ≃ BGL_{1}(A_{ℚ}) ,

with det_{R} from Lemma 4.6. It strictly covers the map det_{π_{0}A_{ℚ}}, so the outer hexagon commutes strictly. This defines the desired map det_{ℚ}.

To compare det_{ℚ} ∘w and q: BGL_{1}(A) → BGL_{1}(A)_{ℚ}, note that both maps have the same composite to BGL_{1}(π_{0}A)_{ℚ}, they have homotopic composites to BGL_{1}(A_{ℚ}), and all composites (and homotopies) to BGL_{1}(π_{0}A_{ℚ}) are equal. Hence the maps to the homotopy pullback are homotopic, too. □

Theorem 4.8. (a) The relative unit map

BBU_{⊗} = BGL_{1}(π_{ku}) ⟶^{w} BGL_{∞}(π_{ku})^{+} ≃ K(π_{ku})

is a rational equivalence, with rational homotopy inverse given by the relative rational determinant map

det_{ℚ}: BGL_{∞}(π_{ku})^{+} → BGL_{1}(π_{ku})_{ℚ} = (BBU_{⊗})_{ℚ} .

(b) The relative unit maps

BBO_{⊗} = BGL_{1}(π_{ko}) → K(π_{ko})
BW_{⊗} = BGL_{1}(π_{ℓ}) → K(π_{ℓ})

are rational equivalences (with rational homotopy inverse det_{ℚ} in each case).

Proof. (a) By Proposition 4.7, the composite

BBU_{⊗} ⟶^{w} K(π_{ku}) ⟶^{det_{ℚ}} (BBU_{⊗})_{ℚ}

is a rational equivalence, so w is rationally injective. Here π_{∗}BBU_{⊗} ≅ π_{∗−1}BU_{⊗} has Poincaré series t^{3}/(1 − t^{2}), just like K(π_{ku}) by Theorem 2.5(a). Thus w is a rational equivalence.

(b) The same proof works for ko and ℓ, using that BBO_{⊗} and BW_{⊗} have Poincaré series t^{5}/(1 − t^{4}) and t^{2p−1}/(1 − t^{2p−2}), respectively. □

Remark 4.9. The analogous map w: BSL_{1}(MU) → K(π_{MU}) is rationally injective, but not a rational equivalence. For the Poincaré series of the source is

t(p(t) − 1) = t^{3} + 2t^{5} + 3t^{7} + 5t^{9} + …,

where p(t) = ∏_{k≥1} 1/(1 − t^{2k}), and the Poincaré series of the target is

(th(t) − t)/(1 + t) = t^{3} + 2t^{5} + 3t^{7} + t^{8} + 5t^{9} + …,

by Theorem 3.4(a). These first differ in degree 8, since π_{8}BSL_{1}(MU) ≅ π_{7}MU is trivial, but K_{8}(MU) and K_{8}(π_{MU}) have rank one. A generator of the latter group maps to db_{1} · db_{2} in ker(B) ⊂ T HH_{∗}(MU) ⊗ ℚ.

In the same way, w: BSL_{1}(MSO) → K(π_{MSO}) and its symplectic variant are rationally injective, but not rational equivalences.

4.10. Traces. Our original strategy for proving that w: BGL_{1}(A) → K(A) is rationally injective for A = ku was to use the trace map tr: K(A) → T HH(A), in place of the rational determinant map. By [Schl04, §4], there is a natural commutative diagram

\xymatrix{BGL_{1}(A) \ar[r]^{w} \ar[d] & K(A) \ar[d] \ar[dr]^{tr} \\ B^{cy}GL_{1}(A) \ar[r] & K^{cy}(A) \ar[r] & T HH(A) \\ GL_{1}(A) \ar[rr] \ar[u] &  & Ω^{∞}A \ar[u]}

where B^{cy} and K^{cy} denote the cyclic bar construction and cyclic K-theory, respectively. The middle row is the geometric realization of two cyclic maps, hence consists of circle equivariant spaces and maps.

When A = ku, the resulting B-operator on H_{∗}(B^{cy}BGL_{1}(A); ℚ) takes primitive classes in the image from H_{∗}(GL_{1}(A); ℚ) ≅ H_{∗}(BU_{⊗}; ℚ) to primitive classes generating the image from H_{∗}(BGL_{1}(A); ℚ) ≅ H_{∗}(BBU_{⊗}; ℚ), so by a diagram chase we can determine the images of the latter primitive classes in H_{∗}(T HH(A); ℚ). By an appeal to the Milnor–Moore theorem [MM65, App.], this suffices to prove that tr ∘ w is rationally injective in this case.

In comparison with the rational determinant approach taken above, this trace method involves more complicated calculations. For commutative S-algebras, it is therefore less attractive. However, for non-commutative S-algebras, the trace method may still be useful, since no (rational) determinant map is likely to exist. We have therefore sketched the idea here, with a view to future applications.

§5. Two-vector bundles and elliptic objects

The following discussion elaborates on the second author’s work with Baas and Dundas in [BDR04]. It is intended to explain some of our interest in Theorem 0.1.

5.1. Two-vector bundles. A 2-vector bundle E of rank n over a base space X is represented by a map X → |BGL_{n}(V)|, where V is the symmetric bimonoidal category of finite dimensional complex vector spaces. A virtual 2-vector bundle E over X is represented by a map X → K(V), where K(V) the algebraic K-theory of the 2-category of finitely generated free V-modules; see [BDR04, Thm. 4.10]. By [BDRR], spectrification induces a weak equivalence Spt: K(V) → K(ku), so the 2vector bundles over X are geometric 0-cycles for the cohomology theory K(ku)^{∗}(X).

5.2. Anomaly bundles. The preferred rational splitting of π: K(ku) → K(ℤ) defines an infinite loop map

det_{ℚ}: K(ku) → (BBU_{⊗})_{ℚ} ,

which extends the rationalization map over w: BBU_{⊗} → K(ku) and agrees with the relative rational determinant on K(π_{ku}). (We do not know if there exists an integral determinant map BGL_{∞}(ku)^{+} → BBU_{⊗} in this case.) We define the rational determinant bundle |E| = det(E) of a virtual 2-vector bundle represented by a map E: X → K(V) ≃ K(ku), as the composite map

|E|: X ⟶^{E} K(ku) ⟶^{det_{ℚ}} (BBU_{⊗})_{ℚ} .

We define the rational anomaly bundle H → LX of E as the composite map

H: LX ⟶^{L|E|} L(BBU_{⊗})_{ℚ} ⟶^{r_{ℚ}} (BU_{⊗})_{ℚ} ,

where r: LBBU_{⊗} → BU_{⊗} is the retraction defined as the infinite loop cofiber of the constant loops map BBU_{⊗} → LBBU_{⊗}. Up to rationalization, H is a virtual vector bundle of virtual dimension +1, i.e., a virtual line bundle. Furthermore, the anomaly bundle relates the composition ⋆ of free loops, when defined, to the tensor product of virtual vector spaces: the square

(5.3)  \xymatrix{LX ×_{X} LX \ar[r]^{(H,H)} \ar[d]_{⋆} & (BU_{⊗})_{ℚ} × (BU_{⊗})_{ℚ} \ar[d]^{⊗} \\ LX \ar[r]^{H} & (BU_{⊗})_{ℚ}}

commutes up to coherent isomorphism.

5.4. Gerbes. A 2-vector bundle of rank 1 over X is the same as a ℂ^{∗}-gerbe G, which is represented by a map G: X → BBU(1). When viewed as a virtual 2vector bundle, via BBU(1) → BBU_{⊗} → K(ku), the associated anomaly bundle is the complex line bundle over LX that is represented by the composite

LX ⟶^{LG} LBBU(1) ⟶^{r} BU(1) .

This is precisely the anomaly line bundle for G, as described in [Br93, §6.2]. Note that the rational anomaly bundles of virtual 2-vector bundles represent general elements in

1 + K̃^{0}(LX) ⊗ ℚ ⊂ K^{0}(LX) ⊗ ℚ ,

whereas the anomaly line bundles of gerbes only represent elements in H^{2}(LX).

5.5. State spaces and action functionals. In physical language, we think of a free loop γ: S^{1} → X as a closed string in a space-time X. For a 2-vector bundle E → X, we think of the fiber H_{γ} (a virtual vector space) at γ of the anomaly bundle H → LX as the state space of that string. Then the state space of a composite of two strings (or a disjoint union of two strings) is the tensor product of the individual state spaces, as is usual in quantum mechanics. Similarly, the state space of an empty set of strings is ℂ. In the special case of anomaly line bundles for gerbes, the resulting state spaces are only complex lines, but in our generality they are virtual vector spaces. These are much closer to the Hilbert spaces usually considered in more analytical approaches to this subject.

There is evidence that a two-part differential-geometric structure (∇_{1}, ∇_{2}) on E over X (somewhat like a connection for a vector bundle, but providing parallel transport both for objects and for morphisms in the 2-vector bundle) provides H → LX with a connection, and more generally an action functional

S(Σ): Hγ̄_{1} ⊗ ⋯ ⊗ Hγ̄_{p} → H_{γ_{1}} ⊗ ⋯ ⊗ H_{γ_{q}} ,

where Σ: F → X is a compact Riemann surface over X, with p incoming and q outgoing boundary circles. The time development of the physical system is then given by the Euler–Lagrange equations of the action functional.

In a little more detail, the idea is that the primary form of parallel transport in (E, ∇_{1}) around γ provides an endo-functor γ̃ of the fiber category E_{x} ≅ V^{n} over a chosen point x of γ. More precisely, parallel transport only provides a zig-zag of functors connecting E_{x} to itself, but the determinant in (BU_{⊗})_{ℚ} is still well-defined. This “holonomy” is then the fiber H_{γ} = det(γ̃) at γ of the anomaly bundle H. For a moving string, say on the Riemann surface F, the secondary form of parallel transport ∇_{2} specifies how the holonomy changes with the string, and this defines the connection ∇ on H → LX. In the gerbe case, this theory has been worked out in [Br93, §5.3], where ∇_{1} is called “connective structure” and ∇_{2} is called “curving”.

For a closed surface F, S(Σ): ℂ → ℂ is multiplication by a complex number, which would only depend on the rational type of E. Optimistically, this association can produce a conformal invariant of F over X, which in the case of genus 1 surfaces would lead to an elliptic modular form. Less naively, additional structure derived from a string structure on X should account for the weight of the modular form. With such structure, a 2-vector bundle E with connective structure ∇ would qualify as a Segal elliptic object over X.

5.6. Open strings. In the presence of D-branes in the space-time X, we can extend the anomaly bundle to also cover open strings with end points restricted to lie on these D-branes; see [Mo04, §3.4]. In this terminology, the (rational) determinant bundle |E| → X plays the role of the B-field.

By a (rational) D-brane (W, E) in X we will mean a subspace W ⊂ X together with a trivialization E of the restriction of the (rational) determinant bundle |E| to W. In terms of representing maps, E is a null-homotopy of the composite map

W ⊂ X ⟶^{E} K(ku) ⟶^{det_{ℚ}} (BBU_{⊗})_{ℚ} .

In similar terminology, we may refer to the determinant bundle |E| → X as the (rational) B-field.

When the B-field |E| is rationally trivial, then a second choice of trivialization E amounts to a choice of null-homotopy of the trivial map W → (BBU_{⊗})_{ℚ}, or equivalently to a map E: W → (BU_{⊗})_{ℚ}. In other words, E is a virtual vector bundle over W of virtual dimension +1, up to rationalization. In this case, the Ktheory class of E → W in 1 + K̃^{0}(W) ⊗ ℚ is the “charge” of the D-brane (W, E). This conforms with the (early) view on D-branes as coming equipped with a charge [E] in topological K-theory.

For a general B-field |E|, the possible trivializations E of its restriction to W instead form a torsor under the group 1+K̃^{0}(W)⊗ℚ. For two such trivializations E and E′ differ by a loop of maps W → (BBU_{⊗})_{ℚ}, or equivalently a map E′−E: W → (BU_{⊗})_{ℚ}. So [E′−E] is a topological K-theory class measuring the charge difference between the two D-branes (W, E) and (W, E′).

Given two D-branes (W_{0}, E_{0}) and (W_{1}, E_{1}) in (X, E), we have a commutative diagram

\xymatrix{W_{0} \ar[r] \ar[d]_{E_{0}} & X \ar[d]^{|E|} & W_{1} \ar[l] \ar[d]^{E_{1}} \\ P (BBU_{⊗})_{ℚ} \ar[r]^{π} & (BBU_{⊗})_{ℚ} & P (BBU_{⊗})_{ℚ} \ar[l]_{π}}

where π: P Y → Y denotes the path space fibration covering a based space Y. An open string in X, constrained to W_{0} and W_{1} at its ends, is a map γ: I → X with γ(0) ∈ W_{0} and γ(1) ∈ W_{1}. In other words, it is an element in the homotopy pullback of the top row in the diagram above. Let Ω(X, W_{0}, W_{1}) denote the space of such open strings. The homotopy pullback of the lower row is Ω(BBU_{⊗})_{ℚ} ≃ (BU_{⊗})_{ℚ}. Hence the 2-vector bundle E and the two D-branes specify a map of homotopy pullbacks

H: Ω(X, W_{0}, W_{1}) → (BU_{⊗})_{ℚ}

that we call the (rational, virtual) anomaly bundle of this space of open strings. Again, we think of each fiber H_{γ} at γ: (I, 0, 1) → (X, W_{0}, W_{1}) as the state space of that open string.

In the presence of a suitable connection (∇_{1}, ∇_{2}) on E → X, parallel transport in (E, ∇_{1}) along γ induces a (zig-zag) functor γ̃ from E_{x} to E_{y}, with determinant det(γ̃) from the fiber of |E| at x = γ(0) to the fiber at y = γ(1). The trivializations of these two fibers provided by the D-brane data E_{0} and E_{1}, respectively, then agree up to a correction term, which is the fiber H_{γ} in the anomaly bundle:

det(γ̃)(E_{0,x}) ≅ H_{γ} ⊗ E_{1,y}

Again, the secondary part of the connection may induce a connection on H over Ω(X, W_{0}, W_{1}), and more generally an action functional S(Σ), where now Σ: F → X and the incoming and the outgoing parts of F are unions of circles and closed intervals. For example, an open string might split off a closed string. One advantage of the above perspective is that the state spaces of open and closed strings arise in a compatible fashion, as the holonomy of parallel transport in the 2-vector bundle E, and this makes the construction of S(Σ) feasible. The gerbe case is discussed in [Br93, §6.6].

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Institute of Mathematics, University of Bonn, DE-53115 Bonn, Germany

E-mail address: ausoni@math.uni-bonn.de

Department of Mathematics, University of Oslo, NO-0316 Oslo, Norway

E-mail address: rognes@math.uio.no