K_{4}(ℤ) IS THE TRIVIAL GROUP

John Rognes

Abstract. We prove that the fourth algebraic K-group of the integers is the trivial group, i.e., that K_{4}(ℤ) = 0. The argument uses rank-, poset- and component filtrations of the algebraic K-theory spectrum K(ℤ) from [R1] and [R2], and a group homology computation of H_{1}(SL_{4}(ℤ); St_{4}) from [S3], to compute the odd primary spectrum homology of K(ℤ) in degrees ≤ 4. This shows that the odd torsion in K_{4}(ℤ) is trivial. The 2-torsion in K_{4}(ℤ) was shown to be trivial in [RW].

Introduction

We prove that K_{4}(ℤ), the fourth algebraic K-group of the integers, is the trivial group. The argument uses the spectrum level rank filtration from [R1], preliminary computations from [R2], and Soulé’s calculation of H_{1}(SL_{4}(ℤ); St_{4}) from [S3], to deduce that the finite Abelian group K_{4}(ℤ) contains no odd torsion.

This result can be combined with the computation by the author and Weibel [We], [RW] of the 2-primary algebraic K-theory of the integers, to deduce that K_{4}(ℤ) contains no 2-torsion. That work uses Voevodsky’s proof of the Milnor conjecture [V], as well as the Bloch–Lichtenbaum spectral sequence [BL] from motivic cohomology to algebraic K-theory. Hence we can state the following theorem:

Theorem 0.1. K_{4}(ℤ) = 0.

This result improves on that of Lee and Szczarba [LS3], asserting that K_{4}(ℤ) is 2-, 3- and 5-torsion, and of Soulé [S1], [S2], asserting that K_{4}(ℤ) is 2- and 3-torsion, with 3-torsion either 0 or ℤ/3.

The paper is organized as follows. Sections 1–4 review the algebraic K-theory spectrum, rank and poset filtrations, stable buildings and stable apartments from [R1], with a view towards the applications in this paper. The reader may refer to [R1, §§1–9] for further discussion. Section 7 introduces the new component filtration of the stable buildings, and Proposition 7.7 describes its filtration subquotients in terms of homotopy types that were introduced and analyzed in sections 5 and 6. This leads to the component filtration spectral sequence in Theorem 7.9. In the final section 8, calculations are made with this spectral sequence, and the rank filtration spectral sequence (8.3). The argument concludes in Theorem 8.5 by citing results from [S3] and [R2] to compute the spectrum homology of K(ℤ) in degrees ≤ 4.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary: 19D50, Secondary: 11F75, 20E42, 55P42, 55T25.

Key words and phrases. Algebraic K-theory of the integers, spectrum level rank filtration, stable building, poset filtration, component filtration.

The author gratefully thanks C. Soulé for writing the paper [S3], and for stimulating discussions regarding the present paper.

1. The rank filtration

Let R be an associative, unital ring. The category P(R) of finitely generated projective R-modules admits the structure of a category with cofibrations and weak equivalences in the sense of Waldhausen [Wa]. The nth space of the algebraic Ktheory spectrum K(R) can be obtained by applying the S_{•}-construction of loc.cit. n times to this category, restricting to the isomorphism subcategory, and taking geometric realization:

K(R)_{n} = |iS_{•}^{n}P(R)| .

The simplices of K(R)_{n} are suitable diagrams in P(R). The homotopy groups of the spectrum K(R) are Quillen’s higher algebraic K-groups [Q] of the ring R.

Definition 1.1. By a spectrum X we mean a sequence of simplicial sets X_{n}, equipped with simplicial structure maps ΣX_{n} = X_{n} ∧ S^{1} → X_{n+1} for all n ≥ 0. A subspectrum Y ⊆ X is a sequence of simplicial subsets Y_{n} ⊆ X_{n} preserved by the structure maps.

Now suppose R is such that R^{i} split injects into R^{j} as an R-module only if i ≤ j. This certainly holds if R is commutative. Then each finitely generated free R-module has a well-defined rank, and there is a sequence of subspectra

(1.2)  ∗ ≃ F_{0}K(R) ⊂ F_{1}K(R) ⊂ ⋯ ⊂ F_{k}K(R) ⊂ ⋯ ⊂ K(R)

called the spectrum level rank filtration [R1, §3]. Here F_{k}K(R) ⊂ K(R) is the subspectrum with nth space built from the simplices of K(R)_{n} that only involve free modules in P(R) of rank ≤ k. When all finitely generated projective R-modules are free the rank filtration exhausts K(R). This holds when R is a PID. To describe the filtration subquotients

F̄_{k}K(R) = F_{k}K(R)/F_{k−1}K(R)

for k ≥ 1 we need to introduce some terminology, and the stable buildings D(R^{k}).

2. Stable buildings

Sets equipped with a partial quasi-ordering (= posets) can be thought of as small categories, with a unique morphism a → b if and only if a ≤ b in the partial quasi-ordering. Conversely small categories with at most one morphism between any pair of objects correspond to such posets. We shall use both points of view. Also, a functor from a small category can be thought of as a (commutative) diagram indexed on that category. We will then call the objects of the indexing category the vertices of the diagram.

Let [q] = {0 → 1 → ⋯ → q} and let [q]^{n} be its n-fold Cartesian power. Viewed as posets, [q] is linearly ordered while [q]^{n} has the product partial ordering. Let Sub(R^{k}) be the subcategory of P(R) of submodules of R^{k} and inclusions among these. By a split injection of R-modules we mean an R-module homomorphism that admits a left inverse. A choice of a left inverse is not part of the structure.

Definition 2.1. Let k ≥ 1. Define D(R^{k})_{n} as the based simplicial set with qsimplices (the base point ∗_{q} and) the diagrams

X: [q]^{n} → Sub(R^{k})

satisfying the following three axioms:

(1) X(i_{1}, …, i_{n}) = 0 if some i_{s} = 0,

(2) X(q, …, q) = R^{k}, and

(3) for each pair of vertices u < v in [q]^{n} the induced map

colim_{u≤w<v} X(w) → X(v)

is a split injection of R-modules. The simplicial structure maps d_{i} and s_{j} act on X by precomposition, except d_{0} which takes any nondegenerate q-simplex to ∗_{q−1}. Its value on a degenerate simplex is determined by the simplicial identities. The spaces D(R^{k})_{n} for n ≥ 0 assemble to a spectrum D(R^{k}), called the kth stable building of R. (See also [R1, §3].)

These axioms are called the lattice conditions, and a diagram X: [q]^{n} → Sub(R^{k}) satisfying them is called a lattice. In a lattice, each X(v) is submodule of R^{k}, and each map X(u) → X(v) is a split injection. By induction each colimit colim_{u≤w<v} X(w) equals the internal sum in X(v) of the submodules X(w) for the w satisfying u ≤ w < v.

The following result is [R1, 3.8].

Proposition 2.2. F̄_{k}K(R) ≃ EGL_{k}(R)_{+} ∧_{GL_{k}(R)} D(R^{k}).

Here the general linear group GL_{k}(R) acts on D(R^{k}) via its natural action on the category of submodules of R^{k}. It remains to determine the naively GL_{k}(R)- equivariant homotopy type of D(R^{k}), in the sense where a GL_{k}(R)-equivariant map which is a non-equivariant homotopy equivalence is considered to be an equivalence. Such maps preserve the homotopy type of the corresponding GL_{k}(R)-homotopy orbit spectra.

3. The poset filtration

Fix a lattice X: [q]^{n} → Sub(R^{k}). The vertices v ∈ [q]^{n} where the induced map

colim_{w<v} X(w) → X(v)

is not an isomorphism will be called special vertices, and the corresponding submodules X(v) ⊆ R^{k} will be called the special modules of X. (These were called ‘pick sites’ and ‘the submodule configuration’ in [R1, §5].) Count each special vertex v with a multiplicity equal to the rank of the cokernel of the map above. Then by [R1, 5.5] there are precisely k special vertices in [q]^{n}, and we can number these as v_{1}, …, v_{k}.

The special vertices and the corresponding special modules uniquely determine the lattice X. At any vertex w ∈ [q]^{n} the module X(w) is the internal sum in R^{k} of the special modules X(v) indexed by the special vertices v with v ≤ w.

Define a partial quasi-ordering ≤ on {1, …, k} by setting i ≤ j if and only if v_{i} ≤ v_{j} in the product partial ordering on [q]^{n}, or equivalently, if X(v_{i}) ⊆ X(v_{j}).

This gives {1, …, k} the structure of a poset, which we denote as ω(X). A different choice of numbering of the k special vertices v_{1}, …, v_{k} may well give rise to a different poset structure on {1, …, k}, but at least its isomorphism class [ω(X)] is well defined. (Two posets are isomorphic if they are isomorphic as categories.) We call [ω(X)] the poset associated to X.

We say that a poset ω_{1} is stronger than a poset ω_{2} with the same underlying set, if each relation i ≤ j in ω_{2} also holds in ω_{1}, or equivalently, if ω_{2} is a subcategory of ω_{1}. Likewise the isomorphism class [ω_{1}] is stronger than [ω_{2}] if ω_{1} is stronger than some poset isomorphic to ω_{2}. This defines partial orderings on the set of posets, resp. the set of isomorphism classes of posets, that have underlying set {1, …, k}. Let ι_{k} = {1 ≅ 2 ≅ ⋯ ≅ k} be the indiscrete poset with k equivalent elements, and let δ_{k} = {1, 2, …, k} be the discrete poset with k unrelated elements. Then ι_{k} is stronger than any other poset ω on {1, 2, …, k}, which in turn is stronger than δ_{k}.

Definition 3.1. Given a poset ω with underlying set {1, …, k} we let

F_{[ω]}D(R^{k}) ⊆ D(R^{k})

be the subspectrum whose nth space has q-simplices (the base point ∗_{q} and) the lattices X: [q]^{n} → Sub(R^{k}) such that [ω(X)] is stronger than [ω]. As [ω] varies, these subspectra filter D(R^{k}). This is the poset filtration on D(R^{k}), which is indexed on the partially ordered set of isomorphism classes [ω] of poset structures on {1, …, k}.

We remark that this filtration is somewhat unusual in that it is not indexed on a linearly ordered set, but a partially ordered one.

Let F_{<[ω]}D(R^{k}) be the subspectrum of F_{[ω]}D(R^{k}) consisting of lattices whose associated poset is strictly stronger than [ω] (not isomorphic to ω), and let

F̄_{[ω]}D(R^{k}) = F_{[ω]}D(R^{k})/F_{<[ω]}D(R^{k})

be the filtration subquotient for the poset filtration. To describe these quotient spectra we compare with the algebraic K-theory of finite sets.

4. Stable apartments

Let E be the category of finite sets. It admits the structure of a category with cofibrations and weak equivalences, and has an associated algebraic K-theory spectrum K(E) equivalent to the sphere spectrum S, by the Barratt–Priddy–Quillen theorem. (See [R1, §4] for more details.) We filter K(E) by subspectra F_{k}K(E) whose nth space has simplices that are diagrams of sets of cardinality ≤ k. Let Sub(k) be the subcategory of E of subsets of {1, …, k} and inclusions among these.

Definition 4.1. Define A(k)_{n} as the based simplicial set with q-simplices (the base point ∗_{q} and) the diagrams

X: [q]^{n} → Sub(k)

satisfying the axioms (1) X(i_{1}, …, i_{n}) = ∅ if some i_{s} = 0,

(2) X(q, …, q) = {1, …, k}, and

(3) for each pair of vertices u < v in [q]^{n} the induced map

colim_{u≤w<v} X(w) → X(v)

is an injection of finite sets.

The face and degeneracy maps are defined as usual by deletions and repetitions, cf. 2.1. The spaces A(k)_{n} for n ≥ 0 assemble to a spectrum A(k), called the kth stable apartment.

For each lattice X: [q]^{n} → Sub(k) we again take note of the special vertices v_{1}, …, v_{k} ∈ [q]^{n} for which the induced map

colim_{w<v} X(w) → X(v)

is not a bijection. This time these can be canonically numbered, by letting v_{i} be the minimal vertex in [q]^{n} such that i ∈ X(v_{i}), for each i ∈ {1, …, k}. Hence to each lattice X of subsets of {1, …, k} we have a well-defined associated poset ω(X), in which i ≤ j if and only if v_{i} ≤ v_{j} in [q]^{n}.

Let F_{ω}A(k) ⊆ A(k) be the subspectrum whose nth space has q-simplices the lattices X with associated poset ω(X) stronger than ω. As ω varies, this defines the poset filtration on A(k). Let F_{<ω}A(k) be the subspectrum of lattices with associated poset strictly stronger than ω, and let

F̄_{ω}A(k) = F_{ω}A(k)/F_{<ω}A(k)

be the filtration subquotient.

There is a functor E → P(R) taking a set U to the free R-module R[U] it generates. We identify R[{1, …, k}] with R^{k} by taking i to the ith basis element e_{i}. There results a functor Sub(k) → Sub(R^{k}) and an embedding A(k) → D(R^{k}). We think of A(k) as a subspectrum of D(R^{k}) by way of this embedding. The embedding preserves associated posets, hence respects the poset filtrations, and maps F̄_{ω}A(k) into F̄_{[ω]}D(R^{k}) for all ω. The symmetric group Σ_{k} acts on A(k) by permuting the elements of {1, …, k}, and the embedding is equivariant with respect to the usual homomorphism Σ_{k} → GL_{k}(R).

Let ω be a poset structure on {1, …, k}. Consider a lattice X: [q]^{n} → Sub(k) determining a q-simplex in the nth space of A(k), thought of as embedded in D(R^{k}), that has associated poset ω(X) = ω. Then the stabilizer for the GL_{k}(R)-action on X is the parabolic subgroup P_{ω} ⊆ GL_{k}(R) leaving invariant the special modules X(v_{i}) of X for i = 1, …, k. These are precisely the submodules R[U_{i}] ⊆ R^{k} spanned by the subsets

U_{i} = {j | j ≤ i in ω}

for i = 1, …, k. In particular this stabilizer is the same for each such X in the stratum F_{ω}A(k)_{n} \ F_{<ω}A(k)_{n}. Also let (Σ_{k})_{ω} ⊆ Σ_{k} be the stabilizer of ω for the Σ_{k}-action on the set of poset structures on {1, …, k}.

The following result is [R1, 8.6].

Proposition 4.2. F̄_{[ω]}D(R^{k}) ≅ GL_{k}(R)/P_{ω+} ∧_{(Σ_{k})_{ω}} F̄_{ω}A(k).

This leaves us with the task of describing the naively (Σ_{k})_{ω}-equivariant homotopy type of F̄_{ω}A(k).

5. (ℓ + 1)-ad homotopy types

A poset is connected if, when viewed as a category its geometric realization is path connected. Its components are the maximal connected sub-posets. Let c(ω) denote the number of components of ω. A poset is linear if i ≤ j or j ≤ i for each pair i, j in its underlying set. It is componentwise linear if each component is linear. The length of a linear poset is the maximal number of composable, non-invertible morphisms in the poset. The length of a componentwise linear poset is the sum of the lengths of its components. Let ℓ(ω) denote the length of a componentwise linear poset ω.

The following result is from [R1, §9].

Proposition 5.1. Let ω be a poset structure on {1, …, k}.

(1) If ω = ι_{k} is indiscrete then F_{ω}A(k) ≅ S is the sphere spectrum with trivial Σ_{k}-action. If ω is not indiscrete then F_{ω}A(k) ≃ ∗ is contractible.

(2) If ω is not componentwise linear then F_{<ω}A(k) ≃ F_{ω}A(k) and F̄_{ω}A(k) ≃ ∗.

Let F_{c}A(k) ⊆ A(k) be the union of the subspectra F_{ω}A(k) over the ω with ≤ c components. Write F_{<c}A(k) = F_{c−1}A(k) for the union over the ω with < c components, and let F̄_{c}A(k) = F_{c}A(k)/F_{<c}A(k) be the filtration subquotient.

The filtration of A(k) by the subspectra F_{c}A(k) for c = 1, …, k will be called the component filtration. We will study its analog for the stable building D(R^{k}) in section 7 below.

If ω has c components, let F_{ω,<c}A(k) = F_{ω}A(k) ∩ F_{<c}A(k), and let F̄_{ω,c}A(k) = F_{ω}A(k)/F_{ω,<c}A(k) be the quotient spectrum. Use similar notation when replacing ω by < ω. Then

F̄_{ω}A(k) = F_{ω}A(k)/F_{<ω}A(k) ≅ F̄_{ω,c}A(k)/F̄_{<ω,c}A(k) .

Now suppose ω is componentwise linear, with c = c(ω) linear components of length ℓ = ℓ(ω). Consider the ℓ componentwise linear posets strictly stronger than ω that are obtained by replacing one of the ℓ non-invertible morphisms in ω by an isomorphism. We denote them by ω_{1}, …, ω_{ℓ}. Each of these has c components and length (ℓ − 1). (There are other posets stronger than ω, and we will consider them below.) Let ϵ be the ‘componentwise indiscrete’ poset with the same component sets as ω, obtained by inverting all the morphisms of ω. Then ϵ is an equivalence relation on {1, …, k}, with c equivalence classes.

Let Δ = Δ{1, …, ℓ} be an affine (ℓ − 1)-simplex with vertices {1, …, ℓ}, and let ∂_{s} = Δ{1, …, ŝ…, ℓ} ⊂ Δ be the codimension 1 face where the vertex s is omitted. Let CΔ be a cone on Δ, and let ΣΔ = CΔ/Δ be the unreduced suspension. It is an ℓ-cell, based at the image of Δ. Similarly, there are ℓ contractible subspaces Σ∂_{s} ⊆ ΣΔ. Any intersection of less than ℓ of these is contractible, while the intersection of all Σ∂_{s} is S^{0}, i.e., two points. The quotient space of ΣΔ by the union of all the Σ∂_{s} is homeomorphic to S^{ℓ}.

Notation 5.2. An (ℓ + 1)-ad of spectra (X, X_{1}, …, X_{ℓ}) is a spectrum X and ℓ subspectra X_{i} ⊆ X for i = 1, …, ℓ. The homotopy type of the union X_{1} ∪ ⋯ ∪ X_{ℓ} within X is determined by the homotopy types of the intersections ⋂_{i∈U} X_{i} as U ranges through the nonempty subsets of {1, …, ℓ}. See [Wl, §0].

Proposition 5.3. Let ω be componentwise linear, as above, with c components and length ℓ. Let ΣΔ be an ℓ-cell, as above.

(1) The ℓ subspectra F̄_{ω_{s},c}A(k) of F̄_{ω,c}A(k) with s = 1, …, ℓ cover F̄_{<ω,c}A(k).

(2) Any intersection of less than ℓ of these subspectra is contractible.

(3) The intersection of all ℓ subspectra is F̄_{ϵ,c}A(k) ≅ F̄_{c}A(c).

Hence the (ℓ + 1)-ad formed by F̄_{ω,c}A(k) and the ℓ subspectra F̄_{ω_{s},c}A(k) is naively (Σ_{k})_{ω}-equivariantly homotopy equivalent to the (ℓ + 1)-ad formed by ΣΔ ∧ F̄_{c}A(c) and the ℓ subspectra Σ∂_{s} ∧ F̄_{c}A(c).

Proof. (1) The subspectrum F_{<ω}A(k) of F_{ω}A(k) is covered by the ℓ subspectra F_{ω_{s}}A(k) together with c(c − 1) subspectra F_{ω′}A(k) where ω′ is obtained from ω by adjoining a relation i ≤ j with i minimal in one component, and j maximal in another component of ω. The latter c(c − 1) posets ω′ all have (c − 1) components, hence are ignored when we pass to the quotient spectrum F̄_{<ω,c}A(k) of F_{<ω}A(k) where simplices with associated poset having fewer than c components are collapsed to the base point. See the proof of [R1, 9.10] for further discussion.

(2) The intersection of less than ℓ of the spectra F̄_{ω_{s},c} is the quotient of a spectrum F_{ω′}A(k) by its subspectrum F_{ω′}A(k) ∩ F_{<c}A(k). Here ω′ is obtained from ω by inverting some, but not all, of its morphisms. Thus ω′ is not indiscrete and F_{ω′}A(k) is contractible. Similarly F_{ω′}A(k) ∩ F_{<c}A(k) is a union of c(c − 1) contractible spectra such that each multiple intersection is contractible. This is because each such intersection has the form F_{ω′′}A(k) where ω′′ is not indiscrete. Hence also F_{ω′}A(k) ∩ F_{<c}A(k) is contractible, and the conclusion follows.

(3) The intersection of all ℓ of the spectra F_{ω_{s}}A(k) is F_{ϵ}A(k), where ϵ is obtained by inverting all morphisms in ω. Collapsing the meet with F_{<c}A(k) to a point identifies the intersection with F̄_{ϵ,c}A(k).

If we number the equivalence classes for ϵ viewed as an equivalence relation as {1, …, c}, then each lattice X: [q]^{n} → Sub(k) with associated poset ω(X) = ϵ takes values that are disjoint unions of such equivalence classes. With the given numbering we can identify these with lattices Y : [q]^{n} → Sub(c) with discrete associated poset ω(Y) = δ_{c}. This provides an isomorphism F̄_{ϵ,c}A(k) ≅ F̄_{c}A(c).

For the final statement, only equivariance needs to be checked. A permutation π ∈ (Σ_{k})_{ω} stabilizing ω acts on F̄_{c}A(c) by way of how it permutes the components of ω. To prove equivariance, construct the (ℓ + 1)-ad homotopy equivalence by adjoining one orbit of (ℓ + 1)-ads at a time, in both F̄_{ω,c}A(k) and ΣΔ ∧ F̄_{c}A(c). This starts with the (Σ_{k})_{ω}-equivariant isomorphism F̄_{ϵ,c}A(k) ≅ F̄_{c}A(c), and ends with a naively equivariant homotopy equivalence of (ℓ + 1)-ads, as claimed. □

Notation 5.4. By [R1, 11.11] the spectrum homology of F̄_{c}A(c) is free Abelian of rank (c−1)! and concentrated in degree (2c−2). It is denoted W_{c} = H_{2c−2}(F̄_{c}A(c)) as a Σ_{c}-module.

6. Suspended Tits buildings

Let F be a field, and consider a finite dimensional F -vector space W. The Tits building B(W) is the geometric realization of the poset of proper nontrivial F -subspaces of W, and inclusions among these. If W has dimension k ≥ 2 then B(W) has the homotopy type of a wedge of (k − 2)-spheres by the Solomon–Tits theorem. The group GL_{k}(F) naturally acts on B(F^{k}), as well as on its homology

St_{k}(F) = H̃_{k−2}(B(F^{k})) .

For k = 1 we define St_{1}(F) = ℤ with trivial GL_{1}(F)-action. We call St_{k}(F) the kth Steinberg module over F.

Let R be a ring, and consider a finitely generated free R-module V. We define the Tits building B(V) as the geometric realization of the poset of proper nontrivial direct summands of V, and split R-module inclusions among these.

Hereafter suppose that F is the quotient field of a PID R. The functor taking an R-module M ⊂ R^{k} = V to F ⊗_{R} M ⊂ F^{k} = W induces a map B(R^{k}) → B(F^{k}).

Lemma 6.1. Let R be a PID with quotient field F. The natural map B(R^{k}) → B(F^{k}) is a simplicial isomorphism.

Proof. When R is a PID and N is a proper nontrivial subspace of F^{k}, then N ∩ R^{k} is a proper nontrivial direct summand of R^{k}. (Proof by induction over the dimension of N.) Then the functors M ↦ F ⊗_{R} M and N ↦ N ∩ R^{k} define inverse isomorphisms between the posets defining B(R^{k}) and B(F^{k}). □

Again let V be a finitely generated free R-module. Let CB(V) be the geometric realization of the poset of nontrivial direct summands of V, and split R-module inclusions among these. This poset adjoins the maximal element V to the poset defining B(V), hence CB(V) is a cone on B(V). Let the suspended Tits building ΣB(V) = CB(V)/B(V) be the quotient space of this cone by its base.

Then ΣB(V) is the simplicial set with q-simplices (a base point ∗_{q} and) the linear chains

0 ≠ M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{q} = V

of split inclusions of submodules of V. The face maps d_{i}, resp. degeneracy maps s_{j}, are given by deleting M_{i}, resp. repeating M_{j}, except d_{0} which takes nondegenerate simplices to the base point.

Clearly ΣB(R^{k}) has the homotopy type of a wedge of (k − 1)-spheres, and

St_{k}(R) = H̃_{k−1}(ΣB(R^{k}))

is the restriction of the kth Steinberg module to GL_{k}(R) for k ≥ 1. More generally we let St(V) = H̃_{k−1}(ΣB(V)) as a GL(V)-module, when V is free of rank k.

7. The component filtration

Definition 7.1. Let F_{c}D(R^{k}) be the subspectrum of D(R^{k}) whose nth space has q-simplices (the base point ∗_{q} and) the lattices X: [q]^{n} → Sub(R^{k}) such that [ω(X)] has at most c components. The sequence of subspectra

∗ ≅ F_{0}D(R^{k}) ⊂ F_{1}D(R^{k}) ⊂ ⋯ ⊂ F_{k}D(R^{k}) = D(R^{k})

is called the component filtration of D(R^{k}). Let F̄_{c}D(R^{k}) = F_{c}D(R^{k})/F_{c−1}D(R^{k}) be the cth subquotient of this filtration.

The poset filtration on D(R^{k}) descends to define a filtration on F̄_{c}D(R^{k}) indexed by the isomorphism classes of posets with c components. Among these, the componentwise linear posets are initial. For adding relations to a componentwise linear poset with c components either yields another poset in this class, or reduces the number of components.

Let F̃_{c}D(R^{k}) ⊆ F̄_{c}D(R^{k}) be the union of the spectra F_{[ω]}D(R^{k}) for which ω is componentwise linear with c components, with their intersection with F_{c−1}D(R^{k}) collapsed to a point. Then the simplices in the nth space of F̃_{c}D(R^{k}) have associated posets that are componentwise linear with c components, together with the base point.

Lemma 7.2. The inclusion F̃_{c}D(R^{k}) ⊆ F̄_{c}D(R^{k}) is a homotopy equivalence.

Proof. The inclusion is filtered by adjoining simplices which have associated posets with c components that are not all linear, and successively allowing more and more such posets. Each filtration subquotient has the form F̄_{[ω]}D(R^{k}) with ω not componentwise linear, which is contractible by 4.2 and 5.1(2). Hence the inclusion at each step of the filtration is a homotopy equivalence, and the result follows. □

Consider a q-simplex X: [q]^{n} → Sub(R^{k}) in the nth space of F̃_{c}D(R^{k}), and number its special vertices v_{1}, …, v_{k}. Then ω = ω(X) has c components, each linear. Let us write ω = λ_{1} ∐ ⋯ ∐ λ_{c}, with each λ_{i} linear. Let k_{i} be the cardinality of λ_{i}, and ℓ_{i} its length. Then ∑_{i=1}^{c} k_{i} = k and ∑_{i=1}^{c} ℓ_{i} = ℓ is the length of ω.

Then the special vertices v_{1}, …, v_{k} of X fall into c linear chains, each unrelated to the others in the product ordering on [q]^{n}. We can reindex them as follows:

v_{11} ≤ v_{12} ≤ ⋯ ≤ v_{1k_{1}}
⋮
v_{c1} ≤ v_{c2} ≤ ⋯ ≤ v_{ck_{c}} .

The corresponding special modules also fall into c linear chains of direct summands of R^{k}:

0 ≠ M_{11} ⊆ M_{12} ⊆ ⋯ ⊆ M_{1k_{1}} = V_{1}
⋮
0 ≠ M_{c1} ⊆ M_{c2} ⊆ ⋯ ⊆ M_{ck_{c}} = V_{c} .

Here M_{ij} = X(v_{ij}) for 1 ≤ i ≤ c, 1 ≤ j ≤ k_{i}, and we write V_{i} = M_{ik_{i}} for 1 ≤ i ≤ c. We call the V_{1}, …, V_{c} the top modules of the simplex X.

Lemma 7.3. The top modules V_{1}, …, V_{c} form a direct sum decomposition of R^{k} into nontrivial submodules:

R^{k} = ⨁_{i=1}^{c} V_{i} .

Proof. The V_{1}, …, V_{c} span R^{k}, because X(q, …, q) = R^{k} is the sum of the special modules X(v) for the special vertices v ≤ (q, …, q), each of which is contained in some X(v_{ik_{i}}) = V_{i}.

The V_{1}, …, V_{c} are pairwise independent, since V_{i} ∩ V_{j} = X(w) where w ∈ [q]^{n} is maximal such that w ≤ v_{ik_{i}} and w ≤ v_{jk_{j}}. When i ≠ j there are no special vertices v_{s} ≤ w, since the special vertices lie in c unrelated linear chains. Thus X(w) = 0. □

Definition 7.4. Let R^{k} = ⨁_{i=1}^{c} V_{i} be a direct sum decomposition of R^{k} into c nontrivial free submodules. Let

E(V_{1}, …, V_{c}) ⊆ F̃_{c}D(R^{k})

be the subspectrum whose nth space has q-simplices the lattices X: [q]^{n} → Sub(R^{k}) with top modules equal to V_{1}, …, V_{c}. The subgroup of GL_{k}(R) preserving the unordered set {V_{1}, …, V_{c}} naturally acts on E(V_{1}, …, V_{c}).

Lemma 7.5. There is a GL_{k}(R)-equivariant splitting of spectra

F̃_{c}D(R^{k}) ≅ ⋁ E(V_{1}, …, V_{c})

where the wedge sum runs over all unordered direct sum decompositions R^{k} = ⨁_{i=1}^{c} V_{i} of R^{k} into c nontrivial free submodules. The action of GL_{k}(R) permutes the summands on the right.

Proof. Any face map that alters the top modules of a simplex must reduce the number of connected components of the associated poset, and hence maps that simplex to the base point in F̃_{c}D(R^{k}). Thus the set level decomposition of the simplices of F̃_{c}D(R^{k}) by their top modules persists to the spectrum level. □

Proposition 7.6. There is a naively equivariant homotopy equivalence

E(V_{1}, …, V_{c}) ≃ ΣB(V_{1}) ∧ ⋯ ∧ ΣB(V_{q}) ∧ F̄_{c}A(c) .

Proof. We shall filter the simplices of E(V_{1}, …, V_{c}) by their associated posets, and the c linear chains of special modules leading up to V_{1}, …, V_{c}. Then we iteratively build E(V_{1}, …, V_{c}) and ΣB(V_{1}) ∧ ⋯ ∧ ΣB(V_{c}) ∧ F̄_{c}A(c) in parallel, by successively allowing more posets ω, together with all ‘compatible’ chains of special modules. Let k_{i} be the rank of V_{i}, for i = 1, …, c.

Consider a poset ω = λ_{1} ∐ ⋯ ∐ λ_{c}, with each λ_{i} linear of cardinality k_{i}. We label the elements of λ_{i} by (i, 1), …, (i, k_{i}). Let each λ_{i} have length ℓ_{i}, and let ℓ = ∑_{i=1}^{c} ℓ_{i}.

Also consider c linear chains of split inclusions

0 ≠ M_{11} ⊆ M_{12} ⊆ ⋯ ⊆ M_{1k_{1}} = V_{1}
⋮
0 ≠ M_{c1} ⊆ M_{c2} ⊆ ⋯ ⊆ M_{ck_{c}} = V_{c}

such that M_{i,j−1} = M_{i,j} if and only if (i, j − 1) ≅ (i, j) in λ_{i}. In particular we assume that M_{i,j−1} ≠ M_{i,j} if (i, j − 1) < (i, j). We then say that these (M_{ij})_{ij} are compatible with ω.

Iteratively build E(V_{1}, …, V_{c}) by adjoining the simplices with associated poset equal to the given ω = λ_{1} ∐ ⋯ ∐ λ_{c}, and special modules the c linear chains displayed above. The specific locations in [q]^{n} of the special vertices may vary, subject to these bounds. This means we are adjoining a copy of F̄_{ω,c}A(k) along the subspectrum F̄_{<ω,c}A(k) where the posets are strictly stronger. Thus the iterative step in building E(V_{1}, …, V_{c}) is to adjoin the first (ℓ + 1)-ad of 5.3.

In parallel build ΣB(V_{1}) ∧ ⋯ ∧ ΣB(V_{c}) ∧ F̄_{c}A(c) by adjoining the subspectrum

ΣΔ^{ℓ_{1}−1} ∧ ⋯ ∧ ΣΔ^{ℓ_{c}−1} ∧ F̄_{c}A(c)

where each ΣΔ^{ℓ_{i}−1} is the suspended simplex represented by M_{i1} ⊆ ⋯ ⊆ M_{ik_{i}} = V_{i} in ΣB(V_{i}).

We naturally identify ΣΔ^{ℓ_{1}−1} ∧ ⋯ ∧ ΣΔ^{ℓ_{c}−1} with ΣΔ where Δ is the (ℓ − 1)- simplex spanned by the ℓ vertices of Δ^{ℓ_{1}−1}, …, Δ^{ℓ_{c}−1}. Thus the iterative step to build ΣB(V_{1}) ∧ ⋯ ∧ ΣB(V_{c}) ∧ F̄_{c}A(c) is to adjoin the second (ℓ + 1)-ad of 5.3. By that proposition these two (ℓ + 1)-ads are homotopy equivalent.

To complete the proof, proceed by induction over the posets ω with c linear components λ_{1}, …, λ_{c} having underlying sets (i, 1), …, (i, k_{i}) as above, partially ordered by strength. For each such ω, adjoin within E(V_{1}, …, V_{c}), resp. ΣB(V_{1}) ∧ ⋯ ∧ ΣB(V_{c}) ∧ F̄_{c}A(c), one copy of the (ℓ + 1)-ad of subspectra in F̄_{ω,c}A(k), resp. the (ℓ + 1)-ad of subspectra in ΣΔ ∧ F̄_{c}A(c), for each c-tuple of chains of special modules (M_{ij})_{ij} compatible with ω. By 5.3 and induction, there results a homotopy equivalence as asserted. Refining the argument to treat one orbit for the action of GL_{k}(R) at a time, we can also obtain the naively equivariant homotopy equivalence we are claiming. □

Bringing these results together, we have proved:

Proposition 7.7. There is a naively GL_{k}(R)-equivariant homotopy equivalence

F̄_{c}D(R^{k}) ≃ ⋁ ΣB(V_{1}) ∧ ⋯ ∧ ΣB(V_{c}) ∧ F̄_{c}A(c)

where the wedge sum runs over the unordered direct sum decompositions R^{k} = ⨁_{i=1}^{c} V_{i} of R^{k} into c nontrivial free submodules. The GL_{k}(R)-action permutes the wedge summands on the right.

Recall from 5.4 that H_{∗}(F̄_{c}A(c)) is concentrated in degree (2c − 2), where it is free Abelian of rank (c − 1)!. As a Σ_{c}-module we denote it W_{c}.

Theorem 7.8. Let R be a PID. There is a complex Z_{∗} = Z_{∗}(R^{k}) of GL_{k}(R)- modules with

Z_{k+c−2} = ⨁ St(V_{1}) ⊗ ⋯ ⊗ St(V_{c}) ⊗ W_{c}

where the sum runs over the unordered direct sum decompositions R^{k} = ⨁_{i=1}^{c} V_{i} into nontrivial submodules. Here 1 ≤ c ≤ k, so Z_{∗} is concentrated in degrees k − 1 ≤ ∗ ≤ 2k − 2. Furthermore,

H_{n}(Z_{∗}) ≅ H_{n}(D(R^{k}))

as GL_{k}(R)-modules.

Proof. The filtration of D(R^{k}) by the F_{c}D(R^{k}) induces a spectral sequence in spectrum homology, with E_{1}-term E_{c,t}^{1} = H_{c+t}(F̄_{c}D(R^{k})). By 7.7 and the Künneth theorem we can express this in terms of the homology of the suspended Tits buildings ΣB(V_{i}) and of F̄_{c}A(c). The homology of ΣB(V_{i}) is concentrated in degree (k_{i} − 1) (by 6.1, since R is a PID), that of F̄_{c}A(c) is concentrated in degree (2c − 2), and both are torsion free. Hence the spectrum homology of F̄_{c}D(R^{k}) is concentrated in degree ∑_{i=1}^{c}(k_{i} − 1) + (2c − 2) = k + c − 2, and so the E^{1}-term above collapses to the single row t = k − 2. Suitably reindexed this is the complex Z_{∗}, and its homology computes H_{∗}(D(R^{k})). □

Theorem 7.9. Let R be a PID. There is a component filtration spectral sequence with

E_{s,t}^{1} = H_{t}(GL_{k}(R); Z_{s})

that is concentrated in the columns k − 1 ≤ s ≤ 2k − 2, and which converges to

H_{s+t}(EGL_{k}(R)_{+} ∧_{GL_{k}(R)} D(R^{k})) ≅ H_{s+t}(F̄_{k}K(R)).

Proof. This is the spectral sequence in spectrum homology for the filtration of EGL_{k}(R)_{+} ∧_{GL_{k}(R)} D(R^{k}) by the subspectra EGL_{k}(R)_{+} ∧_{GL_{k}(R)} F_{c}D(R^{k}) with s = k + c − 2. The spectrum homology of the filtration subquotients is readily computed by a collapsing Serre spectral sequence, since H_{∗}(F̄_{c}D(R^{k})) is concentrated in degree s. Thus the E^{1}-term appears as

E_{s,t}^{1} = H_{s+t}(EGL_{k}(R)_{+} ∧_{GL_{k}(R)} F̄_{c}D(R^{k}))
≅ H_{t}(GL_{k}(R); H_{s}(F̄_{c}D(R^{k})))
≅ H_{t}(GL_{k}(R); Z_{s})

for all s, t ≥ 0. □

8. Calculations

Now specialize the above theory to the case when R = ℤ is the usual ring of integers. We briefly write St_{k} for St_{k}(ℤ) as a GL_{k}(ℤ)-module.

Lee and Szczarba showed in [LS1, 1.3] that H_{0}(GL_{k}(ℤ); St_{k}) = 0 for all k ≥ 2. This has the following consequence:

Lemma 8.1. In the component filtration spectral sequence for R = ℤ we have

E_{s,0}^{1} = H_{0}(GL_{k}(ℤ); Z_{s}) = 0

for all k − 1 ≤ s < 2k − 2.

Proof. For each 1 ≤ c < k the GL_{k}(ℤ)-module Z_{s} = Z_{k+c−2} is a sum of modules co-induced from St_{k_{1}} ⊗ ⋯ ⊗ St_{k_{c}} ⊗ W_{c}, where at least one k_{i} ≥ 2. Hence by Lee and Szczarba’s result its coinvariants vanish. □

Corollary 8.2. For each k ≥ 2 the spectrum F̄_{k}K(ℤ) is at least (k − 1)-connected.

The filtration (1.2) of K(ℤ) by the subspectra F_{s+1}K(ℤ) for s ≥ 0 induces the rank filtration spectral sequence in spectrum homology, with

(8.3)  E_{s,t}^{1} = H_{s+t}(F̄_{s+1}K(ℤ))

converging to H_{s+t}(K(ℤ)).

The spectrum homology H_{∗}(F̄_{k}K(ℤ)) was computed for k ≤ 3 in [R2]. This determines the columns s = 0, 1 and 2 of the rank filtration spectral sequence above. (Working modulo finite 2-groups, only Proposition 3.1, Theorem 3.2(2) and Theorem 3.4(2) of [R2] are needed. These are proven on pages 186 and 198–199 of that paper.) By 8.2 each group E_{s,0}^{1} = 0 for s ≥ 1.

When s = 3, we use the component filtration spectral sequence 7.9 for k = 4 to compute E_{3,1}^{1} ≅ H_{1}(GL_{4}(ℤ); Z_{3}(ℤ^{4})) = H_{1}(GL_{4}(ℤ); St_{4}). By the calculation of Soulé [S3], we have H_{1}(SL_{4}(ℤ); St_{4}) ≡ 0 modulo the Serre class of finite 2-groups. Consequently H_{1}(GL_{4}(ℤ); St_{4}) ≡ 0 modulo the same Serre class, and we deduce that E_{3,1}^{1} ≡ 0 modulo finite 2-groups.

The rank filtration spectral sequence for ℤ thus appears as follows, modulo finite 2-groups:

(8.4)  \begin{aligned}5 & 0 & 0 & 0 & ? & ? & ? \\ 4 & 0 & 0 & 0 & ? & ? & ? \\ 3 & 0 & 0 & ℤ & ? & ? & ? \\ 2 & 0 & ℤ/3 & ℤ/3 & ? & ? & ? \\ 1 & 0 & 0 & 0 & 0 & ? & ? \\ 0 & ℤ & 0 & 0 & 0 & 0 & 0 \\ & 0 & 1 & 2 & 3 & 4 & 5 \end{aligned}

We know that H_{∗}(K(ℤ)) begins (ℤ, 0, 0, ℤ/2, …) by the calculations of Lee and Szczarba [LS2]. Hence there is a bijective differential

d_{2,2}^{1} : E_{2,2}^{1} → E_{1,2}^{1}

modulo finite 2-groups.

Theorem 8.5. The spectrum homology of K(ℤ) begins

H_{∗}(K(ℤ)) ≅ (ℤ, 0, 0, ℤ/2, 0, …)

in degrees 0 ≤ ∗ ≤ 4.

Proof. The 2-primary part of this statement follows from the results of [V], [W] and [RW]. Modulo finite 2-groups the E^{∞}-term of the rank filtration spectral sequence has E_{00}^{∞} ≅ ℤ and trivial groups in total degrees 1 ≤ ∗ ≤ 4, by the discussion above. The result follows. □

Theorem 8.6. The algebraic K-groups of the integers begin

K_{∗}(ℤ) ≅ (ℤ, ℤ/2, ℤ/2, ℤ/48, 0, …)

in degrees 0 ≤ ∗ ≤ 4.

Proof. The 2-primary result follows from the same references as above. Modulo finite 2-groups the unit map S → K(ℤ) is at least 4-connected by the spectrum homology calculation. Hence π_{4}(S) = 0 surjects onto K_{4}(ℤ), which therefore vanishes. □

References

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[LS1] R. Lee and R.H. Szczarba, On the homology and cohomology of congruence subgroups, Invent. Math. 33 (1976), 15–53.

[LS2] R. Lee and R.H. Szczarba, The group K_{3}(ℤ) is cyclic of order forty-eight, Ann. of Math. (2) 104 (1976), 31–60.

[LS3] R. Lee and R.H. Szczarba, On the torsion in K_{4}(ℤ) and K_{5}(ℤ), Duke Math. J. 45 (1978), 101–129.

[Q] D. Quillen, Higher algebraic K-theory. I, Lecture Notes in Math., vol. 341, Springer Verlag, 1973, pp. 85–147.

[R1] J. Rognes, A spectrum level rank filtration in algebraic K-theory, Topology 31 (1992), 813–845.

[R2] J. Rognes, Approximating K_{∗}(ℤ) through degree five, K-Theory 7 (1993), 175–200.

[RW] J. Rognes and C. Weibel, Two-primary algebraic K-theory of rings of integers in number fields, Preprint (1997).

[S1] C. Soule, Addendum to the article “On the torsion in K_{∗}(ℤ)”, Duke Math. J. 45 (1978), 131–132.

[S2] C. Soulé, Groupes arithméthiques et K-théorie des anneaux d’entiers de corps de nombres, Thèse de doctorat d’etat, Université Paris VII (1979).

[S3] C. Soulé, On the 3-torsion in K_{4}(ℤ), Preprint (1998).

[V] V. Voevodsky, The Milnor conjecture, Preprint (1996).

[Wa] F. Waldhausen, Algebraic K-theory of spaces, Lecture Notes in Math., vol. 1126, Springer Verlag, 1985, pp. 318–419.

[Wl] C.T.C. Wall, Surgery on compact manifolds, London Mathematical Society Monographs, vol. 1, Academic Press, 1970.

[We] C. Weibel, The 2-torsion in the K-theory of the integers, C. R. Acad. Sci. Paris 324 (1997), 615–620.

Department of Mathematics, University of Oslo, Norway

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