K_{4}(ℤ) = 0

John Rognes

Paris, July 26th 1994

I will talk about the proof that K_{4} of the integers is the trivial group.

Recall that the algebraic K-groups K_{i}(ℤ) are the homotopy groups of a representing space K(ℤ), which is defined as a loop space. Let P(ℤ) = F(ℤ) denote the category of finitely generated projective (or free) ℤ-modules, let Q(ℤ) = QP(ℤ) be Quillen’s Q-construction applied to this category, and let |Q(ℤ)| = BQ(ℤ) be the classifying space of the Q-construction. Then K(ℤ) = Ω|Q(ℤ)|, and |Q(ℤ)| is a model for the first delooping of K(ℤ).

In 1976 Lee and Sczcarba proved K_{3}(ℤ) = ℤ/48, and in 1979 Soulé (1979) found K_{4}(ℤ) ≅ (two–torsion) ⊕ (0 or ℤ/3). These authors computed the ordinary homology H_{∗+1}(|Q(ℤ)|) of the first delooping of K-theory, in a range, and proceeded by killing homotopy groups à la Serre to extract the desired homotopy groups.

The main new idea is to utilize that the representing space K(ℤ) is an infinite loop space, i.e. can be delooped infinitely often. Instead of working at the level of the first delooping, we will pass to the colimit over higher and higher deloopings, to compute the spectrum homology of the K-theory spectrum, and use the Atiyah– Hirzebruch spectral sequence to extract the homotopy groups of interest.

There is a stabilization map from the singly delooped to the infinitely delooped, or stable, situation, and one point of today’s lecture will be to make a comparison of the methods of Lee, Sczcarba and Soulé with mine, and in particular to point out where we are able to overcome the indeterminacy left in Soulé’s result on K_{4}(ℤ).

1. Filtering singly delooped K-theory

To prove finite generation of the K-theory of the ring of integers in a number field, Quillen introduced a filtration of the Q-construction, by letting Q_{k}(ℤ) be the full subcategory of Q(ℤ) involving only modules of rank at most k. Then there is an increasing filtration {Q_{k}(ℤ)}_{k} of subcomplexes of the first delooping of K-theory. By investigating the internal design of the Q-construction we find

|Q_{k}(ℤ)|/|Q_{k−1}(ℤ)| ≃ Σ^{2}B(ℤ^{k}) ∧_{GL_{k}(ℤ)} EGL_{k}(ℤ)_{+} = Σ^{2}B(ℤ^{k})/hGL_{k}(ℤ)

where B(ℤ^{k}) is the Tits building for ℤ^{k}.

The Tits building is usually defined over fields, and B(ℚ^{k}) is the nerve of the partially ordered set of proper nontrivial sub-vectorspaces of ℚ^{k}, partially ordered by inclusion. We may extend the definition to ℤ, letting B(ℤ^{k}) be the nerve of the poset of proper nontrivial direct summands of ℤ^{k}, partially ordered by splittable

Typeset by AMS-TEX

inclusions There is a homeomorphism B(ℤ^{k}) → B(ℚ^{k}) induced by extension of scalars.

We will need explicit models for the suspensions ΣB(ℤ^{k}) and Σ^{2}B(ℤ^{k}). For ΣB(ℚ^{k}) consider the simplicial set with q-simplices of the form

U_{0} ⊂ U_{1} ⊂ ⋯ ⊂ U_{q} = ℚ^{k},

and identify any faces with U_{q} ≠ ℚ^{k} to a base point ∗. For Σ^{2}B(ℚ^{k}) take the simplicial set with q-simplices

0 = U_{0} ⊂ U_{1} ⊂ ⋯ ⊂ U_{q} = ℚ^{k},

and a base point. Similar models may be given over ℤ in place of ℚ, by insisting that every inclusion be splittable.

The following result determines the homotopy type of the Tits buildings.

Theorem (Solomon–Tits). Σ^{2}B(ℤ^{k}) ≃ ⋁ S^{k}.

There is an obvious action of GL_{k}(ℤ) on Σ^{2}B(ℤ^{k}), and the resulting homology representation H̃_{k}(Σ^{2}B(ℤ^{k})) is the Steinberg representation St_{k} = St(ℤ^{k}).

Then Lee and Sczcarba, as well as Soulé, proceed by computing

H_{∗+1}(|Q_{k}(ℤ)|, |Q_{k−1}|) ≅ H_{∗+1−k}(GL_{k}(ℤ); St_{k})

for k ≤ 3, and partially for k = 4. They then resolve extension problems and find H_{∗+1}(|Q(ℤ)|) for ∗ small. Finally they apply the method of killing homotopy groups à la Serre to find K_{∗}(ℤ) = π_{∗+1}(|Q(ℤ)|) when ∗ ≤ 3 or 4.

These calculations may be summarized in the spectral sequence associated to the filtration {|Q_{k}(ℤ)|}_{k} of |Q(ℤ)|.

(∗)  E_{s,t}^{1} = H_{t}(GL_{s+1}(ℤ); St_{s+1}) ⟹ H_{s+t+1}(|Q(ℤ)|)

which receives a map from the groups of interest K_{s+t}(ℤ) ≅ π_{s+t+1}(|Q(ℤ)|).

5  ℤ/2  ?  ?  ?  ?  ?
4  0  (ℤ/2)^{2}  ℤ/2  ?  ?  ?
3  ℤ/2  (ℤ/2)^{2}  ℤ ⊕ (ℤ/2)^{2}  ?  ?  ?
2  0  #4 ⊕ ℤ/3  #4 ⊕ ℤ/3  T_{2} ⊕ ℤ/3  ?  ?
1  ℤ/2  ℤ/2  0  0  ?  ?
0  ℤ  0  0  0  0  0
0  1  2  3  4  5

H_{0}(GL_{k}(ℤ); St_{k}) = 0 for all k ≥ 0 by a result of Lee and Sczcarba. #n denotes a group of order n, and T_{2} a possible two–torsion group.

Localized at three, Soulé’s indeterminacy in K_{4}(ℤ) comes from the ℤ/3 in E_{2,2}^{1} of (∗), which might (or might not) be a boundary, say from the ℤ/3 in E_{3,2}^{1} = H_{2}(GL_{4}(ℤ); St_{4}).

How are the groups H_{∗}(GL_{k}(ℤ); St_{k}) in the columns of (∗) determined ? One approach is to use the skeleton filtration on ΣB(ℤ^{k}). Its nondegenerate q-simplices divide into GL_{k}(ℤ)-orbits with one representative

0 ≠ ℤ^{i_{0}} ⊂ ⋯ ⊂ ℤ^{i_{q}} = ℤ^{k}

in each orbit, for every increasing sequence I = (0 < i_{0} < ⋯ < i_{q} = k). The stabilizer of this orbit is a parabolic subgroup P_{I} ⊆ GL_{k}(ℤ).

((Sketch some parabolics, e.g. P_{1,12} = P_{I} associated to I = (0 < 1 < 2 < 3 = 3) for k = 3, q = 2.))

Hence there is a spectral sequence

(†) E_{s,t}^{1} = ⨁_{I} H_{t}(P_{I}) ⟹ H̃_{s+t}(ΣB(ℤ^{k})/hGL_{k}(ℤ)) ≅ H_{s+t+1−k}(GL_{k}(ℤ); St_{k})

with the sum running over the sequences I increasing to k with (s + 1) terms.

Examples. In the case k = 2 we have the following E^{1}-term:

H_{∗}(GL_{2}(ℤ)) ⟵^{d^{1}} H_{∗}(P_{1})

The d^{1}-differentials are induced by group inclusions.

Similarly for k = 3, there is a spectral sequence with E^{1}-term

H_{∗}(GL_{3}(ℤ)) ⟵^{d^{1}} H_{∗}(P_{1}) ⊕ H_{∗}(P_{12}) ⟵^{d^{1}} H_{∗}(P_{1,12})

Again the d^{1}-differentials are induced by group inclusions.

((Sketch shapes of parabolic groups used here.))

2. Filtering the K-theory spectrum

We wish to replace these constructions on the singly–delooped spaces |Q(ℤ)| = BK(ℤ) with their infinitely delooped, or stable analogs. We prefer to work with iterated applications of Waldhausen’s S_{•}-construction to describe models for the higher deloopings of K(ℤ). This S_{•}-construction applies to a category with cofibrations and weak equivalences, like F(ℤ), and returns a simplicial category of the same kind. Repeating the process n times, S(n)_{•}F(ℤ) is an n-multisimplicial category, and the classifying space of its isomorphism subcategory is an n-fold delooping of K(ℤ), denoted K(ℤ)_{n}:

K(ℤ)_{n} = |iS(n)_{•}F(ℤ)|

Then ΩK(ℤ)_{n+1} ≃ K(ℤ)_{n} for n ≥ 1.

S(n)_{•}F(ℤ) is a category of diagrams in F(ℤ) and we will specify various sub– prespectra of K(ℤ) by restricting the objects in these diagrams to suitable subclasses of modules.

There is a prespectrum level rank filtration

∗ ≃ F_{0}K(ℤ) → F_{1}K(ℤ) → ⋯ → F_{k}K(ℤ) → ⋯ → K(ℤ).

where F_{k}K(ℤ) is the prespectrum with nth space F_{k}K(ℤ)_{n} defined as the subcomplex of K(ℤ)_{n} = |iS(n)_{•}F(ℤ)| realizing the full subcategory of diagrams of modules of rank at most k. The filtration above arises by admitting diagrams involving larger and larger modules.

Proposition. There are homotopy equivalences of prespectra

F_{k}K(ℤ)/F_{k−1}K(ℤ) ≃ D(ℤ^{k}) ∧_{GL_{k}(ℤ)} EGL_{k}(ℤ)_{+} = D(ℤ^{k})/hGL_{k}(ℤ)

where D(ℤ^{k}) is a spectrum called the stable building of rank k.

For every n ≥ 1 the nth space D(ℤ^{k})_{n} of the stable building realizes an explicit n-multisimplicial category of diagrams of submodules of ℤ^{k}. We omit the explicit description, to avoid getting too technical, but we note that ew can specify subprespectra of D(ℤ^{k}) by restricting which submodules of ℤ^{k} may appear in these diagrams.

We have a spectral sequence analogous to (∗) abutting to the spectrum homology of K(ℤ):

(∗∗)  E_{s,t}^{1} = H_{t}^{spec}(D(ℤ^{s+1})/hGL_{s+1}(ℤ)) ⟹ H_{s+t}^{spec}(K(ℤ))

There is a stabilization map from (∗) to (∗∗), and we list the beginnings of the latter spectral sequence now for comparison.

5  ℤ/2 ⟵ (ℤ/2)^{3}  ?  ?  ?  ?
4  0  (ℤ/2)^{2}  ℤ/2  ?  ?  ?
3  ℤ/2 ⟵ (ℤ/2)^{2} ⟵ ℤ  ?  ?  ?
2  0  ℤ/3 ⊕ ℤ/2 ⟵ ℤ/3  ?  ?  ?
1  ℤ/2 ⟵ ℤ/2  0  0  ?  ?
0  ℤ  0  0  0  0  0
0  1  2  3  4  5

A theorem like the Solomon–Tits theorem might still hold in the stable situation, that is, D(ℤ^{k}) may be homotopy equivalent to the suspension spectrum on a wedge of (2k − 2)-spheres, but this is still conjectural in the general case.

We analyze the stable buildings by means of two filtrations. The first, coarser filtration will be used to prove connectivity results for the stable building.

The second, finer filtration gives rise to a spectral sequence for computing the spectrum homology groups H_{t}^{spec}(D(ℤ^{k})/hGL_{k}(ℤ)), much like the spectral sequence arising from the skeleton filtration on ΣB(ℤ^{k}) used above.

First, recall how one using Γ-spaces may construct deloopings by considering categories of sum diagrams. In an analogous fashion, we will filter D(ℤ^{k}) by a collection of sub–prespectra indexed by the various direct sum decompositions of ℤ^{k} as a sum of one or more nontrivial submodules. This is useful because we can describe the filtration subquotients in terms of Tits buildings, corresponding to the singly delooped case.

So let ℤ^{k} = ⊕_{j=1}^{s}V_{j} be a sum decomposition of ℤ^{k} in terms of 1 ≤ s ≤ k nontrivial submodules V_{j} ⊆ ℤ^{k}. Let σ denote such a sum decomposition. Then there is a sub–prespectrum G_{σ}D(ℤ^{k}) of D(ℤ^{k}) realizing categories of diagrams involving only submodules of ℤ^{k} which are sums of some of the V_{j}.

If σ refines a sum decomposition τ, then G_{τ} D(ℤ^{k}) will be a sub–prespectrum of G_{σ}D(ℤ^{k}). So {G_{σ}D(ℤ^{k})}_{σ} will be our coarse filtration.

Proposition. G_{σ}D(ℤ^{k}), modulo the union of the sub–prespectra G_{τ} D(ℤ^{k}) for all τ refined by σ, is homotopy equivalent to the suspension spectrum on

ΣB(V_{1}) ∧ ⋯ ∧ ΣB(V_{s}) ∧^{s}A_{s}

where ^{s}A_{s} is homotopy equivalent to the suspension spectrum on a wedge of (s − 1)! spheres of dimension (2s − 2).

There is a natural action of Σ_{s} on ^{s}A_{s}. Hence W_{s} = H_{2s−2}^{spec}(^{s}A_{s}) is an integral Σ_{s}-representation of rank (s − 1)!. W_{1} is trivial of rank one, while W_{2} is the rank one sign representation of Σ_{2}.

So the coarse filtration subquotient of D(ℤ^{k}) has spectrum homology isomorphic to St(V_{1}) ⊗ … St(V_{s}) ⊗ W_{s}, concentrated in degree ∑_{j=1}^{s}(dim(V_{j}) − 1) + (2s − 2) = k + s − 2. So there is a chain complex (Z_{∗}, d_{∗}) associated to the filtration, with

Z_{k+s−2} = ⨁_{σ} St(V_{1}) ⊗ … St(V_{s}) ⊗ W_{s}

where the sum runs over sum decompositions σ of ℤ^{k} = ⊕_{j=1}^{s}V_{j} into s summands.

The homology of the chain complex (Z_{∗}, d_{∗}) is thus isomorphic to H_{∗}^{spec}(D(ℤ^{k})). The number of summands s runs from 1 to k, so the chain complex is concentrated in degrees (k − 1) to (2k − 2), and so D(ℤ^{k}) is at least (k − 2)-connected for all k.

We can analyze the boundary maps in (Z_{∗}, d_{∗}) by making a comparison with the K-theory of the category of finite sets, via the usual functor from sets to free ℤ-modules. This gives sufficient control of the differentials to prove:

Proposition. D(ℤ^{k}) is at least (k − 1)-connected for k ≥ 2. D(ℤ^{4}) is at least four–connected.

Hence D(ℤ^{k}) is four–connected for k ≥ 4, so F_{k−1}K(ℤ) → F_{k}K(ℤ) is four– connected in these cases, and F_{3}K(ℤ) → K(ℤ) is four–connected. It might very well be five–connected.

Thus the coarse filtration lets us prove that F_{3}K(ℤ) is a reasonably close approximation to K(ℤ), good enough to give an upper bound on K_{4}(ℤ).

The finer filtration may be considered to arise by filtering the suspended Tits buildings ΣB(V_{j}) in the proposition above by their skeleta. It is indexed by tuples of increasing flags of submodules of ℤ^{k} such that the top modules of each of these flags form a sum decomposition of ℤ^{k}.

{0 ≠ U_{0}^{j} ⊂ ⋯ ⊂ U_{n_{j}}^{j} = V_{j}}_{j=1}^{s} with ⊕_{j=1}^{s} V_{j} = ℤ^{k}.

We will call such collections of submodules constellations. We are interested in the spectrum homology of D(ℤ^{k})/hGL_{k}(ℤ), so again we can do this by considering the GL_{k}(ℤ)-orbits of these constellations of submodules in ℤ^{k}. Each orbit contains a unique representative of the following type.

Let each of I^{1}, …, I^{s} be a sequence of natural numbers increasing to k_{1}, …, k_{s}, with ∑_{j=1}^{s} k_{j} = k. We assume k_{1} ≥ ⋯ ≥ k_{s}. So for all j we have I^{j} = (0 <

i_{0}^{j} < ⋯ < i_{n_{j}}^{j} = k_{j}). Then for each j we have a flag 0 ≠ ℤ^{i_{0}^{j}} ⊂ ⋯ ⊂ ℤ^{i_{n_{j}}^{j}} = ℤ^{k_{j}}, which we identify with a flag of submodules in ℤ^{k} via the usual sum decomposition ℤ^{k} = ⊕_{i=1}^{s}ℤ^{k_{i}}.

Let ω be such a constellation of submodules of ℤ^{k}, and let P_{ω} ⊆ GL_{k}(ℤ) be its stabilizer under the GL_{k}(ℤ)-action. We think of this as a generalized parabolic subgroup. P_{ω} will be a product of wreath products of symmetric groups and parabolic groups of the form P_{I}.

Let the size of such a constellation ω be the sum

(∑_{j=1}^{s} n_{j}) + (2s − 2) = ∑_{j=1}^{s}(n_{j} + 1) + (s − 2).

Note that in the case s = 1, the constellations ω are precisely simplices in the suspended Tits building ΣB(ℤ^{k}), and then the size is one less than the dimension of this simplex, which is the same as the column its group homology appears in in the spectral sequence (†).

We can form the finer filtration {G′_{ω}D(ℤ^{k})}_{ω} of the stable building indexed by the isomorphism classes of constellations ω, and obtain a spectral sequence (††) abutting to H_{∗}^{spec}(D(ℤ^{k})/hGL_{k}(ℤ)) with input the group homology of the parabolic subgroups P_{ω} with coefficients in the representations W_{s}.

(††)  E_{p,q}^{1} = ⨁_{ω} H_{q}(P_{ω}; W_{s}) ⟹ H_{∗}^{spec}(D(ℤ^{k})/hGL_{k}(ℤ))

where ω runs through the constellations of size q, and s = s(ω) is the number of decomposition summands.

It is again possible to determine the d^{1}-differential in this spectral sequence via a comparison with the K-theory of finite sets.

Examples. With k = 2, the possible constellations ω are generated by the following collections of subsets of {1, 2}:

{1, 2}, {1} ⊂ {1, 2} and {1}, {2}. The corresponding parabolic subgroups are GL_{2}(ℤ) itself, P_{1} as in the singly delooped case, and Σ_{2} ≀ ℤ/2 = D_{2}, the dihedral group of order eight. The E^{1}-term abutting to H_{∗}^{spec}(D(ℤ^{2})/hGL_{2}(ℤ)) has three columns:

H_{∗}(GL_{2}(ℤ)) ⟵^{d^{1}} H_{∗}(P_{1}) ⟵^{d^{1}} H_{∗}(D_{2}; W_{2})

The stabilization map H̃_{∗+1}(BK(ℤ)) → H_{∗}^{spec}(K(ℤ)) induces the map

H̃_{∗+1}(Σ^{2}B(ℤ^{2})/hGL_{2}(ℤ)) ⟶ H_{∗}^{spec}(D(ℤ^{2})/hGL_{2}(ℤ))

induced by the inclusion of the first two columns into the three–column spectral sequence.

In the case k = 3, the representing constellations are induced by the following chains of subsets partitioning {1, 2, 3}:

{1, 2, 3} of size zero, {1} ⊂ {1, 2, 3} and {1, 2} ⊂ {1, 2, 3} of size one, {1} ⊂ {1, 2} ⊂ {1, 2, 3} and {1, 2}, {3} of size two, {1} ⊂ {1, 2}, {3} of size three, and {1}, {2}, {3} of size four. The corresponding parabolic subgroups are GL_{3}(ℤ), P_{1}, P_{12}, P_{1,12}, P_{12,3}, P_{1,12,3} and Σ_{3} ≀ ℤ/2 = D_{3}. The spectral sequence abutting to H_{∗}^{spec}(D(ℤ^{3})/hGL_{3}(ℤ)) has E^{1}-term:

H_{∗}GL_{3}(ℤ) ⟵ H_{∗}P_{1} ⊕ H_{∗}P_{12} ⟵ H_{∗}P_{1,12} ⊕ H_{∗}P_{12,3} ⟵ H_{∗}P_{1,12,3} ⟵ H_{∗}(D_{3}; W_{3})

Again the stabilization map from singly delooped homology maps into the three first columns here. The extra differentials in the stable setting are typically not induced by group inclusions, but we can still find expressions for them by means of the case of the K-theory of finite sets.

3. Extracting K-groups

We can compute the behaviour of these spectral sequences in low degrees, giving the information in (∗∗). The known values of K_{∗}(ℤ) imply that H_{∗}^{spec}(K(ℤ)) begins (ℤ, 0, 0, ℤ/2, …) as a graded group. This implies the first few differentials in (∗∗), while the others require careful bookkeeping of represening classes and the nature of the boundary map associated to the spectrum level rank filtration.

The conclusion is H_{∗}^{spec}(K(ℤ)) ≅ (ℤ, 0, 0, ℤ/2, 0, ℤ ⊕ (two–torsion), …), and then a spectral sequence argument with the Atiyah–Hirzebruch spectral sequence for stable homotopy theory implies K_{4}(ℤ) = 0. This uses Bökstedt’s two–complete splitting of ΩK^{et}(ℤ)_{2}^{∧} off from ΩK(ℤ)_{2}^{∧}, where K^{et}(ℤ) = JK(ℤ) is a model for étale K-theory of the integers. Hence we can conclude:

Theorem. K_{4}(ℤ) = 0.

The three–torsion ℤ/3 in E_{2,2}^{1} of (∗) comes from a class surviving in (†) from H_{3}(P_{1})⊕H_{3}(P_{12}) ≅ (ℤ/3)^{4} modulo two–torsion. This class dies when (†) is mapped to its stable analog (††), by a d^{1}-differential from H_{3}(P_{12,3}). So this differential represents a reason why the three–torsion found by Soulé in H_{2}(GL_{3}(ℤ); St_{3}) turns out to be a boundary, and does not contribute to K_{4}(ℤ).

In the stable setting this cancellation takes place entirely within the domain of three–by–three matrices, which is why it is accessible to calculation, in spite of our weak understanding of the homology of GL_{4}(ℤ).

Department of Mathematics, University of Oslo, Postboks 1053, Blindern, 0316 Oslo, Norway.