A SPECTRUM LEVEL RANK

FILTRATION IN ALGEBRAIC K–THEORY

John Rognes

Introduction

This paper introduces a new filtration of the algebraic K–theory spectrum KR of a ring R, and investigates the subquotients of this filtration. KR is constructed from the category P(R) of finitely generated projective R–modules, and its homotopy groups are the algebraic K–groups of R as defined by Quillen [Q2]. There is also a free K–theory K^{f}R, constructed from the weakly cofinal subcategory F(R) of P(R) consisting of finitely generated free R–modules. Inclusion induces a covering map K^{f}R → KR, which in turn induces an isomorphism on π_{i} for i > 0 [Gr,Th-T]. In particular the higher free K–groups π_{i}K^{f}R for i > 0 agree with Quillen’s K–groups.

We construct a sequence of ‘unstable’ algebraic K–theory spectra {F_{k}KR}_{k} filtering K^{f}R. We will assume that R has the invariant dimension property [Mit] so that it makes sense to talk about the rank of a finitely generated free R–module. Then for a fixed rank k ≥ 0, F_{k}KR is constructed as a subspectrum of K^{f}R, built from free R–modules of rank less than or equal to k. As k increases, we obtain an increasing rank filtration {F_{k}KR}_{k} of spectra converging to K^{f}R. It turns out that each subquotient spectrum F_{k}KR/F_{k−1}KR is a homotopy orbit spectrum D(R^{k})/hGL_{k}R for some spectrum–with–GL_{k}R–action D(R^{k}). Furthermore we prove that D(R^{k}) is stably equivalent to the suspension spectrum on a finite dimensional GL_{k}R–complex D(R^{k}), which we call the stable building of R^{k}. Here is a description of the stable building, related to Volodin and Wagoner’s constructions of K–theory [V,Wag] :

Definition 14.5’. Let Σ^{−1}D^{V} (R^{k}) be a simplicial set with q–simplices the (q+1)– tuples {M_{0}, …, M_{q}} of free, proper, nontrivial submodules M_{i} ⊂ R^{k}, satisfying the following condition: There exists an R–basis B for R^{k} for which each submodule M_{i} has a subset of B as an R–basis. The stable building D(R^{k}) is the suspension Σ(Σ^{−1}D^{V} (R^{k})).

Understanding the GL_{k}R–homotopy type of D(R^{k}) is thus sufficient to describe the subquotient spectra F_{k}KR/F_{k−1}KR. (Here, and throughout, we are referring to the weak notion of GL_{k}R–homotopy type, where a GL_{k}R–map which is a nonequivariant homotopy equivalence is treated as an equivalence.) To this end, we explicitly describe a ℤGL_{k}R–complex E_{∗}(k) of length (2k − 2) whose homology computes H_{∗}D(R^{k}) = H_{∗}D(R^{k}). There is then a spectral sequence converging to H_{∗}(F_{k}KR/F_{k−1}KR) with E^{1}–term E_{s,t}^{1} = H_{t}(GL_{k}R; E_{s}(k)). Using it we see that when R is finite the unstable K–groups π_{i}F_{k}KR are |GL_{k}R|–torsion for i > 0.

Typeset by AMS-TEX

From computations, we are led to conjecture that D(R^{k}) is (2k − 3)–connected for R local or Euclidean, so that its homology is concentrated in degree (2k − 2). If so, F_{k}KR → K^{f}R is (2k − 1)–connected, and the free R–modules of rank up through k determine the K–groups up through K_{2k−2}R, except K_{0}R. In the final sections of the paper we give some evidence for this conjecture.

We now outline the constructions involved in our analysis of the GL_{k}R–homotopy type of D(R^{k}). We study D(R^{k}) through the stable GL_{k}R–type of its nth space D^{n}(R^{k}). For n = 1, the complex D^{1}(R^{k}) is the double suspension of the Tits building [Wag] on R^{k}, and this is the motivation for calling D^{n}(R^{k}) the n–dimensional building, and D(R^{k}) the stable building. (Note however that these higher dimensional buildings are not buildings in the abstract sense of Jacques Tits [Ti].) From an explicit description of this n–dimensional building, we find a filtration of D^{n}(R^{k}) indexed by the isomorphism classes [ω] of partial orderings (posets) on the set {1, …, k}, which we call the poset filtration {F_{[ω]}D^{n}(R^{k})}_{[ω]}. This is a filtration because the indexing set is again partially ordered by decreasing strength of the partial orderings involved.

We build a covering of D^{n}(R^{k}) by apartments, which are certain subcomplexes homeomorphic to S^{nk}. We are able to completely compute the behavior of the poset filtration restricted to an apartment A, {F_{ω}A}_{ω}. First we find that each F_{ω}A is stably contractible, except when the poset ω is indiscrete (setting 1 = 2 = ⋯ = k), in which case F_{ω}A ≅ S^{n}. Next let F_{<ω}A ⊂ F_{ω}A denote the subcomplex covered by the F_{ω′}A where ω′ is a strictly stronger ordering than ω, so that F_{ω′}A ⊂ F_{ω}A precedes F_{ω}A in the poset filtration. Then each F_{<ω}A is stably contractible, except when each connected component of ω is a linear ordering. In the latter case, if ω has ℓ components, F_{<ω}A has the stable homotopy type of a wedge of (ℓ−1)! spheres all of dimension size(ω) − 1, where size(ω) is roughly the number of relations lacking for ω to be indiscrete. To summarize, F_{ω}A/F_{<ω}A is stably contractible if ω has some nonlinear component, otherwise its stable homology is concentrated in degree size(ω), and in that degree it is free abelian of rank (ℓ − 1)! .

Lastly we express each subquotient of the poset filtration of D^{n}(R^{k}) as a balanced product of subquotient spaces F_{ω}A/F_{<ω}A and a set of suitable configurations of free submodules of R^{k}. GL_{k}R acts on the whole by permuting these configurations, through the GL_{k}R–action on R^{k}. This gives a complete description of the stable GL_{k}R–type of the subquotients in the poset filtration.

We can now describe the chain complex E_{∗}(k) as the stabilization of the spectral sequence converging to H_{∗}D^{n}(R^{k}), associated to the poset filtration of D^{n}(R^{k}). First we introduce some notation. Let L_{s}(k) denote the set of isomorphism classes of componentwise linear posets of size s on {1, …, k}. Each poset ω determines a parabolic subgroup P_{ω} ⊂ GL_{k}R, and a group of self–equivalences (Σ_{k})_{ω}. Let |ω| the number of components of ω, and let δ_{k} denote the discrete poset with no relations other than the identities. Lastly, write W_{ℓ} for a certain Σ_{ℓ}–representation of rank (ℓ − 1)! over ℤ, namely the ℤ–dual of the free abelian subgroup of the free Lie algebra on ℓ generators x_{1}, …, x_{ℓ} generated by iterated Lie brackets involving each x_{i} exactly once. Then :

Theorem 12.1’. The ℤGL_{k}R–module H_{∗}D(R^{k}) equals the homology of the complex (E_{∗}(k)) of length (2k − 2) below

0 ⟶ ℤGL_{k}R/P_{δ_{k}} ⊗Σ_{k} W_{k} ⟶^{d_{2k−2}} …

⟶^{d_{s+1}} ⨁_{[ω]∈L_{s}(k)} ℤGL_{k}R/P_{ω} ⊗(Σ_{k})_{ω} W_{|ω|} ⟶^{d_{s}} … ⟶^{d_{1}} ℤ ⟶ 0 .

Our result does not explicitly describe the boundary maps d_{s}, but they are easy to write down for small k.

In some cases the rank filtration of a K–theory specializes to classical constructions. The first stage F_{1}KR is always the suspension spectrum of the classifying space of the unit group R^{∗} = GL_{1}R, i.e., F_{1}KR ≅ Σ^{∞}BGL_{1}R_{+}. (The added base point shows that the filtration fails to split for rather trivial reasons.) Also the K–theory of E, the category of based finite sets, maps to KR for each ring R, by taking a finite set to the free R–module generated by its non–basepoint elements. KE has a rank filtration analogous to that for KR, and the above map preserves these filtrations. In particular the rank filtration gives a quick proof of the Barratt–Priddy–Quillen theorem, as S ≅ F_{1}KE ↪ KE is a stable equivalence.

The conjecture mentioned earlier holds for k = 2 when R is a Euclidean domain or a local ring, so we have a short exact sequence :

0 ⟶ H_{2}D(R^{2}) ⟶ ℤGL_{2}R/T_{2} ⊗Σ_{2} ℤ[−1] ⟶^{d_{2}} ℤ̃GL_{2}R/P_{1} ⟶ 0.

Here T_{k} ≅ (R^{∗})^{k} is the diagonal torus, ℤ[−1] the sign representation, P_{1} ⊂ GL_{2}R the upper triangular matrices, and if X is a set, ℤ̃X denotes the kernel of ℤX ⟶^{ϵ} ℤ. The differential d_{2} is given by d_{2}(gT_{2} ⊗ 1) = g(1 − ξ)P_{1} with ξ = ({0 \atop 1} {1 \atop 0}). For k = 3, the complex E_{∗}(3) appears as

0 ⟶ ℤGL_{3}R/T_{3} ⊗_{Σ_{3}} W_{3} ⟶^{d_{4}} ℤGL_{3}R/P_{1,12,3} ⟶^{d_{3}}

ℤGL_{3}R/P_{1,12} ⊕ ℤGL_{3}R/P_{12,3} ⟶^{d_{2}} ℤGL_{3}R/P_{1} ⊕ ℤGL_{3}R/P_{12} ⟶^{d_{1}} ℤ ⟶ 0 ,

with W_{3} ≅ ℤ̃^{3} ⊗ ℤ[−1], where ℤ^{3} is the permutation representation of Σ_{3}, and ℤ̃^{3} its augmentation ideal. See definitions 8.5 and 14.1 for the parabolic subgroups, and section 15 for the boundary maps. This complex is known to be exact at E_{0}, at E_{1} for R local or Euclidean, at E_{2} for R a field, and at E_{3} for R = 𝔽_{2}.

These two complexes can be used for direct group homological computations towards determining the spectrum homology of respectively the second and third stages in the rank filtration of KR for concrete rings R.

Acknowledgements.

This paper is a modified version of my Ph.D. thesis. I thank my inspiring advisor Gunnar Carlsson for friendly and helpful guidance with that entire work, and my parents Bergljot Alnæs Nyhus and Sven Erik Rognes for everything they have meant to me. Thanks are also due to the referee, for precise and lucid comments.

The rank filtration of K–theory

1. The simplicial categories describing K–theory spectra.

We study the algebraic K–theory of a ring R using Waldhausen’s [Wal] construction of the K–theory spectrum KR. This is constructed from the category with cofibrations and weak equivalences P(R) of finitely generated projective R– modules, with split injections as the cofibrations and isomorphisms as the weak equivalences. We also consider the free K–theory of R, K^{f}R, constructed from the subcategory with cofibrations and weak equivalences F(R) of finitely generated free R–modules. In this case, cofibrations are split injections with free quotient modules, while the weak equivalences are still isomorphisms. Reformulated, a morphism f: N → M in F(R) is a cofibration precisely if M ≅ N ⊕ N′ for two free R–modules N and N′, and f is inclusion on the first summand.

For a general category with cofibrations and weak equivalences C, the K–theory spectrum KC has nth space B^{n}KC (n ≥ 1), defined as the geometric realization of the n–multisimplicial category wS_{•}S_{•} … S_{•}C, where S_{•} is Waldhausen’s S_{•}– construction. (KR = KP(R) and K^{f}R = KF(R).) Equivalently we may consider the diagonal simplicial category

[q] ⟼ wS_{q}S_{q} … S_{q}C = wS_{q}^{n}C,

and we shall take the realization of this simplicial category as our model for B^{n}KC.

For concreteness, and to settle notation, the remainder of this section recalls the definition of a category with cofibrations and the S_{•}–construction from [Wal].

Definition 1.1. A category C is pointed if it has a chosen zero object ∗, i.e., an object which is both initial and terminal. A category with cofibrations is a pointed category C with a subcategory coC, the category of cofibrations in C. A morphism in coC is called a cofibration, denoted A ↣ B. coC is assumed to contain the isomorphisms of C, and the morphism ∗ → A for each object A in C. coC is also assumed to be closed under arbitrary cobase change in C, i.e., if A ↣ B is a cofibration and A → C is any morphism in C, the categorical pushout C ∪_{A} B should exist in C, and the canonical morphism C → C ∪_{A} B should be a cofibration.

A category with cofibrations and weak equivalences is a category with cofibrations C, (coC is implicit) together with a subcategory wC, the category of weak equivalences in C. A morphism in wC is called a weak equivalence. wC is assumed to contain the isomorphisms of C. Also, if in a commutative diagram

[figure: C; A; B; C′; A′; B′]

the vertical morphisms are in wC, so is the canonical morphism C ∪_{A}B → C′∪_{A′} B′.

A functor between categories with cofibrations (and weak equivalences) is called exact if it preserves these structures, i.e., the zero object, the subcategory of cofibrations, and the categorical pushout diagrams (and the subcategory of weak equivalences).

Definition 1.2. For q ∈ ℕ let q denote the poset {1 < ⋯ < q} and [q] the poset {0 < ⋯ < q}. View them as categories, and give their n–fold products (q)^{n} and ([q])^{n} the product poset structures. Call (a diagram on) ([q])^{n} a size q n–cube. Also use q and [q] to denote the underlying sets.

For a category D, let the arrow category Ar D have objects the morphisms (i → j) of D, and morphisms from (i → j) to (i′ → j′) the set of commutative diagrams in D :

\xymatrix{i \ar[r] \ar[d] & j \ar[d] \\ i′ \ar[r] & j′}

If C and D are categories, let Fun(C, D) denote the category of functors from C to D and natural transformations of these.

Definition 1.3. For a category with cofibrations C, let S_{q}C denote the category of diagrams (functors) F: Ar[q] → C such that for every j ∈ [q], F (j → j) = ∗, and for every triple i ≤ j ≤ k in [q], F (i → j) ↣ F (i → k) is a cofibration, and the diagram

[figure: F (i → j); F (i → k); F (j → j); F (j → k)]

is a pushout square. The morphisms of S_{q}C are natural transformations of such functors. Now the association [q] ↦ S_{q}C defines a simplicial category S_{•}C. Its simplicial structure is given as a simplicial subcategory of the composite [q] ↦ Ar[q] ↦ Fun(Ar[q], C).

S_{•}C is naturally a simplicial category with cofibrations. This means a simplicial object in the category whose objects are categories with cofibrations, and morphisms the exact functors between these. To see this, let coS_{q}C have morphisms the natural transformations η: F → G such that η(i → j): F (i → j) ↣ G(i → j) is a cofibration for each (i → j) in Ar[q], and for each triple i ≤ j ≤ k the canonical map F (i → k) ∪_{F (i→j)} G(i → j) ↣ G(i → k) is a cofibration.

If C has a category of weak equivalences wC, S_{•}C is similarly a simplicial category with cofibrations and weak equivalences: Let wS_{q}C have morphisms the η: F → G such that each η(i → j) is in wC.

By naturality of these extended structures on S_{•}C, we can iterate the S_{•}–construction, obtaining for each n ≥ 1 an n–multisimplicial category with cofibrations and weak equivalences S_{•}S_{•} … S_{•}C. By inspection, we can identify the 1–skeleton of |wS_{•}C| with the suspension Σ|wC|, yielding an inclusion Σ|wC| ↪ |wS_{•}C|. Applying this to the n–multisimplicial category above, we obtain a (pre–)spectrum

n ⟼ |w \underbrace{S_{•}S_{•} … S_{•}}_{n times} C| ≅ |wSn_{•}C|

which turns out to be an Ω–spectrum after the first stage. Waldhausen defines this to be the K–theory spectrum of C.

After this review of Waldhausen’s definitions, we can now define :

Definition 1.4. The nth space B^{n}KC of the K–theory spectrum of C is the realization of the diagonal simplicial category

[q] ⟼ wS_{q}^{n}C

for n ≥ 1.

2. The lattice conditions on cubical diagrams.

We often prefer to think of an object in S_{q}C in terms of the subdiagram it determines over the full subcategory spanned by the (0 → j) in Ar[q] with j ∈ [q]. Note that the restricted diagram determines the full diagram up to isomorphism, due to the pushout conditions on objects in S_{q}C. We identify [q] with this full subcategory, using the injection r: [q] ↪ Ar[q] given by j ↦ (0 → j). Then for each q, r induces a restriction functor r^{∗} taking a diagram F on Ar[q] to a diagram σ = r^{∗}F on [q]. Explicitly, r^{∗}F is a chain of cofibrations in C of the form

∗ = σ(0) ↣ σ(1) ↣ ⋯ ↣ σ(q).

The following lemmas indicate why this is a workable simplification.

Lemma 2.1. In definition 1.3 of S_{•}C, coS_{•}C and wS_{•}C, it suffices to check the hypotheses as stated when i = 0.

Proof. Use identities of the form A ∪_{B} B ∪_{C} D ≅ A ∪_{C} D repeatedly to reduce to the axioms for categories with cofibrations and weak equivalences. □

Lemma 2.2. For each q and n ≥ 1,

(r^{∗})^{n}: S_{q}^{n}C ⟶ (r^{∗}S_{q})^{n}C
(r^{∗})^{n}: coS_{q}^{n}C ⟶ co(r^{∗}S_{q})^{n}C
(r^{∗})^{n}: wS_{q}^{n}C ⟶ w(r^{∗}S_{q})^{n}C

are all equivalences of categories.

Proof. We view an object F of S_{q}C as a diagram σ = r^{∗}F in coC ⊆ C on [q], with σ(0) = ∗, together with choices of subquotients. The last phrase means choices of pushouts σ(k)/σ(j) = F (j → k) for all the diagrams below

[figure: σ(j); σ(k); ∗; F (j → k)]

with j ≤ k in [q], always choosing F (j → j) = ∗. Then a cofibration σ ↣ τ in coS_{q}C is viewed as a commutative diagram

[figure: ∗ = σ(0); σ(1); …; σ(q); ∗ = τ(0); τ(1); …; τ(q)]

where all the columns are in coC, such that each pushout map τ(j)∪_{σ(j)}σ(k) → τ(k) is a cofibration for j ≤ k (checking for k = j + 1 is sufficient.) Given choices of subquotients for σ, τ, there are then canonical quotient cofibrations extending the transformation diagram above to Ar[q].

A weak equivalence σ → τ in wS_{q}C is a commutative diagram as above, where all the column morphisms are in wC. Again, choosing subquotients for σ and τ determines canonical quotient weak equivalences extending the diagram.

Thus, by lemma 2.1 the forgetful functor r^{∗} is an equivalence when applied to any of the categories S_{q}C, coS_{q}C and wS_{q}C, as any consistent choice of subquotients provides an inverse. This proves the lemma for n = 1, and the general case follows by induction. □

Definition 2.3. Observe that an object σ of w(r^{∗}S_{q})^{n}C is “a diagram on [q], ∗ at 0, in the category of diagrams on [q], ∗ at 0, …, repeated n times, … in the category of diagrams on [q], ∗ at 0, in C”, subject to a hierarchy of conditions requiring certain pushout morphisms to be cofibrations. Clearly this is equivalent to a diagram on ([q])^{n}, ∗ off (q)^{n}, subject to said conditions. Call a cube in C on ([q])^{n}, taking the value ∗ off (q)^{n}, satisfying them a lattice. We refer to these conditions as the lattice conditions, and will make them explicit in lemma 2.5.

A morphism σ → τ in w(r^{∗}S_{q})^{n}C is a morphism of lattices such that σ(x⃗) → τ(x⃗) is in wC for each x⃗ ∈ ([q])^{n}.

To formulate the lattice conditions, we need some notation for cubes and subcubes.

Definition 2.4. Take two sites (vertices) a⃗, b⃗ ∈ ([q])^{n} with b⃗ ≤ a⃗. Write a⃗ = (a_{1}, …, a_{n}) ∈ ([q])^{n}, and similarly for b⃗. a⃗ and b⃗ span a sub–cube of ([q])^{n}, namely the full subcategory of ([q])^{n} with objects x⃗ ∈ ([q])^{n} such that each x_{i} = a_{i} or b_{i}. If m ≤ n is the number of coordinates (i_{1} < i_{2} < … < i_{m}) different in a⃗ and b⃗, this is an m–cube. Alternatively we can index the vertices of this cube by all subsets U ⊆ m. Let x⃗_{U} = (x_{1}, …, x_{n}) with

x_{i} = \{ \begin{matrix}a_{i} & if a_{i} = b_{i}, or if i = i_{j} and j ∈ U, \\ b_{i} & if i = i_{j} and j ∉ U. \end{matrix}

Then U ↦ M_{U} := σ(x⃗_{U}) determines an m–dimensional subcube of σ in C, denoted σ_{a⃗,b⃗}, with a⃗ as the top, b⃗ as the bottom site.

This generalizes to cubes with longer edges. Just as we identified vertices of a size 1 n–cube with subsets U ⊆ n, there is a bijective correspondence taking a vertex a⃗ = (a_{1}, …, a_{n}) ∈ (q)^{n} to the increasing chain of subsets

∅ = U_{0} ⊆ U_{1} ⊆ ⋯ ⊆ U_{q} = n

with U_{i} = U_{i}(a⃗) = { j ∈ n | a_{j} ≤ i }.

Lemma 2.5. Let σ be a diagram in C on ([q])^{n}, such that σ(x⃗) = ∗ for x⃗ ∉ (q)^{n}. σ is a lattice precisely if for each m ≤ n and all m–dimensional subcubes σ_{a⃗,b⃗}, the iterated pushout map

colim_{U⊂m} M_{U} ↣ M_{m} = σ(a⃗)

is a cofibration in C. The colimit, or iterated pushout, is here over the proper subsets U of m, i.e., over the cube σ_{a⃗,b⃗} punctured by removing the top vertex. (These requirements are the lattice conditions.)

Proof. We induct on n. For n = 1, only m = 1 is interesting, when we recover the description of wr^{∗}S_{q}C given in the proof of lemma 2.2.

For n ≥ 2, an object σ of (r^{∗}S_{q})^{n}C is a diagram

∗ = σ(0) ↣ σ(1) ↣ ⋯ ↣ σ(q)

in co(r^{∗}S_{q})^{n−1}C. If a_{n} = b_{n}, m < n, and the m–cube σ_{a⃗,b⃗} is a subcube of σ(a_{n}) on ([q])^{n−1}. Since σ(a_{n}) is a lattice, the iterated pushout map is a cofibration by induction.

If a_{n} > b_{n}, the m–cube σ_{a⃗,b⃗} has a lower (m − 1)–face indexed by U ⊆ m with m ∉ U, and an upper (m − 1)–face indexed by U ⊆ m with m ∈ U. By induction,

A := colim_{m∈U⊂m} M_{U} ↣ M_{m} = σ(a⃗)

and

B := colim_{U⊂m−{m}} M_{U} ↣ M_{m−{m}}

are both cofibrations. Each map M_{U−{m}} ↣ M_{U} for m ∈ U ⊂ m is a cofibration, because σ(b_{n}) ↣ σ(a_{n}) is a cofibration in (r^{∗}S_{q})^{n−1}C. By closure under cobase change, it follows that B ↣ A is also a cofibration. We obtain a commutative square of cofibrations

[figure: B; A; M_{m−{m}}; M_{m}]

where the columns are also cofibrations. Using the second requirement for σ(b_{n}) ↣ σ(a_{n}) to be a cofibration in (r^{∗}S_{q})^{n−1}C, the canonical morphism

A ∪_{B} M_{m−{m}} ↣ M_{m}

must be a cofibration. But the left hand side is exactly colim_{U⊂m} M_{U}.

This proves necessity of the lattice conditions. Sufficiency is proved by reversing the argument. □

Remark 2.6. Note that [q] ↦ w(r^{∗}S_{q})^{n}C is not a simplicial category. The face maps ∂_{i} for i = 1, …, q are well defined as restriction to the size (q − 1) n–subcube of ([q])^{n} where no coordinate is equal to i. Also the degeneracy maps s_{j} inserting identity morphisms restrict well over r^{∗}. However, the face map ∂_{0} of S_{q}C uses the choices of subquotients inherent in a diagram on Ar[q], which are forgotten by r^{∗}. It is possible to define face maps ∂_{0} on the level of lattices, but they will generally only satisfy the simplicial identities up to isomorphism. We shall see in section 3 that this difficulty disappears when we limit attention to the subquotients of the rank filtration.

3. The stable rank filtration.

Hypothesis 3.1. We now turn to the algebraic K–theory of a ring. Let R be an associative ring with unit. Throughout the paper we assume that R satisfies the (strong) invariant dimension property [Mit], i.e., if there exists a split injection of R–modules R^{n} ↪ R^{m} then n ≤ m.

This implies that R^{n} and R^{m} are isomorphic if and only if n = m. Examples of such rings are commutative rings, or algebras over commutative rings which additively are finitely generated and free. This includes finite matrix algebras over commutative rings, and group rings of commutative rings over finite groups. Other examples are those rings which admit a homomorphism onto a skew–field.

For the remainder of the paper, we will only be considering free K–theory, and so let KR denote the free K–theory KF(R) of R in place of Quillen’s K–theory KP(R). As previously noted, these two theories agree in all positive degrees. By our hypothesis, finitely generated free R–modules have a well–defined rank, and π_{0}K^{f}R ≅ ℤ.

We now filter F(R) by the full subcategories F_{k}F(R) of submodules of rank k or less. These are not closed under pushouts, hence do not inherit a structure of category with cofibrations from F(R) :

∗ ≃ F_{0}F(R) ⊂ F_{1}F(R) ⊂ ⋯ ⊂ F_{k}F(R) ⊂ ⋯ ⊂ F(R).

From this we obtain a rank filtration of the nth space B^{n}KR of the (free) K–theory spectrum of R, by restricting to diagrams of submodules in F_{k}F(R) ⊂ F(R), as the rank k varies. This is a special case of an S_{•}–construction with coefficients, where the coefficients are a functor from an arbitrary category to the category with cofibrations and weak equivalences in question.

Definition 3.2. F_{k}B^{n}KR ⊂ B^{n}KR is the subcomplex realizing the simplicial full subcategory of [q] ↦ wS_{q}^{n}F(R), denoted [q] ↦ wS_{q}^{n}F_{k}F(R), consisting of diagrams in F(R) which factor through F_{k}F(R). For each n this gives a filtration

∗ ≃ F_{0}B^{n}KR ⊂ F_{1}B^{n}KR ⊂ ⋯ ⊂ F_{k}B^{n}KR ⊂ ⋯ ⊂ B^{n}KR.

Lemma 3.3. The connecting maps ΣB^{n}KR ↪ B^{n+1}KR restrict to connecting maps ΣF_{k}B^{n}KR ↪ F_{k}B^{n+1}KR making each {F_{k}B^{n}KR}_{n} a prespectrum.

Proof. The connecting map identifies a suspended q–simplex F in ΣB^{n}KR with the (1, q)–bisimplex (0 ↣ F) in wS_{•}Sn_{•}F(R), which has realization B^{n+1}KR. This identification preserves rank, hence is respected by the rank filtration. □

Definition 3.4. Denote the associated spectrum F_{k}KR. It is the kth unstable K–theory of R. The rank filtration of KR is the sequence of spectra:

∗ ≃ F_{0}KR → F_{1}KR → ⋯ → F_{k}KR → ⋯ → KR.

Lemma 3.5. colim_{k→∞} π_{i}F_{k}KR ⟶^{≅} π_{i}KR = K_{i}R for all i, that is, the unstable K–theories approximate KR.

Proof. In spectrum level n, the space |B^{n}KR| is a CW–complex and has an exhaustive filtration by the subcomplexes |F_{k}B^{n}KR|. Hence colim_{k→∞} π_{i}F_{k}B^{n}KR ⟶^{≅} π_{i}B^{n}KR, and the lemma follows as n passes to infinity. □

Remark 3.6. Ω^{∞}KP(R) ≃ K_{0}R×BGL(R)^{+} [Gr], where X^{+} denotes Quillen’s plus construction on X. For finite ranks k, BGL_{k}R^{+} is not generally an infinite loop space, so we can not expect a description of this type for the spaces Ω^{∞}F_{k}KR, (for neither free nor ordinary K–theory). In this sense, our rank filtration differs from the one offered by the plus construction on finite rank matrices.

Next we investigate the subquotients of the nth spaces of the rank filtration.

Definition 3.7. If X is a based G–space, X/hG = EG_{+} ∧_{G} X is its (reduced) homotopy orbit space, also known as the Borel construction.

Proposition 3.8. F_{k}B^{n}KR/F_{k−1}B^{n}KR ≃ D^{n}(R^{k})/hGL_{k}R, where D^{n}(R^{k}) is a finite dimensional GL_{k}R–complex. The sequence of spaces {D^{n}(R^{k})}_{n} inherits the structure of a subspectrum of F_{k}KR/F_{k−1}KR, denoted D(R^{k}). In the category of spectra :

F_{k}KR/F_{k−1}KR ≃ D(R^{k})/hGL_{k}R.

Proof. F_{k}B^{n}KR/F_{k−1}B^{n}KR is the realization of a simplicial category which we denote X′_{•}. In degree q, X′_{q} has objects the diagrams on (Ar[q])^{n} in S_{q}^{n}F(R) where the top (largest) module has rank exactly k, together with a base object ∗_{q}. Morphisms in X′_{q} are isomorphisms of such diagrams.

Let X_{q} be the analogous category of lattices. X_{q} has objects the lattices on ([q])^{n} of free R–modules with top module isomorphic to R^{k}, together with a base object ∗_{q}. Again, morphisms in X_{q} are isomorphisms of such diagrams. Restriction over r: [q] → Ar[q] induces an equivalence of categories r^{∗}: X′_{q} → X_{q} for each q, by a filtered version of lemma 2.2. We claim that [q] ↦ X_{q} is a simplicial category, and that r induces an equivalence of simplicial categories r^{∗}: X′_{•} ⟶^{≃} X_{•}.

Note that on a nondegenerate q–simplex σ of X′_{q}, ∂_{0}(σ) = ∗_{q−1}. On degenerate simplices, ∂_{0} is determined by this observation and the simplicial identities. By remark 2.6, X_{•} supports all the simplicial maps except the 0th face maps. So clearly we can define ∂_{0} by mapping all nondegenerate simplices of X_{•} to ∗, and extend to degenerate simplices as dictated by the simplicial identities. This gives X_{•} the structure of a simplicial category, compatible under the map r^{∗} with the simplicial structure on X′_{•}. Hence r^{∗} is a simplicial equivalence. The simplicial identities referred to in remark 2.6, which in general would hold only up to isomorphism, now reduce to the true identity ∗ = ∗.

Having established the equivalence r^{∗}: F_{k}B^{n}KR/F_{k−1}B^{n}KR ≃ |X_{•}|, we turn to studying X_{•}. It has two simplicial small subcategories Y_{•} and Z_{•}, (X_{•} ⊃ Y_{•} ⊃ Z_{•}) defined as follows. Both Y_{•} and Z_{•} have objects the lattices where the top module is equal to R^{k} (not just isomorphic) and the cofibration morphisms are genuine inclusions (not just injective maps). Y_{•} is the simplicial full subcategory of X_{•}, while Z_{•} has only the identity morphisms, and is effectively a simplicial set, namely D^{n}(R^{k}). (Both Y_{•} and Z_{•} also contain the base object in every degree.)

Every object of X_{•} is isomorphic to one in Y_{•}. Choosing such an isomorphism for every object of X_{•}, taking the identity whenever possible, suffices to describe a deformation retraction from |X_{•}| down to |Y_{•}|.

Furthermore, the morphisms in Y_{•} are precisely determined by the source object and the morphism’s action on the top module R^{k}, i.e., an element of GL_{k}R. This means that Y_{•} is the simplicial based translation category for the GL_{k}R–action on Z_{•}, and we easily obtain |Y_{•}| ≅ |Z_{•}|/hGL_{k}R. Identifying the realization of Z_{•} in the category direction with its simplicial object set, which we call D^{n}(R^{k}), we have now established the needed homotopy equivalence |X_{•}| ≃ D^{n}(R^{k})/hGL_{k}R.

For the statements about spectra, note that the inclusions Z_{•} ⊂ Y_{•} ⊂ X_{•} respect the connecting maps coming from KR. □

By untangling the description of the simplicial set |Z_{•}| in the proof above, we extract the following definition :

Definition 3.9. The n–dimensional building D^{n}(R^{k}) is the simplicial set with q– simplices the lattices on ([q])^{n} of inclusions of free submodules of R^{k}, with those lattices where the top module is not R^{k} identified to a base simplex ∗_{q}.

A face map ∂_{i} for i = 1, …, q deletes the vertices in ([q])^{n} lying in the n ‘hyperplanes’ where at least one coordinate equal to i. ∂_{0} iteratively divides out by the modules at the vertices with at least one coordinate equal to 1. The degeneracies s_{j} insert identity morphisms across the n hyperplanes where at least one coordinate is equal to j.

By inspection, any simplex of dimension greater than nk is degenerate, so D^{n}(R^{k}) is (at most) nk–dimensional. GL_{k}R acts on D^{n}(R^{k}) as automorphisms of R^{k} by permuting the submodules.

Apartments

4. Axial submodules.

Definition 4.1. Let E be the category of finite sets, with injections as cofibrations and bijections as weak equivalences. Its K–theory spectrum has 0th space ℤ×BΣ_{∞}^{+}, and has a rank filtration (filter by cardinality) with analogous properties to those indicated for KR in the previous section. (E should really be the category of based finite sets and functions, but we suppress the base point.)

We have a functor E → F(R) respecting the rank filtration, taking a finite set I to R^{I}, the free R–module generated by I. Identify R^{k} and R^{k}.

Definition 4.2. The axial submodules of R^{k} are the submodules R^{I} ⊆ R^{k}, for I ⊆ k.

Lemma 4.3. D^{n}(k) ↪ D^{n}(R^{k}) is identified with the subcomplex of lattice diagrams in axial submodules of R^{k}. □

Definition 4.4. Call A_{n,k} = D^{n}(k) ⊆ D^{n}(R^{k}) the standard apartment in D^{n}(R^{k}). We will omit the subscripts when n and k are clear from the context. The sequence of spaces {A_{n,k}}_{n} inherits the structure of a subspectrum of D(R^{k}), denoted A_{k} or simply A.

Proposition 4.5. A_{n,k} ≅ S^{nk}.

Corollary 4.6. (Barratt–Priddy–Quillen Theorem) KE ≅ S (the sphere spectrum), i.e., ℤ × BΣ_{∞}^{+} ≃ Q(S^{0}).

Proof of corollary. F_{k}B^{n}KE/F_{k−1}B^{n}KE ≅ D^{n}(k)/hΣ_{k} ≅ S^{nk}/hΣ_{k} is (2n − 1)– connected for k ≥ 2 and homeomorphic to S^{n} for k = 1. Thus S ≅ F_{1}KE ↪ KE is a stable equivalence. □

Definition 4.7. For a set X, let Δ(X) be the complex (simplex) with vertices X, and faces the finite nonempty subsets of X.

Proof of proposition 4.5. First consider the case k = 1. A q–simplex σ ≠ ∗_{q} is a lattice on ([q])^{n} of subsets of 1 = {1}. We claim that the lattice conditions are equivalent to that σ(x⃗) = 1 precisely for the x⃗ ≥ a⃗ for some unique a⃗ ∈ (q)^{n}, and σ(x⃗) = ∅ for all other x⃗.

To prove this take vertices a⃗, b⃗ in the same axial 2–plane of ([q])^{n}. Let c⃗ = min{a⃗, b⃗ } and d⃗ = max{a⃗, b⃗ } (i.e., c_{i} = min{a_{i}, b_{i}} for each i, and similarly for d⃗ ). If σ(a⃗) = σ(b⃗) = 1, necessarily σ(d⃗) = 1, because σ(a⃗) ↪ σ(d⃗) ⊆ 1. So for the pushout map

σ(a⃗) {∪ \atop σ(c⃗)} σ(b⃗) ⟶ σ(d⃗)

to be an injection, we must have σ(c⃗) = 1. The claim now follows by this observation and an induction on n.

Hence we can identify a q–simplex σ ≠ ∗_{q} of D^{n}(1) with the minimal vertex a⃗ ∈ (q)^{n} where σ(a⃗) = 1. Using this we shall now recognize this simplicial set as the smash product of n copies of the simplicial circle Δ(1)/∂Δ(1).

The q–simplices in the product Δ(1)^{n} are all the chains ∅ ⊆ U_{0} ⊆ U_{1} ⊆ ⋯ ⊆ U_{q} ⊆ n. To any such chain with ∅ = U_{0} and U_{n} = n, we can bijectively associate a vertex a⃗ ∈ (q)^{n} as in definition 2.4, with U_{i} = U_{i}(a⃗). The remaining chains correspond to simplices in ∂(Δ(1)^{n}), and are identified to ∗. This induces a simplicial isomorphism (Δ(1)/∂Δ(1))^{n} → D^{n}(1).

For general k, a q–simplex σ ≠ ∗_{q} is a lattice on ([q])^{n} of subsets of k. Any such lattice factors as a product over k of q–simplices in D^{n}(1), with the ith (i ∈ k) factor detecting at what vertex i first appears in the subset. If the base point of D^{n}(1) appears in some factor i, none of the subsets of k occurring can contain i, so the top subset of σ cannot be all of k. Hence the entire fat wedge in (D^{n}(1))^{k} is collapsed to ∗. This proves that D^{n}(k) ≅ D^{n}(1)^{∧k} ≅ S^{nk}. □

5. Pick sites and submodule configurations.

We can characterize the lattice diagrams on ([q])^{n} by certain distinguished vertices of (q)^{n}, called the pick sites, and the R–modules occurring there, called the submodule configuration. This section defines these terms.

Lemma 5.1. The isomorphism class of an object σ in wS_{q}^{n}F(R), i.e., a q–simplex of B^{n}KR, is precisely determined by the ranks { rank(σ(a⃗)) }_{a⃗∈(q)^{n}}.

Proof. Use upward induction in (q)^{n} and the lattice conditions to reduce the lemma to finding isomorphisms f: M′ ⟶ N′ completing the diagram below, where rank(M′) = rank(N′) :

[figure: M; ≅; M′; f; N; N′]

The existence of such f follows from splitness of cofibrations. □

Definition 5.2. The rank data of a simplex σ in B^{n}KR is the n–cube rd(σ) = { a⃗ ↦ rank σ(a⃗) }.

This can be viewed as sitting in the n–fold (diagonal) bar construction B^{n}ℕ ≃ K(ℤ, n) on the additive monoid of the natural numbers ℕ = {0, 1, 2, … }. The simplicial map

rd: B^{n}KR ↠ B^{n}ℕ

can also be viewed, by lemma 5.1, as applying X → π_{0}(X) (path components) levelwise to the simplicial space ([q] ↦ |wS_{q}^{n}F(R)|) = B^{n}KR.

What do the lattice conditions determine about the ranks that can occur as rank data of a σ ∈ B^{n}KR ? Fix a σ, and consider a vertex a⃗ ∈ (q)^{n}. Let b⃗ = (b_{1}, …, b_{n}) with each b_{i} = a_{i} − 1. a⃗ and b⃗ span an elementary subcube of σ as in section 2. Let U ↦ σ(x⃗_{U}) = M_{U} for U ⊆ n be an indexing of this subcube. The lattice condition — that

colim_{U⊂n} M_{U} ↣ M_{n}

is a cofibration — implies that the alternating sum of ranks

∑_{U⊆n}(−1)^{n−|U|} rank(M_{U})

is non–negative; in fact this sum equals the rank of the quotient of the cofibration above.

Definition 5.3. Call the sum above the rank jump of σ at a⃗, denoted rj_{σ}(a⃗).

Lemma 5.4. Let σ be a q–simplex of B^{n}KR. For any x⃗ ∈ (q)^{n}

∑_{a⃗≤x⃗} rj_{σ}(a⃗) = rank(σ(x⃗)).

Proof. Expand the definition of the rank jumps. For each a⃗ ≤ x⃗ in ([q])^{n} the term rank(σ(a⃗)) occurs equally often with opposite signs, except when a⃗ = x⃗ or a⃗ ∉ (q)^{n}. In the latter cases σ(a⃗) = 0. □

Now restrict to D^{n}(R^{k}). Let σ ≠ ∗ be a q–simplex in D^{n}(R^{k}).

Corollary 5.5.

∑_{a⃗∈(q)^{n}} rj_{σ}(a⃗) = k.

Hence rj_{σ}(a⃗) is nonzero only at k vertices of (q)^{n}, counting multiplicities. (a⃗ has multiplicity rj_{σ}(a⃗).) □

Definition 5.6. The vertices a⃗ ∈ (q)^{n} where rj_{σ}(a⃗) > 0 are called the pick sites of σ. The submodules of R^{k} occurring at the pick sites, { σ(a⃗) | rj_{σ}(a⃗) > 0 }, are called the submodule configuration of σ. The pick sites are viewed as an unordered set. The submodule configuration is viewed as a family indexed by the pick sites.

Lemma 5.7. The pick sites and submodule configuration precisely determine σ.

Proof. If we know σ(a⃗) ⊆ R^{k} for all a⃗ where rj_{σ}(a⃗) > 0, we can reconstruct all other σ(a⃗) by upward induction along ([q])^{n}, since for all a⃗ other than the pick sites,

colim_{U⊂n} M_{U} ⟶ M_{n} = σ(a⃗)

is the identity map. This determines all σ(a⃗), and the inclusion morphisms in σ are of course unique. □

Clearly any k–tuple of vertices in (q)^{n} can occur as pick sites. However, given the pick sites there are restrictions on what submodules can appear there. In particular, the rank of σ(a⃗) must equal the number of pick sites p⃗ ∈ (q)^{n} with p⃗ ≤ a⃗.

6. Apartments covering D^{n}(R^{k}).

We now give an alternative description of how to construct apartments in D^{n}(R^{k}). These are subcomplexes of D^{n}(R^{k}) homeomorphic to S^{nk}, obtained as GL_{k}R– translates of the standard apartment discussed in section 4. For n = 1 this recovers the doubly–suspended apartments of the Tits building. We show that the apartments cover the n–dimensional building D^{n}(R^{k}).

Fix a g ∈ GL_{k}R. Its columns g_{1}, …, g_{k} ∈ R^{k} form an R–basis for R^{k}. Each g_{i} spans a line l_{i} = Rg_{i} ⊆ R^{k}, i.e., a rank one free submodule of R^{k} with l_{i} ↣ R^{k}. For any k–tuple of vertices p⃗_{1}, …, p⃗_{k} in (q)^{n}, there is associated a q–simplex σ(g; p⃗_{1}, …, p⃗_{k}) in D^{n}(R^{k}):

a⃗ ⟼ ⨁_{p⃗_{i}≤a⃗} l_{i} ⊆ R^{k}

for a⃗ ∈ ([q])^{n}, with the obvious inclusions.

Definition 6.1. Let A(g) ⊆ D^{n}(R^{k}) be the union of the simplices σ(g; p⃗_{1}, …, p⃗_{k}) as the {p⃗_{i}} vary, together with the base point ∗. These subcomplexes A(g), with g ∈ GL_{k}R, are the apartments of D^{n}(R^{k}).

Lemma 6.2. The q–simplex σ(g; p⃗_{1}, …, p⃗_{k}) has pick sites { p⃗_{1}, …, p⃗_{k} }, and its submodule configuration has the module

⨁_{p⃗_{j}≤p⃗_{i}} l_{j} ⊆ R^{k}.

at p⃗_{i}, i ∈ k. □

Lemma 6.3. Let e ∈ GL_{k}R be the identity. A(e) equals the standard apartment A, and A(g) = g · A ⊆ D^{n}(R^{k}). □

This lemma is clear, and indicates how to check that σ(g; p⃗_{1}, …, p⃗_{k}) is a lattice from the analysis in section 4 for A = D^{n}(k).

Proposition 6.4. The apartments {A(g)}_{g∈GL_{k}R} cover D^{n}(R^{k}).

Proof. Fix a q–simplex σ ∈ D^{n}(R^{k}). If σ = ∗_{q}, it occurs in each A(g). Otherwise σ has k pick sites p⃗_{1}, …, p⃗_{k} ∈ (q)^{n}. At each pick site p⃗_{i} choose a basis B_{i} for the kernel of a splitting of the cofibration

colim_{U⊂n} M_{U} ↣ M_{n} = σ(p⃗_{i})

associated to the elementary cube directly below p⃗_{i}. In other terms, choose a basis for a complementary free summand to the image of the colimit in M_{n}. Use the same basis B_{i} for p⃗_{i} and p⃗_{j} if p⃗_{i} = p⃗_{j} in (q)^{n}.

This gives a collection of k elements B = ⋃_{i} B_{i} = {g_{1}, …, g_{k}} in R^{k}. We claim that for each a⃗, σ(a⃗) has R–basis the {g_{j}}_{j} = ⋃_{p⃗_{i}≤a⃗} B_{i} associated to the pick sites p⃗_{i} ≤ a⃗. This is immediate by upward induction along ([q])^{n}, as each colimit (or iterated pushout) colim_{U⊂n} M_{U} will be spanned by the union of the {B_{i}} associated with pick sites strictly below a⃗. In particular, B = {g_{1}, …, g_{k}} forms an R–basis for R^{k}. Let g ∈ GL_{k}R have ith column g_{i}. Then σ = σ(g; p⃗_{1}, …, p⃗_{k}) ∈ A(g). □

The poset filtration

7. Posets.

The results we need about partially ordered sets (posets) are collected here.

Definition 7.1. A poset ω is a small category where for each pair of objects a and b there is at most one morphism from a to b, denoted a → b. If the morphism exists, we write a ≤ b, and if further there is no morphism from b to a, we write a < b. A poset is depicted by listing its objects and a (minimal) set of generating morphisms, separating components by commas. a ⇄ b is shorthand for a → b and b → a.

Fixing an underlying (object) set k, the set of poset structures ω on k forms a poset again, Po(k). We set ω ≤ ω′ if ω is the stronger poset, i.e., ω′ is a subcategory of ω. The discrete poset on k, δ_{k}, has only the identity morphisms, and is maximal in Po(k). The indiscrete poset on k, ι_{k}, has a morphism from each a to each b, and is minimal in Po(k).

The symmetric group Σ_{k} acts on Po(k). If π ∈ Σ_{k}, and ω is a poset on k, π · ω is the poset on k such that a → b in π · ω if and only if π^{−1}a → π^{−1}b in ω. The Σ_{k}–orbit of ω is its isomorphism class [ω], and the isomorphism classes [ω] of posets on k form a poset IPo(k), analogously to Po(k). Let (Σ_{k})_{ω} = { π ∈ Σ_{k} | π · ω = ω } denote the isotropy subgroup.

Write ω_{i} = { j ∈ k | j → i in ω }. The collection of sets { ω_{i} }_{i} determines ω. The components of ω are the maximal subposets with connected underlying space

(as a category). Let |ω| denote the number of components. ω is linear if for any a and b we have a ≤ b or b ≤ a. The indiscrete poset is thus linear. A poset ω is componentwise linear if each connected component is linear. The size of a finite poset ω is

size(ω) = |ω| + (number of equivalence classes in ω) − 2.

Here a and b in ω are equivalent if a ⇄ b in ω. Let L_{s}(k) ⊆ IPo(k) denote the set of isomorphism classes of componentwise linear posets of size s.

Lemma 7.2. If ω ≤ ω′ then size(ω) ≤ size(ω′). If ω′ is componentwise linear, then ω < ω′ implies size(ω) < size(ω′). □

Definition 7.3. Call i → j in ω indecomposable if j ↛ i and there is no a ∈ k with i → a → j such that a ↛ i and j ↛ a. The length of a linear poset is one less than the number of non–equivalent objects in it, or equivalently its maximal number of composable indecomposable morphisms.

A morphism i → j is extremal with respect to ω if i ↛ j in ω, i is minimal among the a such that a ↛ j in ω, and j is maximal among the b such that i ↛ b in ω. Let ω + (i → j) denote the poset obtained by adjoining the morphism i → j to ω.

Lemma 7.4. The maximal elements in { ω′ ∈ Po(k) | ω′ < ω } are exactly the ω′ = ω + (i → j) where i → j is extremal with respect to ω.

Proof. If ω′ < ω, there exists i → j in ω′ but not in ω. Either i → j is extremal with respect to ω, or we can find i′ → i and j → j′ in ω such that i′ → j′ in ω′, but not in ω, and i′ ⇄̸ i or j ⇄̸ j′. Either i′ → j′ is now extremal with respect to ω, or we can repeat the process. As long as the latter holds, we obtain a decreasing sequence i, i′, i′′, … and an increasing sequence j, j′, j′′, . . .. As k is finite, the process must eventually halt. Hence we can find i → j in ω′ but not in ω which is extremal with respect to ω. Then ω′ ≤ ω + (i → j) < ω. □

Definition 7.5. Suppose the poset ω has underlying set k. A subset I ⊆ k is called an order ideal if ω_{i} ⊆ I for each i ∈ I.

8. The poset filtration of D^{n}(R^{k}).

In this section we introduce the poset filtration on D^{n}(R^{k}) by observing that the pick sites of a lattice inherit a partial ordering from the n–cube in which they are located. A filtration indexed over the isomorphism classes of partial orderings on {1, …, k} results, and we describe the subquotients of this filtration in terms of an analogous filtration on the standard apartment A, and a GL_{k}R–set of submodule configurations.

Consider a q–simplex σ of D^{n}(R^{k}), different from ∗. Choose some numbering of its pick sites (p⃗_{1}, …, p⃗_{k}). This determines a function k → (q)^{n}, and hence induces a poset structure ω_{σ} on k by pullback; this is the strongest ordering for which the function is order preserving. Choosing a different numbering of the pick sites induces an isomorphic poset structure, as the results will differ only by the action of an element in Σ_{k}. Hence σ determines a unique isomorphism class [ω_{σ}] ∈ IPo(k). Thus there is a filtration {F_{[ω]}D^{n}(R^{k})} of D^{n}(R^{k}) indexed by the poset of isomorphism classes of posets, IPo(k) :

Definition 8.1. Let F_{[ω]}D^{n}(R^{k}) consist of (the base simplex ∗ and) the simplices whose pick sites satisfy the relations of [ω], i.e., [ω_{σ}] ≤ [ω]. As [ω] varies, this is the poset filtration of D^{n}(R^{k}).

The standard apartment carries the further structure of a poset filtration indexed by Po(k), i.e., the actual poset structures on k. This is because there is a canonical numbering of the pick sites of a (non–basepoint) simplex σ in A, given by insisting that the ith pick site p⃗_{i} is the initial site x⃗ ∈ (q)^{n} where i ∈ k occurs, i.e., i ∈ σ(x⃗). Thus we have a filtration {F_{ω}A}_{ω∈Po(k)} of the standard apartment, as defined below :

Definition 8.2. Let F_{ω}A consist of (∗ and) the simplicies σ ∈ D^{n}(k) ↪ D^{n}(R^{k}) for which the pick sites with the canonical numbering satisfy p⃗_{i} ≤ p⃗_{j} in (q)^{n} whenever i ≤ j in ω.

Definition 8.3. Let {F_{[ω]}A(g)} denote the restriction of the poset filtration to an apartment A(g) of D^{n}(R^{k}), i.e., F_{[ω]}A(g) = F_{[ω]}D^{n}(R^{k}) ∩ A(g).

Write F_{<[ω]}D^{n}(R^{k}) = ⋃_{ω′<ω} F_{[ω′]}D^{n}(R^{k}) for the subcomplex of F_{[ω]}D^{n}(R^{k}) occurring ‘earlier’ in the filtration, and similarly for F_{<[ω]}A(g) and F_{<ω}A.

Lemma 8.4.

F_{[ω]}A = Σ_{k} · F_{ω}A = ⋃_{ω′≅ω} F_{ω′}A ⊆ A
F_{[ω]}D^{n}(R^{k}) = ⋃_{g∈GL_{k}R} F_{[ω]}A(g)
F_{[ω]}A/F_{<[ω]}A ≅ Σ_{k+} ∧(Σ_{k})_{ω} (F_{ω}A/F_{<ω}A)

□

Note that F_{[ω]}D^{n}(R^{k}) is GL_{k}R–invariant, and preserved by the connecting maps in the quotient spectrum F_{k}KR/F_{k−1}KR. Hence we obtain a spectrum–with– GL_{k}R–action F_{[ω]}D(R^{k}) = {F_{[ω]}D^{n}(R^{k})}_{n}, and similarly there are spectra F_{ω}A_{k}, F_{[ω]}A_{k}, F_{<ω}A_{k}, F_{<[ω]}A_{k} and F_{<[ω]}D(R^{k}).

Next consider a simplex σ in the standard apartment A, with ω_{σ} = ω. Its submodule configuration consists of the axial submodules R^{ω_{i}} in R^{k} as i varies through k. Hence the isotropy of the GL_{k}R–action on this submodule configuration, which fixes the indexing of the axial submodules, is the parabolic subgroup defined below. As usual the orbit of this action can be identified with GL_{k}R/P_{ω}.

Definition 8.5. For any partial ordering ω on k let P_{ω} denote the parabolic subgroup { g ∈ GL_{k}R | gR^{ω_{i}} = R^{ω_{i}} for all i ∈ k }.

We have the following description of the subquotients of the poset filtration.

Proposition 8.6. For each ω,

F_{[ω]}D^{n}(R^{k})/F_{<[ω]}D^{n}(R^{k}) ≅ GL_{k}R/P_{ω+} ∧(Σ_{k})_{ω} (F_{ω}A/F_{<ω}A)

GL_{k}R–equivariantly.

Proof. We use the inclusion A ↪ D^{n}(R^{k}) to identify Y := F_{ω}A/F_{<ω}A with a subcomplex of X := F_{[ω]}D^{n}(R^{k})/F_{<[ω]}D^{n}(R^{k}). Every simplex (except ∗) of X is a lattice on ([q])^{n} whose pick sites precisely satisfy ω when suitably numbered. The simplices in Y are those lattices where furthermore all the modules in the submodule configuration are axial.

As usual GL_{k}R acts on X by moving submodules of R^{k} around. We claim that every simplex of X is in the GL_{k}R–orbit of some simplex in Y.

For any simplex σ ≠ ∗ of Y, the subgroups P_{ω} and (Σ_{k})_{ω} map σ into Y. We claim that the set of elements of GL_{k}R which take σ into Y is precisely the product of subgroups (Σ_{k})_{ω} · P_{ω}. In particular (Σ_{k})_{ω} · P_{ω} acts on Y.

Note that (Σ_{k})_{ω} normalizes P_{ω}, so that (Σ_{k})_{ω} acts on GL_{k}R/P_{ω}. Using the two claims, we can now conclude

X ≅ GL_{k}R_{+} ∧(Σ_{k})_{ω} · P_{ω} Y ≅ GL_{k}R/P_{ω+} ∧(Σ_{k})_{ω} Y

as P_{ω} acts trivially on Y. It remains to prove the two claims.

To prove the first claim, we use proposition 6.4. Any simplex σ ≠ ∗ in X lies in an apartment A(g), for some g ∈ GL_{k}R. Then g^{−1} · σ lies in A, and the partial ordering of its pick sites is isomorphic to ω. The isomorphism corresponds to a renumbering of the pick sites, which can be effected by applying a permutation from Σ_{k} ⊆ GL_{k}R. Hence σ can be moved into Y by an element of GL_{k}R.

To prove the second claim, consider a simplex σ ≠ ∗ in Y, and a g ∈ GL_{k}R such that g · σ is in Y. Both σ and g · σ have the same pick sites and submodule configuration as sets, but the pick sites may be numbered differently. The renumbering corresponds to the action of an element π of Σ_{k}, which actually must come from (Σ_{k})_{ω} as σ and g · σ both have their pick sites ordered precisely as per ω. Then π^{−1} · g fixes both the pick sites and the submodule configuration, and hence must lie in P_{ω}. □

9. Homotopy type of apartment subcomplexes.

This section computes the weak equivariant homotopy type of the various subcomplexes occurring in the poset filtration of the standard apartment, up to one special case. That case will be handled in section 11.

Let A = A_{n,k} ⊆ D^{n}(R^{k}) be the standard apartment, ω a poset on k. Then:

Proposition 9.1. F_{ω}A_{n,k} ≅ S^{nl} if ω has l components, all indiscrete. Otherwise it is contractible.

Corollary 9.2.

F_{ω}A ≅ \{ \begin{matrix}S & if ω = ι_{k}, \\ ∗ & otherwise. \end{matrix}

□

The proof begins with an easy lemma.

Lemma 9.3. If ω has components ω_{1}, …, ω_{l}, (ω = ω_{1} ⊔ … ⊔ ω_{l}) then

F_{ω}A ≅ F_{ω_{1}}A ∧ ⋯ ∧ F_{ω_{l}}A.

□

Proof of proposition 9.1. By the lemma we may assume ω is connected. If ω is indiscrete, all the pick sites of a simplex (≠ ∗) coincide in (q)^{n}, hence carry the full module R^{k} in the submodule configuration. Identify a simplex with this pick site as before, to recover

F_{ι_{k}}A_{n,k} ≅ D^{n}(1) ≅ S^{n}.

On the other hand, if ω is not indiscrete there exists i < j in ω.

Recall that a q–simplex σ ∈ A (σ ≠ ∗) corresponds to its k pick sites (p⃗_{1}, …, p⃗_{k}) where e_{i} first appears in σ(p⃗_{i}). Identify each pick site with an increasing chain of subsets

∅ = U_{0}(p⃗_{i}) ⊆ U_{1}(p⃗_{i}) ⊆ ⋯ ⊆ U_{q}(p⃗_{i}) = n

as in section 2. So σ corresponds to a k–tuple of increasing chains of subsets of n, or equivalently an increasing chain of k–tuples of subsets of n.

To complete the proof, we introduce some notation. Let T′ denote the poset of k–tuples Z = (Z_{1}, …, Z_{k}) of arbitrary subsets of n, ordered by inclusion in each of the k components. T′_{ω} is the subposet of T′ of k–tuples satisfying the reverse inclusions of ω, i.e., Z_{i} ⊇ Z_{j} if i ≤ j in ω. Let ˆ0 = (∅, …, ∅), and n̂ = (n, …, n). Set T = T′ − {ˆ0, n̂}, T_{ω} = T′_{ω} − {ˆ0, n̂}. Then we can identify F_{ω}A with the quotient of the nerve of T′_{ω} obtained by collapsing to ∗ all chains not beginning at ˆ0 and ending at n̂. From this it follows easily that F_{ω}A ≅ Σ^{2}|T_{ω}|.

Now fix i < j in ω. Let T_{1} ⊆ T_{ω} be the subposet consisting of Z = (Z_{1}, …, Z_{k}) such that Z_{a} = n if a ∈ ω_{i}. Then r: T_{ω} → T_{1} defined by

r(Z)_{a} = \{ \begin{matrix}n & if a ∈ ω_{i} \\ Z_{a} & else, \end{matrix}

is a deformation retraction, since Z ⊆ r(Z) for all Z.

Lastly let T_{2} ⊆ T_{1} be the subposet where Z_{a} = ∅ if a ∉ ω_{i}. Then s: T_{1} → T_{2} defined by

s(Z)_{a} = \{ \begin{matrix}n & if a ∈ ω_{i} \\ ∅ & else, \end{matrix}

is again a deformation retraction, as Z ⊇ s(Z) for all Z. Using i < j we see that T_{2} has precisely one element, so |T_{2}| = ∗, and F_{ω}A must be contractible. □

Definition 9.4. Suppose a complex X is covered by subcomplexes (X_{i})_{i∈V}. Consider the diagram of complexes indexed on the poset of nonempty finite subsets U ⊆ V, ordered by reverse inclusion, with the complex X[U] := ⋂_{i∈U} X_{i} at U. The Mayer–Vietoris blowup X^{∧} of X is the homotopy direct limit

X^{∧} = hocolim_{{∅≠U⊆V \atop finite}} X[U].

Explicitly

X^{∧} = ⋃_{{∅≠U⊆V \atop finite}} (Δ(U) × ⋂_{i∈U} X_{i}) ⊆ Δ(V) × X.

There is also a reduced version, for based complexes.

Lemma 9.5. The projection Δ(V) × X ⟶^{pr_{2}} X induces an equivalence X^{∧} ⟶^{≃} X.

Proof. Use Quillen’s theorem A [Q2]. □

We will study F_{<ω}A in terms of the covering given in the lemma below. It is a direct consequence of lemma 7.4.

Lemma 9.6. F_{<ω}A is covered by the F_{ω′}A where ω′ = ω + (i → j) with i → j extremal with respect to ω. Also

⋂_{s=0}^{q} F_{ω+(i_{s}→j_{s})}A = F_{ω′}A

whenever ω′ = ω + (i_{0} → j_{0}) + … + (i_{q} → j_{q}). □

Definition 9.7. Let E_{ω} = { ω + (i → j) | i → j is extremal with respect to ω } denote the indexing set for the covering in lemma 9.6. Let the non–indiscrete complex NI(ω) be the subcomplex of Δ(E_{ω}) with faces σ = {ω +(i_{0} → j_{0}), …, ω + (i_{q} → j_{q})} such that the span ω + (i_{0} → j_{0}) + ⋯ + (i_{q} → j_{q}) is not indiscrete, i.e., not the poset ι_{k}. If i → j is extremal with respect to ω, we will think of i → j as the element ω + (i → j) it determines in E_{ω}.

Lemma 9.8. There exists a 2n–connected map F_{<ω}A^{∧} ⟶ S^{n} ∗ NI(ω) for each n, making F_{<ω}A ≃ Σ^{∞}Σ NI(ω) as spectra. Here ∗ denotes the join of spaces.

Proof. We construct a map from X := F_{<ω}A^{∧} to Y := NI(ω)×D^{n+1} ∪Δ(E_{ω})×S^{n}. Here D^{n+1} denotes an n+1–ball with S^{n} as its boundary. As Δ(E_{ω}) is contractible, Y is homotopy equivalent to the join S^{n} ∗ NI(ω). View X and Y as spaces over Δ(E_{ω}), by projection on the first factor.

Every F_{ω′}A contains F_{ι_{k}}A ≅ S^{n}, so by lemma 9.6 both X and Y contain Δ(E_{ω})× S^{n}. We define f: X → Y to be the identity on this part. The remainder sits above NI(ω), and here we extend f over one simplex at a time, by increasing dimension. This is possible as the relevant target is contractible. The resulting map is 2n– connected by a spectral sequence argument, as above each simplex of NI(ω) the ‘fiber’ in X is some F_{ω′}A which is (2n − 1)–connected by proposition 9.1, while the ‘fiber’ in Y is contractible. □

Proposition 9.9. If ω is not componentwise linear, NI(ω) ≃ ∗.

Proof. Take unrelated a, b in the same component of ω. Pick a path

(a = c_{0}, c_{1}, …, c_{l} = b)

of minimal length connecting a to b, such that for each s = 1, …, l, c_{s−1} and c_{s} are related by an indecomposable morphism c_{s−1} → c_{s} or c_{s} → c_{s−1}. Since a and b are unrelated, the path must ‘change direction’ at least once. Hence we can replace a and b with vertices adjacent to such a turn, and assume there exists a c ∈ k with c → a, c → b both indecomposable. There is also an entirely equivalent case with the arrows going in the opposite direction.

Let ω′ be ω with all the extremal morphisms of E_{ω} adjoined. We now claim that a ↛ c in ω′.

If a → c in ω′ there must be a shortest chain a = d_{0} → d_{1} → ⋯ → d_{m} = c with each d_{s−1} → d_{s} in ω or E_{ω}. Clearly no d_{s−1} = d_{s}, since the chain is as short as possible. So if d_{s−1} → d_{s} is in ω and d_{s} → d_{s+1} is in E_{ω}, by minimality of d_{s} the composite d_{s−1} → d_{s+1} must be in ω. Similarly if d_{s−1} → d_{s} is in E_{ω} and d_{s} → d_{s+1} is in ω, d_{s−1} → d_{s+1} is in ω.

Hence either a = d_{0} → d_{1} = c in ω, contradicting the choice of a and c, or each d_{s−1} → d_{s} is in E_{ω}. If m = 1, this states that a = d_{0} → d_{1} = c is in E_{ω}, but since a ↛ b in ω and c → b, c cannot satisfy the maximality condition in the definition of extremal morphisms with respect to ω.

The case m ≥ 2 remains. Since each d_{s−1} → d_{s} is in E_{ω}, by upward induction on s, c → d_{s} is in ω for all s = 1, …, m. Also the composite d_{m−1} → d_{m} = c → a must be in ω. But then the factorisation c → d_{m−1} → a in ω contradicts the indecomposability of c → a. This proves the claim.

Hence ω′ > ι_{k}, and so for each face of Δ(E_{ω}) the span

ω + (i_{0} → j_{0}) + ⋯ + (i_{q} → j_{q}) ≥ ω′ > ι_{k}

is not indiscrete. Therefore NI(ω) = Δ(E_{ω}) ≃ ∗. □

Proposition 9.10. If ω is componentwise linear, then

NI(ω) ≃ Σ^{m} NI(δ_{l}),

where l = |ω| is the number of components of ω and m is the sum of the lengths of these linear components. We have the formula size(ω) = 2l + m − 2.

Proof. The extremal morphisms i → j in E_{ω} divide into two disjoint classes. On one hand there are l(l − 1) ‘minmax’ morphisms X_{ω} ⊆ E_{ω} for which i is minimal in one component and j is maximal in some other component. On the other hand there are m ‘reverse jump’ morphisms Y_{ω} ⊆ E_{ω} for which j is the immediate predecessor of i in ω, i.e., j → i in ω is indecomposable. Let X denote the simplex spanned by the former set, Y the simplex spanned by the latter set. Clearly X ∗Y ≅ Δ(E_{ω}). By collapsing the l components of ω to single objects we can identify X with Δ(E_{δ_{l}}).

To obtain ι_{k} by adjoining morphisms to ω, it is necessary and sufficient to adjoin all the ‘reverse jump’ morphisms and then enough of the ‘minmax’ morphisms to generate the indiscrete poset ι_{l} on the set of components. So a face {ω + (i_{0} → j_{0}), …, ω + (i_{q} → j_{q})} in Δ(E_{ω}) ≅ X ∗ Y is in NI(ω) precisely when either some morphism in Y_{ω} is missing, or the morphisms in X_{ω} are not indiscete on the collapsed components l:

NI(ω) ≅ (X ∗ ∂Y) ∪ (NI(δ_{l}) ∗ Y)
≅ Σ(NI(δ_{l}) ∗ ∂Y)
≅ Σ^{m} NI(δ_{l}).

□

We have now reduced the determination of the stable homotopy type of F_{ω}A/F_{<ω}A to that of A/F_{<δ_{l}}A ≃ Σ^{2} NI(δ_{l}) for varying l, i.e., only the top subquotient of the poset filtration of A remains. This is resolved in section 11.

The spectrum homology of D(R^{k})

10. Stable buildings.

To show that D(R^{k}) is equivalent to a suspension spectrum, we consider the covering of D^{n}(R^{k}) by the apartments {A(g)} as g ∈ GL_{k}R varies, and let n tend to infinity. The Mayer–Vietoris blowup of this covering will provide a model for the stable building D(R^{k}).

Definition 10.1. Let T_{k} = P_{δ_{k}} ⊆ GL_{k}R denote the diagonal torus. T_{k} ≅ (R^{∗})^{k}.

Lemma 10.2. GL_{k}R acts transitively on the set of apartments { A(g) | g ∈ GL_{k}R }. The isotropy of the standard apartment A is T_{k}Σ_{k} ⊆ GL_{k}R. □

First we need to investigate the intersections g_{0}A ∩ ⋯ ∩ g_{q}A. These are related to a poset determined by the g_{0}, …, g_{q}.

Let g ∈ GL_{k}R. Choose a π ∈ Σ_{k} such that gπ has nonzero entries on the diagonal, i.e., all (gπ)_{ii} ≠ 0. To see that this can be done, note that every set of c columns of g contains nonzero entries in at least c distinct rows, by the assumed strong invariant dimension property, and apply Hall’s theorem [Mir] (also known as the solution to the marriage problem).

Definition 10.3. Let ω(g) denote the poset on k generated by the morphisms i → j where (gπ)_{ij} ≠ 0.

Lemma 10.4. ω(g) is independent of the choice of π.

Proof. Momentarily let ω(g) be defined only for g ∈ GL_{k}R with nonzero diagonal entries, as the poset generated by the i → j with g_{ij} ≠ 0. Suppose g and gπ both have nonzero diagonal entries, with g ∈ GL_{k}R, π ∈ Σ_{k}. For all j, 0 ≠ g_{jj} = (gπ)_{jπ^{−1}j}, so j → π^{−1}(j) in ω(gπ). Since π is of finite order, π^{−1}(j) → j too in ω(gπ). So if 0 ≠ g_{ij} = (gπ)_{iπ^{−1}j}, then i → π^{−1}(j) → j in ω(gπ). Hence ω(gπ) ≤ ω(g), and the same argument for gπ and gπ ·π^{−1} = g shows that ω(g) = ω(gπ). □

Definition 10.5. If g_{1}, …, g_{q} ∈ GL_{k}R, let ω(g_{1}, …, g_{q}) = ω(g_{1}) + … + ω(g_{q}) be the span of the posets on the right.

Using this definition we can state:

Lemma 10.6. g_{0}A ∩ g_{1}A ∩ ⋯ ∩ g_{q}A = g_{0}F_{ω}A, where ω = ω(g_{0}^{−1}g_{1}, …, g_{0}^{−1}g_{q}).

Proof. We prove the special case A ∩ gA = F_{ω(g)}A, from which the general case follows by the second part of lemma 9.6.

We can assume g has nonzero diagonal entries. A q–simplex σ ≠ ∗_{q} in A∩gA ⊆ A is an axial lattice on ([q])^{n}, such that each submodule σ(x⃗) = R^{I} ⊆ R^{k} occurring is also the span of a set of column ‘vectors’ of g, i.e., σ(x⃗) = R^{I} = gR^{J} for some J ⊆ k. Since g_{ii} ≠ 0 for all i, I = J and so gR^{I} = R^{I}.

First we prove A ∩ gA ⊆ F_{ω(g)}A. Let σ ∈ A ∩ gA have pick sites (p⃗_{i}) with σ(p⃗_{i}) = R^{I_{i}}, with i ∈ I_{i} = { s ∈ k | p⃗_{s} ≤ p⃗_{i} }. Suppose g_{ij} ≠ 0 so i → j in ω(g). From σ ∈ A ∩ gA we saw that gR^{I_{j}} = R^{I_{j}}, so g · e_{j} ∈ gR^{I_{j}} = R^{I_{j}} has nonzero ith coordinate, whence i ∈ I_{j} and p⃗_{i} ≤ p⃗_{j}. So the pick sites of σ satisfy ω(g), and σ ∈ F_{ω(g)}A.

Conversely, if σ ∈ F_{ω(g)}A, again write (p⃗_{i}) for its pick sites, with I_{i} as above. We claim that gR^{I_{i}} = R^{I_{i}} for all i. For if some g_{st} ≠ 0 with t ∈ I_{i}, then s → t in ω(g), so p⃗_{s} ≤ p⃗_{t}. Thus I_{s} ⊆ I_{t} ⊆ I_{i} and s ∈ I_{i}, which proves the claim. Hence g^{−1}σ is also axial, and σ ∈ A ∩ gA. □

Corollary 10.7. When ω(g_{0}^{−1}g_{1}, …, g_{0}^{−1}g_{q}) = ι_{k}, (g_{0}A ∩ ⋯ ∩ g_{q}A) ≅ S^{n}. Otherwise (g_{0}A ∩ ⋯ ∩ g_{q}A) is at least (2n − 1)–connected. □

In view of this high connectivity result, we obtain a good approximation to the Mayer–Vietoris blowup associated to this covering as follows :

Definition 10.8. Let Σ^{−1}D(R^{k}) ⊆ Δ(GL_{k}R/T_{k}Σ_{k}) denote the GL_{k}R–subcomplex with faces {g_{0}T_{k}Σ_{k}, …, g_{q}T_{k}Σ_{k}} such that ω(g_{0}^{−1}g_{1}, …, g_{0}^{−1}g_{q}) ≠ ι_{k}. The stable building is the suspension D(R^{k}) = Σ(Σ^{−1}D(R^{k})).

Theorem 10.9. Σ^{∞}(D(R^{k})) ≅ D(R^{k}) GL_{k}R–equivariantly.

Proof. Let D^{n}(R^{k})^{∧} be the Mayer–Vietoris blowup associated to the covering {A(g)} of D^{n}(R^{k}). By an argument similar to that in the proof of lemma 9.8, and corollary 10.7, there is a 2n–connected GL_{k}R–map D^{n}(R^{k})^{∧} → S^{n} ∗Σ^{−1}D(R^{k})^{∧} ≅ Σ^{n}D(R^{k}). □

11. Configuration spaces.

In sections 8 and 9 we described the subquotients of the poset filtration on D^{n}(R^{k}) in terms of submodule configurations and the complex F_{<δ_{l}}A_{n,l} ⊂ A_{n,l} for various l. Here we relate this space to configuration spaces.

F_{<δ_{k}}A ⊂ A ≅ S^{nk} is the subcomplex of k–tuples of pick sites which satisfy some relation in (q)^{n}. It has a geometric description.

Definition 11.1. Let the constellation space C(ℝ^{n}, k) ⊆ (ℝ^{n})^{k} be the space of k–tuples (x⃗_{1}, …, x⃗_{k}) in ℝ^{n} such that no x⃗_{i} ≤ x⃗_{j} in the product ordering on ℝ^{n} when i ≠ j.

C(ℝ^{n}, k) embeds as an open submanifold of (S^{n})^{∧k} ≅ S^{nk}.

Definition 11.2. Let the singular space S(ℝ^{n}, k) be the subcomplex S^{nk}−C(ℝ^{n}, k) of (S^{n})^{∧k}, consisting of ∗ and (x⃗_{1}, …, x⃗_{k}) in ℝ^{n} such that some x⃗_{i} ≤ x⃗_{j} with i ≠ j.

Lemma 11.3. Σ_{k}–equivariantly F_{<δ_{k}}A_{n,k} ≅ S(ℝ^{n}, k).

Proof. Using the simplicial isomorphism (I = Δ(1), the unit interval) A_{n,k} ≅ (I^{n}/∂I^{n})^{∧k} we obtain homeomorphisms for each poset ω :

f: F_{ω}A ⟶^{≅} F_{ω}(I^{n}/∂I^{n})^{∧k}.

Here F_{ω}(I^{n}/∂I^{n})^{∧k} ⊆ (I^{n}/∂I^{n})^{∧k} is the space of k–tuples (y_{1}, …, y_{k}) of points in I^{n} with y_{i} ≥ y_{j} whenever i → j in ω, identified to ∗ if some y_{i} ∈ ∂I^{n}. Gluing shows that f: F_{<δ_{k}}A → F_{<δ_{k}}(I^{n}/∂I^{n})^{∧k} is a homeomorphism. Clearly F_{<δ_{k}}(I^{n}/∂I^{n})^{∧k} ≅ S(ℝ^{n}, k) by a suitable homeomorphism. □

Definition 11.4. Suppose W is some Euclidean space. Let F(W, k) denote the configuration space of ordered k–tuples of distinct points in W.

Lemma 11.5. Σ_{k}–equivariantly C(ℝ^{n}, k) ≃ F(ℝ^{n−1}, k).

Proof. Let s: ℝ^{n} → ℝ take x⃗ = (x_{i})_{i} to ∑_{i=1}^{n} x_{i}. Set W = ker(s) ⊂ ℝ^{n}, and project π: ℝ^{n} → W along lines x⃗ + ℝ · (1, …, 1). (π, s) determines an isomorphism ℝ^{n} ≅ W ⊕ ℝ.

We obtain a deformation retraction from C(ℝ^{n}, k) to F(W, k) by Σ_{k}–equivariantly deforming each k–tuple of unrelated points (x⃗_{1}, …, x⃗_{k}) along the lines x⃗_{i} + ℝ · (1, …, 1) to W. The condition that two a⃗, b⃗ ∈ ℝ^{n} are unrelated amounts to |s(a⃗) − s(b⃗)| < \sqrt{n} · ∥π(a⃗) − π(b⃗)∥. Here | | denotes the absolute value in ℝ, and ∥ ∥ the Euclidean norm in W ⊂ ℝ^{n}. An explicit deformation down to F(W, k) is then given by:

(x⃗_{i})_{i}, t ⟼ (π(x⃗_{i}), (1 − t)s(x⃗_{i}))_{i}

as t goes from 0 to 1. □

Definition 11.6. Let V = ℝ^{k} be the Σ_{k}–representation given by permutation of the coordinates.

Proposition 11.7. Σ_{k}–stably A/F_{<δ_{k}}A is S^{nV} –dual to F(ℝ^{n−1}, k).

Proof. Σ_{k}–stably A/F_{<δ_{k}}A ≃ ΣF_{<δ_{k}}A. Also F_{<δ_{k}}A_{n,k} and C(ℝ^{n}, k) are complementary Σ_{k}–subspaces of A_{n,k}, which is Σ_{k}–homeomorphic to S^{nV}. So the proposition follows directly from lemma 11.5. □

Fred Cohen [C1,C2] has computed the cohomology algebra of configuration spaces:

Computation 11.8. H^{∗}F(ℝ^{n}, k) ≅ A(n, k) as ℤΣ_{k}–algebras, where A(n, k) has generators a_{uv} in degree (n − 1) for k ≥ u > v ≥ 1, and relations a_{uv}^{2} = 0 and a_{uw}a_{uv} = a_{vw}(a_{uv} − a_{uw}) if u > v > w. Set a_{vu} = (−1)^{n}a_{uv}. Then π ∈ Σ_{k} acts by π · a_{uv} = a_{π(u)π(v)}.

Remark 11.9. This is a left Σ_{k}–representation on cohomology, ℤ–dual to the right representation on homology.

Hence the cohomology of F(ℝ^{n−1}, k) sits in degrees 0, n − 2, …, (k − 1)(n − 2), with top group free on (k − 1)! generators a_{2j_{2}}a_{3j_{3}} · . . . · a_{kj_{k}} with each i > j_{i} ≥ 1. By duality, the homology of A/F_{<δ_{k}}A sits in degrees n + (2k − 2), n + (2k − 2) + (n − 2), …, nk.

Definition 11.10. Let W_{k} be the ℤΣ_{k}–representation given by the bottom homology group of A/F_{<δ_{k}}A, for n large.

Comparison with lemma 9.8 shows that this is well–defined. W_{1} = ℤ, the trivial representation, and W_{2} = ℤ[−1], the sign representation. As an abelian group W_{k} is free of rank (k − 1)! . Cohen’s computation gives an explicit presentation, including the Σ_{k}–action, which we will find a simple description of in section 13.

Corollary 11.11. The stable homology of A/F_{<δ_{k}}A is isomorphic to W_{k}, and concentrated in degree (2k − 2). □

Proposition 11.12. Stably H_{∗}(F_{ω}A/F_{<ω}A) is concentrated in degree size(ω). It is W_{l} if ω has l components, all linear, else the homology is 0.

Proof. By lemma 9.8, the homology of NI(δ_{k}) is concentrated in degree 2k −4. The proposition then follows from propositions 9.9 and 9.10. □

12. The complex of length (2k − 2).

From the complete homological analysis of the subquotients of the poset filtration, we can now write down an expression for the spectrum homology of D(R^{k}). Our main results follow :

Theorem 12.1. The ℤGL_{k}R–module H_{∗}D(R^{k}) equals the homology of the complex (E_{∗}^{1}(R^{k})) of length (2k − 2) below

0 ⟶ (ℤGL_{k}R/P_{δ_{k}}) ⊗Σ_{k} W_{k} ⟶^{d_{2k−2}} (ℤGL_{k}R/P_{1→2}) ⊗(Σ_{k})_{1→2} W_{k−1} ⟶^{d_{2k−3}} …

⟶^{d_{s+1}} ⨁_{[ω]∈L_{s}(k)}(ℤGL_{k}R/P_{ω}) ⊗(Σ_{k})_{ω} W_{|ω|} ⟶^{d_{s}} … ⟶^{d_{1}} ℤ ⟶ 0 .

Thus the homology is concentrated in degrees 0 through (2k − 2).

Proof. Filter {D^{n}(R^{k})}_{n} by Φ_{s}, the union of the {F_{[ω]}D^{n}(R^{k})}_{n} for which size(ω) ≤ s. Here s varies from 0 through (2k − 2).

By propositions 8.6, 9.1 and 9.9, and lemma 9.8, {F_{[ω]}D^{n}(R^{k})/F_{<[ω]}D^{n}(R^{k})}_{n} is stably contractible when [ω] ∉ L_{s}(k). If [ω] ≠ [ω′] ∈ L_{s}(k), then by lemma 7.2 [ω] and [ω′] are unrelated, i.e., [ω] ≰ [ω′] and [ω′] ≰ [ω]. Hence Φ_{s}/Φ_{s−1} stably splits into the summands {F_{[ω]}D^{n}(R^{k})/F_{<[ω]}D^{n}(R^{k})}_{n} over the [ω] ∈ L_{s}(k). So associated to this filtration, by proposition 11.12, we obtain a one–line (t = 0) spectral sequence with E^{1}–term:

E_{s,0}^{1}(R^{k}) = ⨁_{[ω]∈L_{s}(k)} ℤGL_{k}R/P_{ω} ⊗(Σ_{k})_{ω} W_{|ω|}

converging to H_{∗}(E_{∗}^{1}(R^{k}), d^{1}) ≅ H_{∗}D(R^{k}) as a ℤGL_{k}R–module. □

Remark 12.2. The inclusion A_{k} ↪ D(R^{k}) allows a similar filtration by lemma 8.4, and induces a map of spectral sequences E_{∗∗}^{∗}(k) → E_{∗∗}^{∗}(R^{k}). The analogous complex E_{∗}^{1}(k) of length (2k − 2) is exact for k ≥ 2, as its homology computes H_{∗}A_{k} = 0:

0 ⟶ W_{k} ⟶^{d_{2k−2}} ℤΣ_{k} ⊗(Σ_{k})_{1→2} W_{k−1} ⟶^{d_{2k−3}} …

⟶^{d_{s+1}} ⨁_{[ω]∈L_{s}(k)} ℤΣ_{k} ⊗(Σ_{k})_{ω} W_{|ω|} ⟶^{d_{s}} … ⟶^{d_{1}} ℤ ⟶ 0 .

Furthermore the differentials in E_{∗}^{1}(R^{k}) are left adjoint to those of E_{∗}^{1}(k); hence recognizing the differentials on the standard apartment is sufficient to describe the differentials in the theorem.

The exactness of the complex above is also sufficient to reprove by induction on k that W_{k} is free abelian and, in principle, to recover that its rank is (k − 1)! .

We will see computational evidence for the following connectivity conjecture in sections 14 and 15.

Conjecture 12.3. Suppose R is a Euclidean domain or a local ring. Then the ℤGL_{k}R–complex E_{∗}(k) := (E_{∗}^{1}(R^{k}), d^{1}) is exact except at ∗ = (2k − 2).

This conjecture implies that H_{∗}D(R^{k}) is concentrated in degree (2k − 2), and equal to the kernel of d_{2k−2}^{1}. In this case

H_{s+(2k−2)}(D(R^{k})/hGL_{k}R) ≅ H_{s}(GL_{k}R; H_{2k−2}E_{∗}(k)) ,

so F_{k}KR/F_{k−1}KR is at least (2k − 3)–connected, and K_{i}R is realized as π_{i}F_{k}KR for k ≥ i/2 + 1.

The following three results do not depend on this conjecture :

Proposition 12.4. There exists a spectral sequence with E^{1}–term

E_{s,t}^{1} = H_{t}(GL_{k}R; E_{s}(k)) ≅ ⨁_{[ω]∈L_{s}(k)} H_{t}(P_{ω} · (Σ_{k})_{ω}; W_{|ω|})

converging to H_{s+t}(F_{k}KR/F_{k−1}KR).

Proof. This is a special case of a based G–complex X, with a finite G–filtration ∗ = X_{−1} ⊆ X_{0} ⊆ ⋯ ⊆ X_{2k−2} = X such that H̃_{∗}(X_{s}/X_{s−1}) is concentrated in degree s for each s. The G–cofibration sequences X_{s−1} ↪ X_{s} ↠ X_{s}/X_{s−1} induce long exact sequences

⟶ H̃_{i}(X_{s−1}/hG) ⟶ H̃_{i}(X_{s}/hG) ⟶ H̃_{i}((X_{s}/X_{s−1})/hG) ⟶^{∂}

forming an exact couple with

E_{s,t}^{1} = H̃_{s+t}((X_{s}/X_{s−1})/hG)

converging to H̃_{s+t}(X/hG). Furthermore

H̃_{s+t}((X_{s}/X_{s−1})/hG) ≅ H_{t}(G; H̃_{s}(X_{s}/X_{s−1})) ,

as H̃_{∗}(X_{s}/X_{s−1}) is concentrated in degree s. Hence the E^{1}–term can be written E_{s,t}^{1} = H_{t}(G; H̃_{s}(X_{s}/X_{s−1})). In our situation, G = GL_{k}R, X = D^{n}(R^{k}) for n large, X_{s} = Φ_{s}, and H̃_{s}(Φ_{s}/Φ_{s−1}) = E_{s}(k). Recall that

E_{s}(k) = ⨁_{[ω]∈L_{s}(k)} ℤGL_{k}R/P_{ω} ⊗(Σ_{k})_{ω} W_{|ω|}

and use

H_{∗}(GL_{k}R; ℤGL_{k}R/P_{ω}⊗(Σ_{k})_{ω}W_{|ω|})
= H_{∗}(GL_{k}R; ℤGL_{k}R⊗P_{ω} · (Σ_{k})_{ω}W_{|ω|})
≅ H_{∗}(P_{ω} · (Σ_{k})_{ω}; W_{|ω|})

where P_{ω} · (Σ_{k})_{ω} ⊆ GL_{k}R is a subgroup because (Σ_{k})_{ω} acts on P_{ω} by conjugation. □

Corollary 12.5. Suppose R is finite. Then the higher unstable K–groups π_{i}F_{k}KR, i > 0, are |GL_{k}R|–torsion (k < ∞).

Proof. By induction, starting at F_{0}KR ≅ ∗, it suffices to prove that the spectrum homology groups H_{∗}(F_{k}KR/F_{k−1}KR) are |GL_{k}R|–torsion, except for k = 1, ∗ = 0. For k ≥ 1, π_{0}F_{k}KR ≅ ℤ. We will suppress that summand for the remainder of the proof. Let E_{∗∗}^{∗} denote the spectral sequence in proposition 12.4. There exists an analogous spectral sequence for KE, Ē_{∗∗}^{∗}, converging to H_{∗}(F_{k}KE/F_{k−1}KE) = 0, with

Ē_{s,t}^{1} = ⨁_{[ω]∈L_{s}(k)} H_{t}((Σ_{k})_{ω}; W_{|ω|}),

and KE → KR induces a map of spectral sequences.

For t > 0, E_{s,t}^{1} is |GL_{k}R|–torsion and Ē_{s,t}^{1} is k!–torsion. For t = 0 note that the terms E_{s,0}^{1} are independent of the ring R. Furthermore

Ē_{s,0}^{1} = H_{0}((Σ_{k})_{ω}; W_{|ω|}) ⟶ H_{0}(P_{ω} · (Σ_{k})_{ω}; W_{|ω|}) = E_{s,0}^{1}

is an isomorphism for each s, as P_{ω} acts trivially on W_{|ω|}. Hence Ē_{s,0}^{2} ≅ E_{s,0}^{2}.

But since Ē_{∗∗}^{∞} = 0, Ē_{s,0}^{2} is k!–torsion. k! divides |GL_{k}R|, so all of E_{s,t}^{2} is |GL_{k}R|– torsion, and as E_{∗∗}^{2} converges to H_{∗}(F_{k}KR/F_{k−1}KR), this proves the corollary. □

Corollary 12.6. For each k, π_{i}F_{k}Kℤ/p^{n} → π_{i}F_{k}K𝔽_{p} is an isomorphism away from the prime p (k ≤ ∞).

Proof. Let φ: R = ℤ/p^{n} → 𝔽_{p}. Write P_{ω}R ⊆ GL_{k}R and P_{ω}𝔽_{p} ⊆ GL_{k}𝔽_{p} for the parabolic subgroups. The surjection

P_{ω}R · (Σ_{k})_{ω} ⟶^{φ} P_{ω}𝔽_{p} · (Σ_{k})_{ω}

has a kernel which is a finite p–group. Hence φ induces an isomorphism away from p of the spectral sequences in proposition 12.4 for R and 𝔽_{p}, and so F_{k}KR ⟶^{φ} F_{k}K𝔽_{p} is a homology equivalence away from p. □

Assuming the connectivity conjecture 12.3, we obtain a surjection

π_{2i−1}F_{i}KR ↠ π_{2i−1}F_{i+1}KR ≅ K_{2i−1}R,

and π_{2i−2}F_{i}KR ≅ K_{2i−2}R. So K_{2i−1}R and K_{2i−2}R must be |GL_{i}R|–torsion. Note how this fits neatly with Quillen’s computation for R = 𝔽_{q} [Q1]. If l is a prime dividing q^{i} − 1 but not q^{i−1} − 1, a nontrivial l–component of π_{2i−1}F_{k}K𝔽_{q} will first appear in F_{i}K𝔽_{q} by corollary 12.5, then may be hit from the bottom (2ith) homotopy of F_{i+1}K𝔽_{q}/F_{i}K𝔽_{q}, but already at F_{i+1}K𝔽_{q} the unstable K–theory realizes actual K–theory; K_{2i−1}𝔽_{q} ≅ ℤ/(q^{i}−1). Similarly, a nontrivial l–component may first appear in π_{2i}F_{i}K𝔽_{q}, but then dies again in π_{2i}F_{i+1}K𝔽_{q} ≅ K_{2i}𝔽_{q} = 0, for i > 0.

Computations

13. (k + 1)–ad homotopy.

We will reexpress the top homology of a configuration space as the bottom (k + 1)–ad homotopy group of a wedge of spheres. This argument is based on an approach due to Gunnar Carlsson. The proof we give uses a construction related to Thomas Goodwillie’s kth derivative of an analytic functor [Go]. Using this, we give a simple explicit description of the Σ_{k}–representation W_{k} in terms of free Lie algebras.

First we recall the definition of (k + 1)–ad homotopy [To,B-W]. A (k + 1)–ad of spaces Y = (Y ; Y_{1}, …, Y_{k}) is a based space Y with k subspaces (Y_{1}, …, Y_{k}) such that ∗ ∈ ⋂_{i∈k} Y_{i}. Y determines an inclusion k–cube indexed by U ⊆ k :

U ⟼ σ(U) = ⋂_{i∉U} Y_{i} ⊆ Y

with σ(k) = Y, and the obvious inclusion maps σ(U) ↪ σ(U′) for U ⊆ U′. Let F: Spaces_{∗} → Spaces_{∗} be a covariant functor from the category of based topological spaces to itself.

Definition 13.1.

S(Y, F) = hofib(F σ(∅) ⟶ holim_{∅≠U⊆k} F σ(U)),

i.e., the homotopy fiber of the canonical map from F σ(∅) = F (⋂_{i∈k} Y_{i}) to the homotopy inverse limit over the cube punctured at U = ∅, with F applied. Equivalently, this is the k–fold iterated homotopy fiber over a k–cube of spaces indexed by U ⊆ k.

Lemma 13.2. The homotopy of S(Y, id) is the classical (k + 1)–ad homotopy of Y, shifted down k degrees, i.e., π_{i−k}S(Y, id) = π_{i}Y.

Proof. To see this, note that S(Y, id) is the k–fold iterated homotopy fiber over the cube σ, and compare with Toda’s definition using a function space. □

Next we consider a particular (k + 1)–ad with Σ_{k}–action for which there is a dual formulation of (k + 1)–ad homotopy.

Given a based space X let X_{i} = X for all i ∈ k. There is a natural (k + 1)–ad Y = (Y ; Y_{1}, …, Y_{k}) on the k–fold wedge Y = ⋁_{i∈k} X_{i}, given by Y_{i} = ⋁_{j≠i} X_{j}.

Definition 13.3. S_{k}(X, F) = S(Y, F) with Y as above.

Explicitly σ(U) = ⋁_{i∈U} X_{i}. There is a natural Σ_{k}–action on Y and S_{k}(X, F), which permutes the wedge summands. Furthermore, each inclusion map σ(U) ↪ σ(U′) for U ⊆ U′ is naturally split by a map ⋁_{i∈U′} X_{i} ↠ ⋁_{i∈U} X_{i} collapsing each X_{i} with i ∈ U′ − U to ∗. Hence we can form a contravariant k–cube

U ⟼ τ(U) = ⋁_{i∈U} X_{i}

with collapse maps τ(U′) ↠ τ(U) for U ⊆ U′.

Definition 13.4.

T_{k}(X, F) = hofib(F τ(k) ⟶ holim_{U⊂k} F τ(U)) .

Lemma 13.5. Ω^{k}T_{k}(X, F) ≃ S_{k}(X, F).

Proof. Note that if i: A ↪ B is split by p: B ↠ A, there is an equivalence (the composite map)

Ω hofib(p) → ΩB → hofib(i).

Hence both spaces in the lemma are k–fold iterated homotopy fibers of split maps F σ(U) ↪ F σ(U′) for U ⊆ U′. □

So T_{k}(X, F) recovers the (k + 1)–ad homotopy without a dimension shift.

When X is a suspension, and F = id, by the Hilton–Milnor theorem [H,Mil,Wh], π_{∗}T_{k}(X, id) is the summand of π_{∗}Y which maps to 0 by each of the collapse maps Y ↠ Y_{i}, i ∈ k.

Definition 13.6. Let F L_{k} denote the free Lie algebra on k generators. Let XL_{k} ⊂ F L_{k} denote the free abelian subgroup generated by the k–fold iterated brackets in F L_{k} involving each generator exactly once. It has rank (k − 1)! . Σ_{k} acts on both F L_{k} and XL_{k} by permuting the generators.

From now on we will analyze the case X = S^{m}. The Hilton–Milnor theorem gives an explicit description of π_{∗}T_{k}(S^{m}, id), and by the lemma below, also of π_{∗}T_{k}(S^{m}, Ω^{n}Σ^{n}).

Lemma 13.7. T_{k}(X, Ω^{n}Σ^{n}) = Ω^{n}T_{k}(Σ^{n}X, id). □

Lemma 13.8. π_{n}T_{k}(S^{m}, id) is the direct summand of π_{n}(\overbrace{S^{m} ∨ ⋯ ∨ S^{m}}^{k times}) generated by composition of π_{n}(S^{q}) with basic iterated Whitehead brackets p ∈ π_{q}(S^{m} ∨ ⋯ ∨ S^{m}) involving each generator γ_{i} ∈ π_{m}(S_{i}^{m}) at least once. (Here q is the dimension of the iterated bracket p.)

Proof. See [H,C1] for the definition of basic products. Clearly an iterated bracket maps to 0 by every collapsing map Y ↠ Y_{i} precisely if it involves each generator at least once. □

Corollary 13.9. The first nonzero group of π_{∗}T_{k}(S^{m}, Ω^{n}Σ^{n}) = π_{∗+n}T_{k}(S^{m+n}, id) sits in degree b = k(m + n − 1) + 1 − n = km + (k − 1)(n − 1). If m + n is even, the iterated Whitehead brackets on the k generators γ_{1}, …, γ_{k} ∈ π_{m}Y ↪ π_{m}Ω^{n}Σ^{n}Y generate a copy of F L_{k} in π_{∗}Ω^{n}Σ^{n}Y, and π_{b}T_{k}(S^{m}, Ω^{n}Σ^{n}) ≅ XL_{k}.

Proof. Refer to [H,C1] for the anti–commutativity and Jacobi identities for the Whitehead bracket. □

We will now detect the image of the bottom (k + 1)–ad group π_{b}T_{k}(S^{m}, Ω^{n}Σ^{n}) in H_{b}Ω^{n}Σ^{n}Y by the Hurewicz map h, using the Browder operations λ_{n−1} on H_{∗}(Ω^{n}Σ^{n}Y; 𝔽_{p}) for each prime p [B].

There is a Σ_{k}–equivariant commutative diagram :

\xymatrix{0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ π_{∗}T_{k}(S^{m}, Ω^{n}Σ^{n}) \ar[r]^{h} \ar[d] & ker(g) \ar[r]^{≅} \ar[d] & ⨁_{j≥1} ker(g_{j}) \ar[d] \\ π_{∗}Ω^{n}Σ^{n}Y \ar[r]^{h} \ar[d]^{f} & H̃_{∗}Ω^{n}Σ^{n}Y \ar[r]^{≅} \ar[d]^{g} & _{⨁_{j≥1} H̃_{∗}(F(ℝ^{n}, j)_{+} ∧Σ_{j} Y^{∧j})} \ar[d]^{⊕g_{j}} \\ ∏_{i∈k} π_{∗}Ω^{n}Σ^{n}Y_{i} \ar[r]^{h} & ∏_{i∈k} H̃_{∗}Ω^{n}Σ^{n}Y_{i} \ar[r]^{≅} & _{∏_{i∈k} ⨁_{j≥1} H̃_{∗}(F(ℝ^{n}, j)_{+} ∧Σ_{j} Y_{i}^{∧j})}}

where f, g, and the g_{j}s are induced by the collapse maps Y ↠ Y_{i}, and the columns are exact. The isomorphisms on the right follow from the Snaith splitting [S,C-M-T].

Lemma 13.10. In degree b = km+(k−1)(n−1), ker(g) = ker(g_{k}) for m sufficiently large.

Proof. If j < k, g_{j} has no kernel, since a summand X_{i_{1}} ∧ ⋯ ∧ X_{i_{j}} of Y^{∧j} can only involve j of the {X_{i}}, and thus sits in some Y_{i}^{∧j}. If j > k, F(ℝ^{n}, j)_{+} ∧_{Σ_{j}} Y^{∧j} is at least (jm − 1)–connected. For large m this exceeds b. □

Lemma 13.11. ker(g_{k}) ≅ H̃_{∗}(F(ℝ^{n}, k)_{+} ∧ (X_{1} ∧ ⋯ ∧ X_{k})). In degree b,

ker(g_{k})_{b} ≅ H_{(k−1)(n−1)}F(ℝ^{n}, k) ⊗ ℤ[(−1)^{m}]

as ℤΣ_{k}–modules.

Proof. Y^{∧k} splits as a wedge sum ⋁_{i_{1},... ,i_{k}∈k} X_{i_{1}} ∧⋯∧X_{i_{k}}. If any i ∉ {i_{1}, …, i_{k}}, the summand X_{i_{1}} ∧ ⋯ ∧ X_{i_{k}} survives to Y_{i}^{∧k}. Otherwise j ↦ i_{j} is a permutation of k. Hence

ker(g_{k}) = H̃_{∗}(F(ℝ^{n}, k)_{+} ∧Σ_{k} ⋁_{π∈Σ_{k}}(X_{π(1)} ∧ ⋯ ∧ X_{π(k)}))
≅ H̃_{∗}(F(ℝ^{n}, k)_{+} ∧ (X_{1} ∧ ⋯ ∧ X_{k})).

Here the Σ_{k}–action permuting wedge summands of Y translates to the diagonal action on F(ℝ^{n}, k)_{+} and X_{1} ∧ ⋯ ∧ X_{k} ≅ S^{mV}. Clearly H̃_{∗}(S^{mV}) ≅ ℤ[(−1)^{m}] in degree km. □

Lastly we check that :

Lemma 13.12. h: π_{∗}T_{k}(S^{m}, Ω^{n}Σ^{n}) → ker(g) is a Σ_{k}–isomorphism in degree b.

Proof. Both sides are free abelian of rank (k − 1)! . By [C1] there exists operations for each prime p

λ_{n−1}: H_{i}(Ω^{n}Σ^{n}Y; 𝔽_{p}) ⊗ H_{j}(Ω^{n}Σ^{n}Y; 𝔽_{p}) ⟶ H_{i+j+n−1}(Ω^{n}Σ^{n}Y; 𝔽_{p}),

the Browder operations [B], commuting with the Whitehead bracket under Ω^{n} followed by the mod p Hurewicz map h_{p}. The generators γ_{i} in π_{m}S_{i}^{m} ↪ π_{m}Ω^{n}Σ^{n}Y, i ∈ k, map to generators ζ_{i} = h_{p}(γ_{i}) ∈ H_{m}(Ω^{n}Σ^{n}Y; 𝔽_{p}). Furthermore, iterated basic Whitehead brackets in the γ_{i} map to iterated basic λ_{n−1} products in the ζ_{i}. Hence h_{p} takes the generators for XL_{k} ≅ π_{b}T_{k}(S^{m}, Ω^{n}Σ^{n}) to iterated basic λ_{n−1} products in H_{b}(Ω^{n}Σ^{n}Y; 𝔽_{p}). By theorem 3.2 and the formulas preceding lemma 3.8 of [C1], these are linearly independent. Therefore each h_{p} is injective, and h must be an isomorphism. □

Corollary 13.13. W_{k} ≅ (XL_{k})^{∗} as ℤΣ_{k}–modules.

Proof.

W_{k} = H_{n+2k−2}(A/F_{<δ_{k}}A)
≅ H^{(k−1)(n−2)}F(ℝ^{n−1}, k) ⊗ ℤ[(−1)^{n}]

by S^{nV} –duality [L-M-S]

≅ (H_{(k−1)(n−2)}F(ℝ^{n−1}, k))^{∗} ⊗ ℤ[(−1)^{n}]
≅ (XL_{k})^{∗} ⊗ ℤ[(−1)^{m}] ⊗ ℤ[(−1)^{n}]

for m + n even and m sufficiently large

≅ (XL_{k})^{∗} .

□

With this, we have completely determined the modules in the complex E_{∗}^{1}(R^{k}) in theorem 12.1. Note that we have also effectively computed the coefficient spectra in the Taylor series for the identity functor viewed as an analytic functor in the sense of [Go].

14. Connectivity results.

In this section we prove the connectivity conjecture 12.3 for the case k = 2. We also give an alternative description of the stable building, which shows that it has the type of a finite GL_{k}R–complex. Lastly we obtain partial results towards the connectivity conjecture for higher k when R = ℤ/p^{2}.

First we extend definition 8.5 :

Definition 14.1. For I ⊆ k, P_{I} = (GL_{k}R)_{R^{I}}, the isotropy of R^{I} under GL_{k}R– action, is a parabolic subgroup. P_{I_{0}···I_{q}} = P_{I_{0}} ∩ ⋯ ∩ P_{I_{q}}. Note that P_{ω} = P_{ω_{1}···ω_{k}}.

Lemma 14.2. If R is a Euclidean domain or a local ring, D(R^{k}) is simply– connected for k ≥ 2.

Proof. We prove that Σ^{−1}D(R^{k}) is connected. There is an edge connecting hT_{k}Σ_{k} to hgT_{k}Σ_{k} whenever ω(h^{−1}hg) = ω(g) is not indiscrete. In particular this is the case when g is maximal parabolic (i.e., g ∈ P_{I} for some proper, nonempty subset I ⊂ k) or a permutation matrix. But when R is Euclidean or local, it is easy to see that the subgroups {P_{I}}_{∅≠I⊂k} and Σ_{k} generate GL_{k}R by using (the Euclidean algorithm followed by) Gauss elimination. □

Next we construct a Volodin– or BN–type description [V,Wag] of the stable building. By lemma 10.6, g_{0}A ∩ ⋯ ∩ g_{q}A is highly connected precisely when ω = ω(g_{0}^{−1}g_{1}, …, g_{0}^{−1}g_{q}) is not indiscrete, i.e., when ω has some proper nontrivial order ideal I ⊂ k. We can assume I = ω_{i} for some i.

Lemma 14.3. With the notation above, for every s = 0, …, q, g_{0}R^{I} is spanned by a subset of column ‘vectors’ of g_{s}.

Proof. I is an order ideal under ω(g_{0}^{−1}g_{s}). Thus R^{I} occurs in the ([q])^{n}–diagram for a simplex σ ∈ F_{ω(g_{0}^{−1}g_{s})}A, and g_{0}R^{I} occurs in g_{0}σ ∈ g_{0}A ∩ g_{s}A ⊆ g_{s}A. □

Corollary 14.4. {g_{0}, …, g_{q}} spans a simplex in Σ^{−1}D(R^{k}) if and only if there exists a proper, nontrivial submodule M ⊂ R^{k} such that for each g_{s}, M = g_{s}R^{J} for some J ⊂ k. □

Hence Σ^{−1}D(R^{k}) is covered by the contractible subcomplexes

C(M) = Δ({ g ∈ GL_{k}R/T_{k}Σ_{k} | M = gR^{J} for some J ⊂ k })

as M varies through the proper nontrivial M ↣ R^{k}.

Furthermore, if {M_{0}, …, M_{q}} is a collection of such submodules, C(M_{0}) ∩ ⋯ ∩ C(M_{q}) is the span on the set of vertices gT_{k}Σ_{k} for which each M_{s} is spanned by some columns of g. Hence each such intersection is either contractible or empty. So the Mayer–Vietoris blowup of Σ^{−1}D(R^{k}), covered by the {C(M)}, is homotopy equivalent to the complex defined below :

Definition 14.5. Let Σ^{−1}D^{V} (R^{k}) ⊆ Δ({ M | proper nontrivial M ↣ R^{k} }) be the subcomplex with faces {M_{0}, …, M_{q}} such that there exists a g ∈ GL_{k}R with each M_{i} spanned by column ‘vectors’ of g.

We have proved

Lemma 14.6. GL_{k}R–equivariantly D(R^{k}) ≃ Σ(Σ^{−1}D^{V} (R^{k})). □

Let us now consider R = ℤ/p^{2}, p a prime. Write φ: R → 𝔽_{p} for the residue field map. Let N_{k} denote the kernel of the induced homomorphism φ: GL_{k}R → GL_{k}𝔽_{p}. It is an elementary abelian p–group on k^{2} generators. There is a GL_{k}R–map φ: D^{n}(R^{k}) → D^{n}(𝔽_{p}^{k}).

Definition 14.7. Let the small n–dimensional building d^{n}(R^{k}) = φ^{−1}(A) ⊆ D^{n}(R^{k}) be the preimage of the standard apartment A ⊆ D^{n}(𝔽_{p}^{k}). It is an N_{k}– complex. Let Σ^{−1}d(R^{k}) in Δ(N_{k}/(N_{k} ∩ T_{k})Σ_{k}) be the subcomplex with faces {n_{0}, …, n_{q}} whose associated poset ω(n_{0}^{−1}n_{1}, …, n_{0}^{−1}n_{q}) is not indiscrete. As before we write d(R^{k}) = Σ(Σ^{−1}d(R^{k})). Let Σ^{−1}d^{V} (R^{k}) ⊆ Σ^{−1}D^{V} (R^{k}) be the complex with q–simplices {M_{0} …, M_{q}} with every M_{i} spanned by some columns of one g ∈ N_{k}.

The analogous result to lemma 14.6 is still true. Also :

Lemma 14.8. d^{n}(R^{k}) = ⋃_{g∈N_{k}} A(g) ⊂ D^{n}(R^{k}). □

Remark 14.9. D^{n}(R^{k}) is covered by {g·d^{n}(R^{k})} for g ∈ GL_{k}R/N_{k} ≅ GL_{k}𝔽_{p}, sitting over the covering of D^{n}(𝔽_{p}^{k}) by A(g) for g ∈ GL_{k}𝔽_{p}. So since we understand K𝔽_{p} and the stable covering by apartments of D^{n}(𝔽_{p}^{k}), in principle we should be able to reduce the study of D(R^{k}) to that of d(R^{k}).

Definition 14.10. View H_{t}(N_{k}; 𝔽_{p}) for p = 2 as the symmetric coalgebra on the 𝔽_{p}–vector space N_{k} with basis vectors {x_{i,j}} corresponding to the i, jth matrix entries (resp. for p odd, the analogous exterior tensor symmetric coalgebra). A monomial z ∈ H_{t}(N_{k}; 𝔽_{p}) determines a poset ω_{z} on k, generated by (i → j) if z involves x_{ij} for p = 2 (resp. the reduced homology of the i, jth factor if p is odd). z is mixing if and only if ω_{z} is indiscrete.

Proposition 14.11. For R = ℤ/p^{2}, d(R^{k})/hN_{k} is (k − 1)–connected.

Proof. The q–simplices of the simplicial N_{k}–set Σ^{−1}d^{V} (R^{k}) is an N_{k}–set with orbit representatives {{R^{I_{0}}, …, R^{I_{q}}}} and associated N_{k}–isotropy subgroups Q_{I_{0}···I_{q}} = P_{I_{0}···I_{q}} ∩ N_{k}. Suspending the skeletal filtration for ∅ ⊂ Σ^{−1}d^{V} (R^{k}) once, and applying ( )/hN_{k}, induces a spectral sequence for H̃_{∗}(d(R^{k})/hN_{k}; 𝔽_{p}) with

E_{s,t}^{1} = \{ \begin{matrix}H_{t}(N_{k}; 𝔽_{p}) & for s = 0, \\ ⨁_{I_{1}···I_{s}} H_{t}(Q_{I_{1}···I_{s}} ; 𝔽_{p}) & for s ≥ 1, \end{matrix}

and differentials d^{1} commuting with the inclusions H_{t}(Q_{I_{1}···I_{s}} ; 𝔽_{p}) ↪ H_{t}(N_{k}; 𝔽_{p}). Notice that the (E^{1}, d^{1})–term of the spectral sequence splits additively over the basis monomials of H_{t}(N_{k}; 𝔽_{p}).

Fix a basis monomial z ∈ H_{t}(N_{k}; 𝔽_{p}). By the splitting, the summand of E^{1} corresponding to z is a horizontal chain complex. It is equal to the augmented chain complex of a complex Δ_{z}, shifted upward (to the right) one degree. Here Δ_{z} has a q–simplex for every {I_{0}, …, I_{q}} such that z ∈ H_{t}(Q_{I_{0}···I_{q}} ; 𝔽_{p}), i.e., z occurs at position (q + 1, t) in the spectral sequence. It follows that Δ_{z} is the simplex spanned by its vertices. So Δ_{z} ≃ ∗ unless Δ_{z} = ∅, which happens precisely when z is mixing.

Clearly E^{2} is the sum of the homologies of the augmented chain complexes of the Δ_{z} :

E_{s,t}^{2} = \{ \begin{matrix}J_{t} & if s = 0, \\ 0 & if s > 0. \end{matrix}

where J_{t} ⊆ H_{t}(N_{k}; 𝔽_{p}) is the submodule generated by mixing monomials z ∈ H_{t}(N_{k}; 𝔽_{p}). Since J_{t} = 0 for t < k, this completes the proof. □

15. Computations for low ranks.

We gather here some notes and added details about theorem 12.1 for k ≤ 3.

Lemma 15.1. For any R, F_{0}KR ≃ ∗ and F_{1}KR ≃ Σ^{∞}BGL_{1}R_{+}.

Proof. For each n, D^{n}(R^{1}) = A_{n,1} ≅ S^{n} by proposition 6.4, so F_{1}KR has nth space S^{n}/hGL_{1}R ≅ Σ^{n}(BGL_{1}R_{+}) by proposition 3.8. □

Lemma 15.2. For R local or Euclidean

0 ⟶ H_{2}D(R^{2}) ⟶ ℤGL_{2}R/T_{2} ⊗Σ_{2} ℤ[−1] ⟶^{d_{2}} ℤGL_{2}R/P_{1} ⟶^{d_{1}} ℤ ⟶ 0

is exact. Here d_{2}(gT_{2} ⊗1) = g(1−ξ)P_{1}, d_{1}(gP_{1}) = 1, with ξ = (12) ∈ Σ_{2} ↪ GL_{2}R. In terms of submodule configurations, gT_{2} ∈ GL_{2}R/T_{2} determines a pair of basis lines (l_{1}, l_{2}). Then d_{2}(l_{1}, l_{2}) = l_{1} − l_{2}. d_{1}(l_{1}) = d_{1}(l_{2}) = 1.

Proof. The exactness of the sequence is the content of theorem 12.1 and lemma 14.2 for k = 2. To evaluate the differentials we consider the sequence in remark 12.2 and employ adjointness. We omit the details.

The note about submodule configurations is immediate from the comment preceding definition 8.5. □

Let φ = (12) and ψ = (123) generate Σ_{3} ↪ GL_{3}R.

Lemma 15.3. The complex E_{∗}(3) is

0 ⟶ ℤGL_{3}R/T_{3} ⊗Σ_{3} W_{3} ⟶^{d_{4}} ℤGL_{3}R/P_{1,12,3} ⟶^{d_{3}} ℤGL_{3}R/P_{1,12} ⊕ ℤGL_{3}R/P_{12,3}

⟶^{d_{2}} ℤGL_{3}R/P_{1} ⊕ ℤGL_{3}R/P_{12} ⟶^{d_{1}} ℤ ⟶ 0.

W_{3} ≅ ℤ̃^{3} ⊗ℤ[−1] ≅ (ℤ̃^{3})^{∗} ≅ (XL_{3})^{∗} ≅ ℤ\{w_{1}, w_{2}\}. In the basis \{w_{1}, w_{2}\}, φ = ({0 \atop 1} {1 \atop 0}), and ψ = ({0 \atop 1} {−1 \atop −1}).

d_{4}(gT_{3} ⊗ w_{1}) = g(−e + φ + ψ^{2} − φψ)P_{1,12,3}
d_{4}(gT_{3} ⊗ w_{2}) = g(+e − φ − ψ + ψφ)P_{1,12,3}
d_{3}(gP_{1,12,3}) = (g(+e − φψ + ψ^{2})P_{1,12}, gP_{12,3})
d_{2}(gP_{1,12}, 0) = (−gP_{1}, gP_{12})
d_{2}(0, gP_{12,3}) = (gψ^{2}P_{1}, −gP_{12})
d_{1}(gP_{1}, 0) = d_{1}(0, gP_{12}) = 1.

Note that T_{3} = P_{δ_{3}}, P_{1,12,3} = P_{1→2}, P_{1,12} = P_{1→2→3}, P_{12,3} = P_{1=2}, P_{1} = P_{1→2=3} and P_{12} = P_{1=2→3}.

Proof. As for the previous lemma, the proof reduces to a detailed inspection of the poset filtration on A_{n,3} ≅ S^{3n}. We find that E_{∗}^{1}(3) is the exact complex :

0 ⟶ ℤ\{w_{1}, w_{2}\} ↪ ℤΣ_{3} ⟶^{( {1−φψ+ψ^{2} \atop 1} )} ℤΣ_{3} ⊕ ℤΣ_{3}/(φ)

⟶^{( {−1 \atop 1} {ψ^{2} \atop −1} )} ℤΣ_{3}/(φψ) ⊕ ℤΣ_{3}/(φ) ⟶^{ϵ} ℤ ⟶ 0

and the lemma follows by adjointness. □

Lemma 15.4. E_{∗}(3) is exact at E_{0}, at E_{1} if R is local or Euclidean, at E_{2} if R is a field, and at E_{3} if R = 𝔽_{2}.

Proof. The augmentation is surjective for any ring with units, which amounts to exactness at E_{0}.

ker(d_{1}) is generated by the (e, −g) := (eP_{1}, −gP_{12}) for g ∈ GL_{3}R, as a ℤGL_{3}R– module. im(d_{2}) contains (e, −g) for g ∈ P_{1}, g ∈ P_{12} and g = ψ by lemma 15.3. Furthermore, the g ∈ GL_{3}R with (e, −g) ∈ im(d_{2}) form a subgroup, as (e, −g^{−1}h) = (e, −e)−g^{−1}(e, −g)+g^{−1}(e, −h). For R local or Euclidean, P_{1}, P_{12} and ψ generate all of GL_{3}R, so im(d_{2}) = ker(d_{1}).

We call a free submodule of R^{k} of rank two, included by a cofibration, a plane. We interpret a GL_{3}R–coset of P_{1}, P_{12}, P_{1,12}, P_{12,3}, P_{1,12,3} and T_{3} as a line l, a plane α, a line in a plane (l ⊂ α) (l is included into α by a cofibration), a line transverse to a plane (l ⋔ α) (R^{3} is the direct sum (l + α)), a line in a plane with a second transverse line (l_{1} ⊂ α ⋔ l_{2}), and three basis lines (l_{1}, l_{2}, l_{3}) respectively. Then d_{1}(l) = d_{1}(α) = 1, d_{2}(l ⊂ α) = −d_{2}(l ⋔ α) = α − l, and d_{3}(l_{1} ⊂ α_{1} ⋔ l_{2}) = (l_{1} ⊂ α_{1}) − (l_{1} ⊂ α_{2}) + (l_{2} ⊂ α_{2}) + (l_{2} ⋔ α_{1}) where α_{2} = l_{1} + l_{2} is a plane.

Let (l, α) denote (l ⊂ α) or −(l ⋔ α). Then ker(d_{2}) consists of sums ∑_{i} n_{i}(l_{i}, α_{i}) with each l or α occurring algebraically zero times. By induction on ∑_{i} |n_{i}| we see that these sums are generated by expressions □(l_{1}, l_{2}, α_{1}, α_{2}) = +(l_{1}, α_{1})−(l_{1}, α_{2})− (l_{2}, α_{1}) + (l_{2}, α_{2}) with l_{1} ≠ l_{2} and α_{1} ≠ α_{2}. im(d_{3}) contains □(l_{1}, l_{2}, α_{1}, α_{2}) when l_{1} ⊂ α ⋔ l_{2} and α_{2} = l_{1} + l_{2}. We claim that if R is a field, so that for any l, α either l ⊂ α or l ⋔ α, then these terms generate all of ker(d_{2}).

Let l_{3} = α_{1} ∩ α_{2}. If l_{1} ≠ l_{3}, use

□(l_{1}, l_{2}, α_{1}, α_{2}) = □(l_{3}, l_{2}, α_{1}, α_{2}) − □(l_{3}, l_{1}, α_{1}, α_{2})

to reduce to the case l_{1} = α_{1} ∩ α_{2}. Since l_{2} ≠ l_{1}, we may assume l_{2} ⋔ α_{1}. If l_{2} ⊂ α_{2}, α_{2} = l_{1} + l_{2} and □(l_{1}, l_{2}, α_{1}, α_{2}) = d_{3}(l_{1} ⊂ α_{1} ⋔ l_{2}). Otherwise l_{2} ⋔ α_{2} and □(l_{1}, l_{2}, α_{1}, α_{2}) = d_{3}(l_{1} ⊂ α_{1} ⋔ l_{2}) − d_{3}(l_{1} ⊂ α_{2} ⋔ l_{2}). This proves the claim of exactness at E_{2}.

The check that E_{∗}(3) is exact at ∗ = 3 for R = 𝔽_{2} was done using Mathematica^{T M}, a formula manipulator program, by listing a basis for E_{4}, E_{3} and E_{2}, writing d_{4} and d_{3} in matrix form, and solving the necessary linear systems. ker(d_{4}) ≅ H_{4}D(𝔽_{2}^{3}) is free abelian of rank 8. Hence the connectivity conjecture holds for k = 3, R = 𝔽_{2}. □

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