THE TITS BUILDINGS FOR ℤ/p^{n}

John Rognes

April 1991

Abstract. We study the algebraic K–theory of a cyclic ring R of the form ℤ/p^{n}. The idea is to consider the so–called rank filtration on a space representing the first delooping of K–theory. Each subquotient of this filtration is equivalent to the homotopy quotient of an action of the general linear group GL_{r}R on the double suspension of a finite complex B_{r}R, which is the rank r Tits building of R. To understand the homotopy type of these subquotients, we equivalently study the homotopy type of the Tits buildings together with the GL_{r}R–action.

Our results include 1) a description of the non–equivariant homotopy type of each Tits building, as a bouquet of spheres all in the same dimension, 2) a computation of the equivariant homology of a subcomplex of B_{r}R called the small building, which captures the difference between the K–theory of R and the known K–theory of 𝔽_{p}, the finite field with p elements, and 3) a description of the homotopy type of each fixed subcomplex of the Tits building under the action of a subgroup of the general linear group, in a (sufficiently) large class of such subgoups.

Introduction

Not yet written

K–theory and Tits Buildings

In this section we will review definitions of K–theory and the rank filtration, and fix some notation.

The K–theory of a ring. Let R be an associative ring with unit. The algebraic K–theory of R can be defined as follows. Let P(R) be the additive category of finitely generated projective R–modules, viewed as a category with cofibrations and weak equivalences as per [Quillen, §2] and [Waldhausen, §1.9]. Explicitly, a cofibration is the inclusion of a direct summand in the category, and a weak equivalence is an isomorphism. Then following [Waldhausen], we can apply the S_{•}– construction to P(R), obtaining a simplicial category wS_{•}P(R) whose geometric realization is the (first) delooping of K–theory. This means that the higher algebraic K–groups of R can be defined as:

K_{i}R = π_{i+1}|wS_{•}P(R)| for i ≥ 0.

This definition agrees with that of [Quillen] using the Q–construction. Equivalently, we may think of the K–groups of R as the homotopy groups of the loopspace Ω|wS_{•}P(R)|.

Typeset by AMS-TEX

Let F(R) ⊆ P(R) denote the full subcategory of finitely generated free R– modules. It is a subcategory with cofibrations and weak equivalences in the sense of [Waldhausen, p. 321], and furthermore it is (weakly) cofinal in P(R) [Grayson], so the induced map

|wS_{•}F(R)| → |wS_{•}P(R)|

is a covering map. If we define the free K–theory of R by

K_{i}^{f}R = π_{i+1}|wS_{•}F(R)| for i ≥ 0,

it follows that K_{i}^{f}R → K_{i}R is an isomorphism for i ≥ 1, while it is an injection for i = 0. We shall be considering the free K–theory of R in this paper, but by the above argument this is the same as the usual K–theory in all positive degrees, so we shall henceforth suppress the difference in notation, and use K_{i}R to denote the free K–groups. Let BK(R) denote the first delooping of free K–theory; BK(R) = |wS_{•}F(R)|.

The rank filtration. Now suppose that the rank of a finitely generated free R– module is well defined. Concretely, we will assume that R has the invariant dimension property [Mitchell], i.e. R^{n} and R^{m} are isomorphic as R–modules only if n = m. This automatically holds if R is commutative [Atiyah–Macdonald]. It is also clear that R has this property if it is an algebra over a commutative ring, for which it is finitely generated as a module. The examples we have in mind are matrix algebras over commutative rings. Denote the rank of a finitely generated free R–module M by rank(M). Then if M′ ↣ M ↠ M′′ is a cofibration sequence in F(R), i.e. a split short exact sequence of free R–modules, we have

rank(M) = rank(M′) + rank(M′′).

We wish to approximate the K–theory of R by introducing a filtration of the delooped K–theory space, such that each successive stage of the filtration is a better approximation to the full K–theory. We do this, following [Quillen] and [Mitchell], by a rank filtration of the space BK(R). Inspecting the S_{•}–construction used to define this space, we see that its simplices correspond to diagrams in the category F(R). For a fixed rank r we can consider the subcomplex F_{r}BK(R) consisting of simplices corresponding to diagrams involving only free R–modules of rank less then or equal to r. This gives an increasing filtration of spaces {F_{r}BK(R)}_{r≥0} exhausting BK(R), and we call F_{r}BK(R) the rth stage of the rank filtration on the first delooping of the K–theory of R.

There are also other definitions of a rank filtration on K–theory. By analogy to the plus-construction definition of K–theory [Quillen], one can define

K_{i,r}^{Q}R = π_{i}(BGL_{r}R)^{+} for i > 0,

as soon as a suitable perfect subgroup can be found, for instance if r ≥ 3. There is also a Volodin construction [Suslin], where

K_{i,r}^{V}R = π_{i+1}V (GL_{r}R, {T^{α}}) for i ≥ 0,

and Suslin shows that these two agree through a range. See Suslin’s paper for a definition of the expression above. Lastly there is a spectrum level rank filtration of the K–theory spectrum, different from all the above, which is developed in [Rognes, thesis].

The subquotients. Our approach to understanding the K–groups of R will be to investigate the homotopy type of the subquotients of the rank filtration, i.e. the spaces F_{r}BK(R)/F_{r−1}BK(R). There is a description of this in terms of the equivariant homotopy type of the rank r Tits building of R [Tits], as noted in [Quillen] and [Mitchell]. We now define the Tits buildings, state the description (proposition 1), and give a proof.

Definition. Suppose R satisfies the invariant dimension property, and fix a rank r. Consider the set of proper, nontrivial free R–submodules M of R^{r}: 0 ≠ M ⊂ R^{r}. Give this set a partial ordering by setting M ≺ N if M ⊆ N and the inclusion map is a cofibration in F(R), i.e. M is included as a direct summand in N with a free complementary summand. Denote this partially ordered set by O(R^{r}), and define the rank r Tits building of R, B_{r}R, to be its nerve [Quillen] :

B_{r}R = NO(R^{r}).

Hence B_{r}R is a simplicial set, with q–simplices the sequences of cofibrations

0 ≠ M_{0} ⊆ ⋯ ⊆ M_{q} ⊂ R^{r}

of free R–modules, also known as flags.

Proposition 1 [Quillen] [Mitchell].

F_{r}BK(R)/F_{r−1}BK(R) ≃ EGL_{r}R_{+} ∧_{GL_{r}R} Σ^{2}B_{r}R.

Proof. Review the S_{•}–construction from [Waldhausen, §1.3]. BK(R) = |wS_{•}F(R)| where wS_{•}F(R) is the simplicial category [q] ↦ wS_{q}F(R). An object in S_{q}F(R) is a diagram Ar[q] → F(R) satisfying certain extra hypotheses. Here [q] = {0 < 1 < ⋯ < q} is thought of as a category with q + 1 objects, and Ar[q] is the arrow category associated with [q]. Specifically, a diagram A : Ar[q] → F(R) associates to each i ≤ j a free R–module A_{i,j} = A(i → j), and A lies in S_{q}F(R) if the following three conditions hold: 1) A_{i,i} = 0 for each i, 2) the natural map A_{i,j} → A_{i,k} is a cofibration for each i ≤ j ≤ k, and 3) the commutative square

A_{i,j} ⟶ A_{i,k}
↓  ↓
A_{j,j} ⟶ A_{j,k}

is a pushout square for each i ≤ j ≤ k. Such a diagram A is determined up to isomorphism by its restriction across [q] ↪ Ar[q] taking i in [q] to (0 → i) in Ar[q]. The restriction of a diagram A is then a sequence of cofibrations

0 = A_{0,0} → A_{0,1} → ⋯ → A_{0,q}.

Let F_{r}(R) ⊂ F(R) denote the full subcategory where the objects have rank at most r. Then F_{r}BK(R) realizes the simplicial subcategory wF_{r}S_{•}F(R) where F_{r}S_{q}F(R) ⊂ S_{q}F(R) has objects the diagrams Ar[q] → F(R) which factor through F_{r}(R) ⊂ F(R). Similarly, F_{r}BK(R)/F_{r−1}BK(R) realizes the simplicial (quotient) category [q] ↦ wF_{r}S_{q}F(R)/wF_{r−1}S_{q}F(R). For fixed q, the latter category has a base point object ∗_{q}, together with the diagrams Ar[q] → F(R) in S_{q}F(R) which factor through F_{r}(R), but not through F_{r−1}(R). These are precisely the diagrams A such that the ‘top’ module A_{0,q} has rank r.

Inspecting the simplicial structure on wF_{r}S_{•}F(R)/wF_{r−1}S_{•}F(R), we see that the restriction of diagrams over [q] ↪ Ar[q] as above induces an equivalence of simplicial categories. Furthermore, the target simplicial category is equivalent to its full simplicial subcategory where the top module A_{0,q} actually is R^{r}. Thus F_{r}BK(R)/F_{r−1}BK(R) is homotopy equivalent to the realization of the simplicial category Y_{•}, which in simplicial degree q has the objects ∗_{q} and the cofibration sequences (flags)

0 = A_{0,0} → A_{0,1} → ⋯ → A_{0,q} = R^{r},

and morphisms the isomorphisms of such diagrams.

Let X_{•} be the simplicial set which has as q–simplices the object set of Y_{q}. For any flag A ∈ X_{q}, the morphisms in Y_{q} originating at A are precisely characterized by their effect on the top module A_{0,q} = R^{r}, i.e. an element of GL_{r}R. In this situation we call Y_{•} the (based) GL_{r}R–translation category on X_{•}. Using the usual simplicial model for EGL_{r}R it is straightforward to check that the nerve of Y_{q} is isomorphic to EGL_{r}R_{+} ∧_{GL_{r}R} X_{q}, and upon realizing in the q–direction we obtain:

|Y_{•}| ≅ EGL_{r}R_{+} ∧_{GL_{r}R} |X_{•}|.

It remains to recognize |X_{•}| as Σ^{2}B_{r}R.

Let O_{∗}^{∗}(R^{r}) denote the partially ordered set of (not necessarily proper or nontrivial) free R–submodules M of R^{r}, with M ≺ N if M ⊆ N and the inclusion map is a cofibration. Let O_{∗}(R^{r}) denote the partially ordered subset of proper submodules, and let O^{∗}(R^{r}) denote the partially ordered subset of nontrivial submodules. The nerve of either of these three partially ordered sets is contractible, due to the presence of initial and/or terminal elements.

Now note that O(R^{r}) is the intersection of O_{∗}(R^{r}) and O^{∗}(R^{r}). Furthermore, |X_{•}| is obtained from the nerve of O_{∗}^{∗}(R^{r}) by identifying any flags

0 ⊆ M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{q} ⊆ R^{r}

not beginning with M_{0} = 0 or ending with M_{q} = R^{r} to ∗_{q}, which amounts to contracting the nerves of O_{∗}(R^{r}) and O^{∗}(R^{r}) to a point. We conclude :

|X_{•}| = NO_{∗}^{∗}(R^{r})/(NO_{∗}(R^{r}) ∪ NO^{∗}(R^{r}))
≃ Σ(NO_{∗}(R^{r}) ∪ NO^{∗}(R^{r}))
≃ Σ^{2}(NO_{∗}(R^{r}) ∩ NO^{∗}(R^{r}))
= Σ^{2}B_{r}R. □

Thus an understanding of a weak form of the GL_{r}R–homotopy type of B_{r}R suffices to describe the subquotients of the rank filtration, in the sense that a GL_{r}R– map which is a non–equivariant homotopy equivalence is viewed as an equivalence.

Tits Buildings for ℤ/p^{n}

We now specialize to studying the K–theory of the ‘cyclic rings’ ℤ/p^{n}, with p a prime. The K–groups of the finite fields 𝔽_{p} = ℤ/p are known, as computed by Quillen [Quillen]. Furthermore, the unique ring homomorphism ℤ/p^{n} → 𝔽_{p} induces an isomorphism on K–groups away from the prime p, in the sense of localization [Bousfield and Kan]. Thus we are really only interested in the p–component of the K–groups K_{i}ℤ/p^{n} for n ≥ 2. [Aisbett] and [Evens and Friedlander] have computations of these groups for most cases with i ≤ 4, while K_{4}ℤ/4 and K_{4}ℤ/9 remain unknown.

A transfer argument. For the remainder of this section let R = ℤ/p^{n}. On the level of the subquotients of the rank filtration, we are interested in the homotopy type of EGL_{r}R_{+} ∧_{GL_{r}R} Σ^{2}B_{r}R localized at p. In this situation, there is a standard transfer argument which allows us to reduce from studying the p–localized equivariant GL_{r}R–homotopy type of B_{r}R to the analogous thing for the action on B_{r}R by a p–Sylow subgroup of GL_{r}R.

Definition. In the case of GL_{r}𝔽_{p}, the upper triangular matrices with 1’s on the diagonal constitute a p–Sylow subgroup, denoted U_{r} ⊂ GL_{r}𝔽_{p}. In general, let G_{r} = G_{r}R ⊂ GL_{r}R be the pullback of U_{r} ⊂ GL_{r}𝔽_{p} over the canonical map GL_{r}R → GL_{r}𝔽_{p}. Explicitly, G_{r} consists of the matrices with entries which are 0 modulo p below the diagonal, 1 modulo p on the diagonal, and arbitrary above the diagonal. G_{r} is a p–Sylow subgroup of GL_{r}R.

There is a bundle map π:

EG_{r+} ∧_{G_{r}} Σ^{2}B_{r}R ≃ EGL_{r}R_{+} ∧_{G_{r}} Σ^{2}B_{r}R ⟶^{π} EGL_{r}R_{+} ∧_{GL_{r}R} Σ^{2}B_{r}R,

and a stable equivariant transfer map τ [Adams] going the other way, such that the composite π ∘ τ induces multiplication by the index of G_{r} in GL_{r}R on homology or stable homotopy. This index is prime to p, so it follows that π is a split surjection when localized at p. Thus to understand EGL_{r}R_{+} ∧_{GL_{r}R} Σ^{2}B_{r}R, we will look at the G_{r}–homotopy type of B_{r}R, and appeal to the methods of [Feshbach] to compute what is lost by mapping along π.

The Tits building as a simplicial G_{r}–set. We have a description of the Tits building B_{r}R as a simplicial G_{r}–set.

Definition. Let r = {1, …, r}. For any indexing set I ⊆ r let R^{I} ⊆ R^{r} be the axial submodule where only the coordinates in I ⊆ r may be nonzero. Let P_{I} ⊆ G_{r} denote the parabolic subgroup consisting of matrices g = (g_{ij}) ∈ G_{r} such that g_{ij} = 0 if i ∉ I and j ∈ I. P_{I} stabilizes R^{I} ⊆ R^{r} for the action of G_{r} on subspaces of R^{r}. If σ = (I_{0} ⊆ ⋯ ⊆ I_{q}) is a chain of subsets of r, then P_{σ} is the parabolic subgroup defined by P_{σ} = P_{I_{0}...I_{q}} = P_{I_{0}} ∩ ⋯ ∩ P_{I_{q}}. P_{σ} stabilizes the flag R^{I_{0}} ⊆ ⋯ ⊆ R^{I_{q}} in R^{r}.

Proposition 2. As a simplicial G_{r}–set, the Tits building B_{r}R is isomorphic to

[q] ⟼ ∐_{σ} G_{r}/P_{σ}

where σ runs through the chains ∅ ≠ I_{0} ⊆ ⋯ ⊆ I_{q} ⊂ r. The ith face map ∂_{i} is induced by P_{σ} ⊂ P_{∂_{i}σ} where ∂_{i}σ = I_{0} ⊆ ⋯ ⊆ I_{i−1} ⊆ I_{i+1} ⊆ ⋯ ⊆ I_{q}, while the jth degeneracy map s_{j} is induced by P_{σ} = P_{s_{j}σ} where s_{j}σ = I_{0} ⊆ ⋯ ⊆ I_{j} = I_{j} ⊆ ⋯ ⊆ I_{q}.

Proof. We will show that an arbitrary q–simplex in B_{r}R

0 ≠ M_{0} ⊆ ⋯ ⊆ M_{q} ⊂ R^{r}

is in the G_{r}–orbit of an axial flag

0 ≠ R^{I_{0}} ⊆ ⋯ ⊆ R^{I_{q}} ⊂ R^{r}

for some ∅ ≠ I_{0} ⊆ ⋯ ⊆ I_{q} ⊂ r. Next we will check that there is only one such axial flag in each G_{r}–orbit, which proves that σ = (I_{0} ⊆ ⋯ ⊆ I_{q}) is uniquely determined. Lastly we observe that the stabilizer of R^{I_{0}} ⊆ ⋯ ⊆ R^{I_{q}} is P_{I_{0}...I_{q}} ⊆ G_{r}, from which the proposition follows.

For convenience, let us reindex the flag, so as to be considering a (q −2)–simplex

0 = M_{0} ⊂ M_{1} ⊆ ⋯ ⊆ M_{q−1} ⊂ M_{q} = R^{r}.

Let φ : R → 𝔽_{p} denote the ring homomorphism, as well as the homomorphisms induced by it. For each i ∈ r there is a minimal j such that 𝔽_{p}^{i} ⊆ φ(M_{j}). Then 𝔽_{p}^{i} ∩φ(M_{j−1})+𝔽_{p}^{i−1} ∩φ(M_{j}) is a codimension one subspace of 𝔽_{p}^{i} ∩φ(M_{j}) = 𝔽_{p}^{i}. Choose a b_{i} ∈ M_{j} such that φ(b_{j}) ∈ 𝔽_{p}^{i} is not in this subspace. Then b_{i} is part of an extension of an R–basis for M_{j−1} to one for M_{j}, and we may assume that b_{i} has ith coordinate equal to 1 modulo p.

For each j, let I_{j} = {i ∈ r | b_{i} ∈ M_{j}}. Proceeding by induction on j we see that {b_{i}}_{i∈I_{j}} is an R–basis for M_{j}. Then set g = (b_{1}, …, b_{r}) ∈ G_{r}, and observe that g · R^{I_{j}} = M_{j} for each j. This proves the first statement

Next we prove that if g ∈ G_{r} and g · R^{I} = R^{J} then I = J. For each i, φ(g) ∈ U_{r} takes the ith elementary vector e_{i} ∈ 𝔽_{p}^{i} into φ(g) · e_{i} with nonzero ith coordinate. Thus if i ∈ I, φ(g) · e_{i} ∈ φ(g) · 𝔽_{p}^{I} = 𝔽_{p}^{J} and so i ∈ J. Thus I ⊆ J, and by the same argument for g^{−1}, I = J. This proves the second statement, concluding the proof. □

The non–equivariant homotopy type. Next we will analyze the non–equivariant homotopy type of B_{r}R. First we will prove two properties of cofibrations in F(R) which are special to R = ℤ/p^{n}. These will be precisely what are needed in the proof of proposition 3, below. Neither lemma holds true for arbitrary local rings R.

Lemma 1. Each inclusion M ⊆ N of finitely generated free R–modules is a cofibration.

Proof. Choose an R–basis {b_{1}, …, b_{j}} for M, and assume N = R^{r}. Write the coordinates of b_{s} as (b_{s}^{1}, …, b_{s}^{r}). By rescaling, and permuting the coordinates of R^{r}, we may assume b_{j}^{r} = 1. Proceed by induction on j.

If j = 1, R^{r−1} ⊂ R^{r} is a complementary free summand to M = Rb_{1} ⊂ R^{r}, as required.

If j > 1, change the basis {b_{1}, …, b_{j}} for M to {c_{1}, …, c_{j−1}, b_{j}} with

c_{s} = b_{s} − b_{s}^{r} · b_{j}

for s < j. Then {c_{1}, …, c_{j−1}} spans a free submodule of R^{r−1}. Using induction, it has a complementary free submodule, which works for M ⊆ N = R^{r} too. □

Definition. If M ⊆ R^{r} is a free submodule of rank s such that the inclusion map is a cofibration, we call M an s–plane in R^{r}. If s = 1, we call M a line. A line L is transverse to an s–plane M ⊂ R^{r} (s < r) if L + M is a direct sum in R^{r}, i.e. L ∩ M = 0 and L + M is an (s + 1)–plane.

Lemma 2 (Graph lemma). Write R^{r} = R^{r−1} ⊕ R. For any plane M ⊆ R^{r}, either M contains a line transverse to R^{r−1} ⊂ R^{r}, or M is the graph of an R– homomorphism N → pR ⊂ R, with N ⊆ R^{r−1}.

Proof. Let {b_{1}, …, b_{j}} be an R–basis for M, and write b_{s} = (b_{s}^{1}, …, b_{s}^{r}) as before. Let π and π′ denote the projections R^{r} → R and R^{r} → R^{r−1} respectively. It suffices to prove that if all b_{s}^{r} ∈ pR, then π′(b_{1}), …, π′(b_{j}) are linearly independent.

So suppose ∑_{i=1}^{j} r_{i}π′(b_{i}) = 0 for some r_{i} ∈ R. Then π(∑_{i=1}^{j} r_{i}b_{i}) ∈ pR. If π(∑_{i=1}^{j} r_{i}b_{i}) ≠ 0, there is a maximal e > 0 such that π(∑_{i=1}^{j} r_{i}b_{i}) ∈ p^{e}R. Then ∑_{i=1}^{j}(p^{n−e}r_{i})b_{i} = 0 and by freeness p^{n−e}r_{i} = 0 for all i. Thus we can choose t_{i} ∈ R with p^{e}t_{i} = r_{i}, and π(p^{e} ∑_{i=1}^{i} t_{i}b_{i}) ∈ p^{e}R − p^{e+1}R. Hence π(∑_{i=1}^{i} t_{i}b_{i}) ∉ pR, contradicting the hypothesis b_{s}^{r} ∈ pR for all s. We conclude that ∑_{i=1}^{j} r_{i}b_{i} = 0 and all r_{i} = 0, proving independence. □

We now come to the principal result of this section.

Proposition 3. B_{r}R has the homotopy type of a bouquet (one–point wedge) of (r − 2)–spheres.

Proof. To begin with, we introduce some notation. If N ⊆ M is an inclusion of arbitrary finitely generated R–modules, let B(N, M) denote the partially ordered set of finitely generated free R–modules U such that N ⊂ U ⊂ M, where both inclusions are cofibrations. The partial ordering is given by U ≺ V if U ⊆ V and the inclusion is a cofibration. Also let B(N, M) denote the nerve of this partially ordered set, so that B_{r}R = B(0, R^{r}).

Write R^{r} = R^{r−1} ⊕ R as before, and note that there is a filtration of R by the powers of the maximal ideal pR :

R ⊃ pR ⊃ ⋯ ⊃ p^{l}R ⊃ ⋯ ⊃ p^{n}R = 0.

Let π : R^{r} → R be the projection on the last coordinate, as before. We filter the partially ordered set B(0, R^{r}), and hence its nerve, as follows :

B_{r}^{l} = {U ∈ B(0, R^{r}) | π(U) ⊆ p^{l}R}

with the induced ordering. Then

B(0, R^{r}) = B_{r}^{0} ⊃ B_{r}^{1} ⊃ ⋯ ⊃ B_{r}^{n} ≃ ∗.

B_{r}^{n} is contractible as it has a terminal element. We filter B_{r}^{l} ⊃ B_{r}^{l+1} for l = 0, 1, …, n − 1 by

F_{s} = {U ∈ B_{r}^{l} | U ∈ B_{r}^{l+1} or rank(U) ≥ s},

suppressing r and l from the notation. Then

B_{r}^{l} = F_{1} ⊃ F_{2} ⊃ ⋯ ⊃ F_{r−1} ⊃ F_{r} = B_{r}^{l+1}.

The partially ordered set F_{s} is obtained from F_{s+1} by adjoining the s–planes V of R^{r} in B_{r}^{l} but not in B_{r}^{l+1}. The nondegenerate simplices in the nerve of F_{s}, not in the nerve of F_{s+1}, are thus flags of the form :

0 ≠ U_{0} ⊂ U_{1} ⊂ ⋯ ⊂ U_{q} ⊂ R^{r}

where some U_{i} = V. Specifically U_{0}, …, U_{i−1} lie in B(0, V) ∩ B_{r}^{l+1} = B(0, V ∩ (R^{r−1} ⊕ p^{l+1}R)), and U_{i+1}, …, U_{q} lie in B(V, R^{r}) ∩ B_{r}^{l} = B(V, R^{r−1} ⊕ p^{l}R). The graph lemma implies the following two claims :

Claim 1. B(0, V ∩ (R^{r−1} ⊕ p^{l+1}R)) ≅ B(0, R^{s−1} ⊕ pR) ≅ B_{s}^{1}.

Claim 2. B(V, R^{r−1} ⊕ p^{l}R) ≅ B(0, R^{k−s−1} ⊕ p^{l}R) ≅ B_{r−s}^{l}.

Hence a simplex in the quotient of nerves F_{s}/F_{s+1} can be viewed as the join of a simplex in B_{s}^{1}, a vertex V, and a simplex in B_{r−s}^{l}, with the faces opposite to V collapsed to ∗ :

F_{s}/F_{s+1} ≅ ⋁_{V} Σ(B_{s}^{1} ∗ B_{r−s}^{l}) ≅ ⋁_{V} Σ^{2}(B_{s}^{1} ∧ B_{r−s}^{l})

where V runs over the s–planes of R^{r} in B_{r}^{l} − B_{r}^{l+1}.

Now we can prove by induction that all B_{r}^{l} are (r − 3)–connected. This is trivial to check for r ≤ 2, and each B_{r}^{n} ≃ ∗ is certainly (r − 3)–connected. By the formula above, with 0 < s < r, F_{s}/F_{s+1} is 2 + (s − 3) + 1 + (r − s + 3) = (r − 3)–connected for each s. Thus by downward induction on l, the result holds for each r.

Thus each B_{r}^{l} is an (r − 3)–connected (r − 2)–complex, and must have the homotopy type of a bouquet of (r − 2)–spheres. The case l = 0 is the statement of the proposition. □

It remains to prove the two claims from the graph lemma.

Proof of claim 1. Let V ⊂ R^{r} be an s–plane in B_{r}^{l} − B_{r}^{l+1}. In the case l = 0 we can choose a basis {b_{1}, …, b_{s}} for V with π(b_{s}) = 1 and π(b_{i}) = 0 for i < s. Then V ∩ (R^{r−1} ⊕ pR) = R{b_{1}, …, b_{s−1}} ⊕ pR{b_{s}} ≅ R^{s−1} ⊕ pR.

If l > 0, V is the graph of an R–homomorphism f : W → pR ⊂ R with W ⊆ R^{r−1}. We can take a basis {w_{1}, …, w_{s}} for W with f(w_{s}) = p^{l} and f(w_{i}) = 0 for i < s. Then clearly V ∩ (R^{r−1} ⊕ p^{l+1}R) ≅ R{w_{1}, …, w_{s−1}} ⊕ pR{w_{s}} ≅ R^{s−1} ⊕ pR. □

Proof of claim 2. If l = 0, the statement amounts to B(R^{s}, R^{r}) ≅ B(0, R^{r−s}), which is clear.

If l > 0, V is the graph of f : W → pR ⊂ R as in the preceding proof. Extend the basis {w_{1}, …, w_{s}} for W to {w_{1}, …, w_{r}}, a basis for R^{r}, with w_{r} = e_{r}. We can then change basis for R^{r} by replacing w_{s} with w_{s} + p^{l} · e_{r} to reduce to the case f = 0. This case is clear. □

To complete the description of the non–equivariant homotopy type of B_{r}R we give a counting argument for finding the number of wedges in the bouquet.

Notation. Let <r>_{p,n} = (p^{rn} − p^{r(n−1)})/(p^{n} − p^{n−1}) denote the number of lines in R^{r}. Then

\left⟨{r \atop s}\right⟩_{p,n} = \frac{<r>_{p,n} · . . . · <r − s + 1>_{p,n}}{<s>_{p,n} · . . . · <1>_{p,n}}

is the number of s–planes in R^{r}.

Proposition 4. For 0 = s_{0} ≤ s_{1} ≤ ⋯ ≤ s_{q−1} ≤ s_{q} = r there are

\left⟨{s_{1} \atop s_{0}}\right⟩_{p,n}· . . . · \left⟨{s_{q} \atop s_{q−1}}\right⟩_{p,n} = ∏_{i=1}^{q} \left⟨{s_{i} \atop s_{i−1}}\right⟩_{p,n}

flags 0 ≠ V_{1} ⊆ ⋯ ⊆ V_{q−1} ⊂ R^{r} with rank(V_{i}) = s_{i} representing (q − 2)–simplices in B_{r}R. Counting nondegenerate simplices, the Euler characteristic is

χB_{r}R = ∑_{0=s_{0}<···<s_{q}=r}(−1)^{q} ∏_{i=1}^{q} \left⟨{s_{i} \atop s_{i−1}}\right⟩_{p,n}.

Consequently, B_{r}R ≃ ⋁_{β_{r}} S^{r−2} with β_{r} = (−1)^{r}(χB_{r}R − 1). □

It is also possible to give a recursive formula for the number of wedges, denoted β_{r} above, by following the proof of proposition 3. Lastly, one can produce a formula based on proposition 2, by expressing the order of P_{σ} for varying σ.

Small Buildings

In this section, we will construct a covering of B_{r}R by subcomplexes homeomorphic to S^{r−2} called apartments. In the case R = 𝔽_{p}, these are the subcomplexes of B_{r}𝔽_{p} making this complex into a building in the abstract sense [Tits]. With R = ℤ/p^{n}, the ring homomorphism R → 𝔽_{p} maps the apartments of B_{r}R onto those of B_{r}𝔽_{p}. As the covering of B_{r}𝔽_{p} is well understood, we choose to organize the covering apartments in B_{r}R by what apartments they map to in B_{r}𝔽_{p}. The preimage of one apartment in B_{r}𝔽_{p} is then the union of such a gathering of apartments in B_{r}R, which forms a subcomplex of B_{r}R called the small building b_{r}R. We begin an analysis of the equivariant homotopy type of B_{r}R through that of b_{r}R.

Apartments.

Definition. Let the standard apartment A = A_{r}R ⊂ B_{r}R denote the subcomplex consisting of simplices which are axial flags in R^{r} :

0 ≠ R^{I_{0}} ⊆ ⋯ ⊆ R^{I_{q}} ⊂ R^{r}.

The apartments of B_{r}R are the translates g · A, for g ∈ GL_{r}R. They form a covering of B_{r}R. We often identify a simplex in the standard apartment A with the corresponding chain (I_{0} ⊆ ⋯ ⊆ I_{q}) of subsets of r.

Lemma. A ≅ S^{r−2}.

Proof. A is isomorphic to the nerve of the partially ordered set of proper nontrivial subsets of r, i.e. the boundary of an (r − 1)–simplex. □

To analyze B_{r}R in terms of a covering by apartments, it is convenient to introduce some notation for various subcomplexes of A.

Notation. If ω is a partial ordering on r, call I ⊆ r convex [Goodwillie] if every element of r less than an element of I, in the ordering ω, already is in I. Let A_{ω} ⊆ A denote the subcomplex consisting of axial flags

0 ≠ R^{I_{0}} ⊆ ⋯ ⊆ R^{I_{q}} ⊂ R^{r}

where each I_{i} is convex under ω. In the special case that ω partitions r into a collection of unrelated equivalence classes, we use π to denote the partition, and write A_{π} ⊆ A as above.

If T is a collection of r × r R–matrices, let A_{T} = A_{ω_{T}} where ω_{T} is the partial ordering on r generated by i ≺ j whenever g_{ij} ≠ 0 for some g ∈ T.

Lemma 3. If π partitions r into |π| equivalence classes, A_{π} ≅ S^{|π|−2}. If ω is not a partition, i.e. there exists i ≺ j in ω with j ⊀ i, then A_{ω} ≃ ∗. If e ∈ T, then A_{T} = ⋂_{g∈T} g · A.

Proof. See [Rognes, thesis, proposition 9.1 and lemma 10.6]. □

Let R = ℤ/p^{n}. Later in the section we will restrict to R = ℤ/p^{2}.

Notation. Let N_{r} = N_{r}R be the kernel of GL_{r}R ↠ GL_{r}𝔽_{p}, or equivalently of G_{r} ↠ U_{r}. Let Q_{σ} = N_{r} ∩ P_{σ}, where σ = (I_{0} ⊆ ⋯ ⊆ I_{q}) can be thought of as a simplex of A.

Lemma. The G_{r}–translates {g · A}_{g∈G_{r}} of A cover B_{r}R. Similarly {u · A}_{u∈U_{r}} cover B_{r}𝔽_{p}. The preimage of A ⊂ B_{r}𝔽_{p} by B_{r}R → B_{r}𝔽_{p} is the union of the apartments {g · A} with ∈ N_{r}.

Proof. Clear from proposition 2. □

Corollary [Quillen]. There is a U_{r}–homotopy equivalence :

B_{r}𝔽_{p} ≃ ⋁_{u∈U_{r}} S^{r−2}.

Proof. B_{r}𝔽_{p} is covered by the apartments u · A ≅ S^{r−2} with u ∈ U_{r}. Consider a (multiple) intersection ⋂_{u∈T} u · A = A_{T}, where we may assume e ∈ T. If T also contains upper triangular matrices not equal to the identity, ω_{T} cannot possibly be a partition of r, whence by lemma 3, A_{T} ≃ ∗. □

The small buildings.

Definition. The small building b_{r}R is the union of the apartments g · A ⊂ B_{r}R for g ∈ N_{r}. If u ∈ U_{r}, the translate u · b_{r}R ⊂ B_{r}R is well defined, and these cover B_{r}R.

We now look at the N_{r}–homotopy type of B_{r}R, or rather the homotopy type of EN_{r+} ∧_{N_{r}} Σ^{2}B_{r}R, through the covering of B_{r}R by the N_{r}–complexes u · b_{r}R.

Lemma. As a simplicial N_{r}–set, b_{r}R equals

[q] ⟼ ∐_{σ} N_{r}/Q_{σ}

where σ runs through the chains ∅ ≠ (I_{0} ⊆ ⋯ ⊆ I_{q}) ⊂ r.

Proof. Immediate from proposition 2. □

Corollary 1. There is a spectral sequence, with

E_{s,t}^{1} = ⨁_{σ} H_{t}(Q_{σ}; 𝔽_{p})

(group homology) where σ runs through the chains ∅ = I_{0} ⊂ ⋯ ⊂ I_{s} = r, converging to

H̃_{∗}(EN_{r+} ∧_{N_{r}} Σ^{2}b_{r}R; 𝔽_{p}).

The differentials d^{1}|_{σ} : H_{∗}(Q_{σ}; 𝔽_{p}) ↪ H_{∗}(Q_{∂_{i}σ}; 𝔽_{p}) commute with the inclusion into H_{∗}(N_{r}; 𝔽_{p}).

Proof. Σ^{2}b_{r}R has nondegenerate q–simplices ∐_{σ} N_{r}/Q_{σ}, with σ = (∅ = I_{0} ⊂ ⋯ ⊂ I_{q} = r) a nondegenerate simplex in Σ^{2}A, for q > 0. There is also a single base point in degree 0. The spectral sequence is that associated to the skeletal filtration on Σ^{2}b_{r}R, with EN_{r+} ∧_{N_{r}} (−) applied, and we recognize the E^{1}–term by

E_{s,t}^{1} = H̃_{s+t}(EN_{r+} ∧_{N_{r}} (Σ^{2}b_{r}R^{(s)}/Σ^{2}b_{r}R^{(s−1)}); 𝔽_{p})

≅ H̃_{s+t}(EN_{r+} ∧_{N_{r}} Σ^{s}(∐_{σ} N_{r}/Q_{σ} _{+}); 𝔽_{p})

≅ ⨁_{σ} H_{t}(EN_{r} ×_{N_{r}} N_{r}/Q_{σ}; 𝔽_{p})

≅ ⨁_{σ} H_{t}(Q_{σ}; 𝔽_{p}).

The observation about the differentials is immediate from the simplicial structure on b_{r}R. □

For the remainder of this section, assume R = ℤ/p^{2}. Then N_{r} is an elementary abelian p–group of rank r^{2}, with one ℤ/p–factor corresponding to each matrix entry (i, j). In this case we can compute the E^{2}–term of the spectral sequence above, and prove that it collapses there.

Definition. Let C_{∗} = H_{∗}(ℤ/p; 𝔽_{p}) denote the graded Hopf algebra. As a 𝔽_{p}–vector space it has a generator in each nonnegative degree. By the Künneth formula, H_{∗} = H_{∗}(N_{r}; 𝔽_{p}) is additively generated by monomials {z} which are (tensor) products of one of the generators in C_{∗} for each matrix entry (i, j) in N_{r}. We say that a monomial z involves the matrix entries (i, j) for which a generator of positive degree occurs.

Let A_{z} ⊆ A denote the subcomplex A_{ω_{z}} where ω_{z} is generated by i ≺ j if z involves the (i, j)th entry. If π is a partition of r, let J_{∗}(π) ⊂ H_{∗} denote the H_{∗}– comodule which is additively generated by the monomials z such that A_{z} = A_{π}, i.e. ω_{z} = π. These are the π–mixing monomials in H_{∗}.

Proposition 5. The spectral sequence in corollary 1 collapses at the E^{2}–term :

E_{s,∗}^{2} = ⨁_{π} J_{∗}(π)

where π runs over the partitions of r into |π| = s equivalence classes. Hence

H̃_{∗}(EN_{r+} ∧_{N_{r}} Σ^{2}b_{r}R; 𝔽_{p}) ≅ ⨁_{π} Σ^{|π|}J_{∗}(π).

Proof. The E^{1}–term (E^{1}, d^{1}) splits over the monomials z. Fix a z, and consider the corresponding summand of E^{1}. It contains a copy of 𝔽_{p}{z} precisely for each σ ≠ ∗ in Σ^{2}A_{z}, and d^{1} agrees with the differential in the reduced simplicial chain complex C̃_{∗}(Σ^{2}A_{z}; 𝔽_{p}). Hence the contribution to the E^{2}–term is H̃_{∗}(Σ^{2}A_{z}; 𝔽_{p}). By lemma 3, this is 0 unless ω_{z} is a partition π, in which case the homology is a copy of 𝔽_{p} in degree |π|. Hence

E_{∗∗}^{2} ≅ ⨁_{z} H̃_{∗}(Σ^{2}A_{z}) ⊗ 𝔽_{p}{z}

equals the expression in the statement in the proposition, with the sum reindexed by π in place of z.

It remains to prove that the spectral sequence collapses. Consider a partition π of r, and a monomial z in J_{∗}(π) ⊂ H_{∗} representing a generator (π, z) in E_{|π|,|z|}^{2}. Let Q_{z} ⊆ N_{r} be the intersection of all Q_{σ} for σ ∈ A_{z}. Then (π, z) is realized in the homology of BQ_{z+} ∧ Σ^{2}A_{z}. Also there is for each σ ∈ A_{z} an inclusion :

BQ_{z} ⊆ BQ_{σ} ⟶^{≃} EN_{r} ×_{N_{r}} N_{r}/Q_{σ}

extending to

BQ_{z+} ∧ Σ^{2}A_{z} ↪ EN_{r+} ∧_{N_{r}} Σ^{2}b_{r}R .

Associated to the skeletal filtration on Σ^{2}A_{z} and Σ^{2}b_{r}R, we have two spectral sequences (the ‘left’ and the ‘right’ ones, respectively) with a natural map between them, corresponding to the last inclusion above. The spectral sequence on the left side collapses at the E^{2}–term, and as the class (π, z) is realized on this E^{2} term, it supports no higher differentials on either side. As this holds for all π and z, all these classes survive to E^{∞} on the right side, and the proposition follows. □

Example. Let r = 2 and R = ℤ/4. There are two partitions π_{1} = {{1, 2}} and π_{2} = {{1}, {2}} of r, and J_{∗}(π_{1}) ≅ H_{∗} · x_{12}x_{21}, J_{∗}(π_{2}) ≅ 𝔽_{2}[x_{11}, x_{22}]. Here x_{ij} denotes the generator in degree 1 of H_{∗}(ℤ/2; 𝔽_{2}) ≅ 𝔽_{2}[x], corresponding to the (i, j)th entry in N_{2}. Then

H̃_{∗}(EN_{2+} ∧_{N_{2}} Σ^{2}b_{2}ℤ/4; 𝔽_{2}) ≅ Σ^{1}H_{∗} · x_{12}x_{21} ⊕ Σ^{2}𝔽_{2}[x_{11}, x_{22}].

The covering. We would now like to extend this result to Σ^{2}B_{r}R. Recall the covering of B_{r}R by {u · b_{r}R} for u ∈ U_{r}. Let T ⊆ U_{r} be nonempty. Consider the (multiple) intersection

⋂_{u∈T} EN_{r+} ∧_{N_{r}} Σ^{2}(u · b_{r}R) = EN_{r+} ∧_{N_{r}} Σ^{2} ⋂_{u∈T} u · b_{r}R .

If t ∈ T, ⋂_{u∈T} u · b_{r}R = t · ⋂_{u∈t^{−1}T} u · b_{r}R and e ∈ t^{−1}T, so let us assume e ∈ T. By lemma 3 we have the following extension of corollary 1 :

Lemma. Let e ∈ T ⊆ U_{r}. There is a spectral sequence with

E_{s,t}^{1} = ⨁_{σ} H_{t}(Q_{σ}; 𝔽_{p})

where σ runs through the nondegenerate s–simplices of Σ^{2}A_{T}, converging to

H̃_{∗}(EN_{r+} ∧_{N_{r}} Σ^{2} ⋂_{u∈T} u · b_{r}R; 𝔽_{p}). □

We extend the notion of mixing monomials :

Definition. Let e ∈ T ⊆ U_{r} and let π be a partition of r. Let J_{∗}(π, T) ⊂ H_{∗} denote the H_{∗}–comodule which is additively generated by the monomials z such that A_{z} ∩ A_{T} = A_{π}. These are the (π, T)–mixing monomials in H_{∗}. If e ∉ T but T is nonempty, pick t ∈ T and set J_{∗}(π, T) = J_{∗}(π, t^{−1}T).

Proposition 6. Let e ∈ T ⊆ U_{r}. The spectral sequence in the lemma above collapses at the E^{2}–term :

E_{s,∗}^{2} = ⨁_{π} J_{∗}(π)

where π runs over the partitions of r into |π| = s equivalence classes. Hence

H̃_{∗}(EN_{r+} ∧_{N_{r}} Σ^{2} ⋂_{u∈T} u · b_{r}R; 𝔽_{p}) ≅ ⨁_{π} Σ^{|π|}J_{∗}(π, T).

This isomorphism is also true if e ∉ T.

Proof. Redo the proof of proposition 5, but replace all references to σ running through the simplices of Σ^{2}A with σ running through the simplices of Σ^{2}A_{T}. □

Example. We continue the example of r = 2 and R = ℤ/4. U_{2} = {e, u}, where u = ( \begin{matrix}1 & 1 \\ 0 & 1 \end{matrix} ). Then J_{∗}(π, \{e\}) = J_{∗}(π, \{u\}) = J_{∗}(π) for either partition π, while J_{∗}(π_{1}, {e, u}) = H_{∗} · x_{21} and J_{∗}(π_{2}, {e, u}) = 0.

Corollary 2. There is a Mayer–Vietoris spectral sequence (see e.g. [Rognes, thesis, definition 9.4]) with E^{1}–term

E_{s,∗}^{1} = ⨁_{T} ⨁_{π} Σ^{|π|}J_{∗}(π, T)

where the sum runs over T ⊆ U_{r} with (s + 1) elements, converging to the N_{r}– homology of Σ^{2}B_{r}R, namely H̃_{∗}(EN_{r+} ∧_{N_{r}} Σ^{2}B_{r}R; 𝔽_{p}). □

Example. Still r = 2 and R = ℤ/4. The only differential in the spectral sequence above is d^{1} : Σ^{1}J_{∗}(π_{1}, {e, u}) → Σ^{1}J_{∗}(π_{1}, {e}) ⊕ Σ^{1}J_{∗}(π_{1}, {u}), mapping H_{∗} · x_{21} into the diagonal of H_{∗} · x_{12}x_{21} ⊕ H_{∗} · x_{12}x_{21} by the map commuting with the inclusion into H_{∗} for each summand.

Dept. of Math, Univ. of Oslo, Norway