K-Theory and the Enriched Tits Building

To A. A. Suslin with admiration, on his sixtieth birthday.

M. V. Nori and V. Srinivas

Received: December 31, 2009 Revised: May 22, 2010

Abstract. Motivated by the splitting principle, we define certain simplicial complexes associated to an associative ring A, which have an action of the general linear group GL(A). This leads to an exact sequence, involving Quillen’s algebraic K-groups of A and the symbol map. Computations in low degrees lead to another view on Suslin’s theorem on the Bloch group, and perhaps show a way towards possible generalizations.

2010 Mathematics Subject Classification: 19D06, 51A05, 55P10 Keywords and Phrases: K_{i}(A), geometries, homotopy types

The homology of GL_{n}(A) has been studied in great depth by A.A. Suslin. In some of his works ([20] and [21] for example), the action of GL_{n}(A) on certain simplicial complexes facilitated his homology computations. We introduce three simplicial complexes in this paper. They are motivated by the splitting principle. The description of these spaces is given below. This is followed by the little information we possess on their homology. After that comes the connection with K-theory. These objects are defined quickly in the context of affine algebraic groups as follows. Let G be a connected algebraic group^{1} defined over a field k. The collection of minimal parabolic subgroups P ⊂ G is denoted by F L(G) and the collection of maximal k-split tori T ⊂ G is denoted by SP L(G). The simplicial complex 𝔽𝕃(G) has F L(G) as its set of vertices. Minimal parabolics P_{0}, P_{1}, ..., P_{r} of G form an r-simplex if their intersection contains a maximal k-split torus^{2}. The dimension of 𝔽𝕃(G) is one less than the order of the Weyl group of any T ∈ SP L(G). Dually, we define 𝕊ℙ𝕃(G) as the simplicial complex with SP L(G) as its set of vertices, and T_{0}, T_{1}, ..., T_{r} forming an r-simplex if they are all contained in a minimal parabolic. In general, 𝕊ℙ𝕃(G) is infinite dimensional. That both 𝕊ℙ𝕃(G) and 𝔽𝕃(G) have the same homotopy type can be deduced from corollary 7, which is a general principle. A third simplicial complex, denoted by 𝔼𝕋(G), which we refer to as the enriched Tits building, is better suited for homology computations. This is the simplicial complex whose simplices are (nonempty) chains of the partially ordered set E(G) whose definition follows. For a parabolic subgroup P ⊂ G, we denote by U(P) its unipotent radical and by j(P) : P → P/U(P) the given morphism. Then E(G) is the set of pairs (P, T) where P ⊂ G is a parabolic subgroup and T ⊂ P/U(P) is a maximal k-split torus. We say (P′, T′) ≤ (P, T) in E(G) if P′ ⊂ P and j(P)^{−1}(T) ⊂ j(P′)^{−1}T′. Note that dim E(G) is the split rank of the quotient of G/U(G) by its center. Assume for the moment that this quotient is a simple algebraic group. Then (P, T) ↦ P gives a map to the cone of the Tits building. The topology of 𝔼𝕋(G) is more complex than the topology of the Tits building, which is well known to be a bouquet of spheres. When G = GL(V), we denote the above three simplicial complexes by 𝔽𝕃(V), 𝕊ℙ𝕃(V) and 𝔼𝕋(V). These constructions have simple analogues even when one is working over an arbitrary associative ring A. Their precise definition is given with some motivation in section 2. Some basic properties of these spaces are also established in section 2. Amongst them is Proposition 11 which shows that 𝔼𝕋(V) and 𝔽𝕃(V) have the same homotopy type. 𝔼𝕋(A^{n}) has a polyhedral decomposition (see lemma 21). This produces a spectral sequence (see Theorem 2) that computes its homology. The E_{p,q}^{1} terms and the differentials d_{p,q}^{1} are recognisable since they involve only the homology groups of 𝔼𝕋(A^{a}) for a < n. The differentials d_{p,q}^{r} for r > 1 are not understood well enough, however. There are natural inclusions 𝔼𝕋(A^{n}) ↪ 𝔼𝕋(A^{d}) for d > n, and the induced map on homology factors through

1Gopal Prasad informed us that we should take G reductive or k perfect. The standard classification, via root systems, of all parabolic subgroups containing a maximal split torus, requires this hypothesis.

2John Rognes has an analogous construction with maximal parabolics replacing minimal parabolics. His spaces, different homotopy types from ours, are connected with K-theory as well, see [15] .

Documenta Mathematica · Extra Volume Suslin (2010) 459–513

H_{m}(𝔼𝕋(A^{n})) → H_{0}(E_{n}(A), H_{m}(𝔼𝕋(A^{n}))) → H_{m}(𝔼𝕋(A^{d})).

where E_{n}(A) is the group of elementary matrices (see Corollary 9). For the remaining statements on the homology of 𝔼𝕋(A^{n}), we assume that A is a commutative ring with many units, in the sense of Van der Kallen. See [12] for a nice exposition of the definition and its consequences. Commutative local rings A with infinite residue fields are examples of such rings. Under this assumption, E_{n}(A) can be replaced by GL_{n}(A) in the above statement. We have observed that 𝔼𝕋(A^{m+1}) has dimension m. Thus it is natural question to ask whether

H_{0}(GL_{m+1}(A), H_{m}(𝔼𝕋(A^{m+1}))) ⊗ ℚ → H_{m}(𝔼𝕋(A^{d})) ⊗ ℚ

is an isomorphism when d > m + 1. Theorem 3 asserts that this is true for m = 1, 2, 3. The statement is true in general (see Proposition 29) if a certain Compatible Homotopy Question has an affirmative answer. The higher differentials of the spectral sequence can be dealt with if this is true. Proposition 22 shows that this holds in some limited situations. The computation of H_{0}(GL_{3}(A), H_{2}(𝔼𝕋(A^{3}))) ⊗ ℚ is carried out at in the last lemma of the paper. This is intimately connected with Suslin’s result (see [21]) connecting K_{3} and the Bloch group. A closed form for H_{0}(GL_{4}(A), H_{3}(𝔼𝕋(A^{4})) is awaited. This should impact on the study of K_{4}(A). We now come to the connection with the Quillen K-groups K_{i}(A) as obtained by his plus construction. GL(A) acts on the geometric realisation |𝕊ℙ𝕃(A^{∞})| and thus we have the Borel construction, namely the quotient of |𝕊ℙ𝕃(A^{∞})| × EGL(A) by GL(A), a familiar object in the study of equivariant homotopy. We denote this space by 𝕊ℙ𝕃(A^{∞})//GL(A). We apply Quillen’s plus construction to 𝕊ℙ𝕃(A^{∞})//GL(A) and a suitable perfect subgroup of its fundamental group to obtain a space Y (A). Proposition 17 shows that Y (A) is an H-space and that the natural map Y (A) → BGL(A)^{+} is an H-map. Its homotopy fiber, denoted by 𝕊ℙ𝕃(A^{∞})^{+}, is thus also a H-space. The n-th homotopy group of 𝕊ℙ𝕃(A^{∞})^{+} at its canonical base point is denoted by L_{n}(A). There is of course a natural map 𝕊ℙ𝕃(A^{∞}) → 𝕊ℙ𝕃(A^{∞})^{+}. That this map is a homology isomorphism is shown in lemma 16. This assertion is easy, but not tautological: it relies once again on the triviality of the action of E(A) on H_{∗}(𝕊ℙ𝕃(A^{∞})). As a consequence of this lemma, L_{n}(A) ⊗ ℚ is identified with the primitive rational homology of 𝕊ℙ𝕃(A^{∞}), or equivalently, that of 𝔼𝕋(A^{∞}). We have the inclusion N_{n}(A) ↪ GL_{n}(A), where N_{n}(A) is the semidirect product of the permutation group S_{n} with (A^{×})^{n}. Taking direct limits over n ∈ ℕ, we obtain N(A) ⊂ GL(A). Let H′ ⊂ N(A) be the infinite alternating group and let H be the normal subgroup generated by H′. Applying Quillen’s plus construction to the space BN(A) with respect to H, we obtain BN(A)^{+}. Its n-th homotopy group is defined to be H_{n}(A^{×}). From the Dold-Thom theorem, it is easy to see that H_{n}(A^{×}) ⊗ ℚ is isomorphic to the group homology H_{n}(A^{×}) ⊗ ℚ. When A is commutative, this is simply ∧_{ℚ}^{n}(A^{×} ⊗ ℚ). Proposition 20 identifies the groups H_{n}(A^{×}) with certain stable homotopy groups. Its proof was shown to us by J. Peter May. It is sketched in the text of the paper after the proof of the Theorem below.

Theorem 1. Let A be a Nesterenko-Suslin ring. Then there is a long exact sequence, functorial in A:

⋯ → L_{2}(A) → H_{2}(A^{×}) → K_{2}(A) → L_{1}(A) → H_{1}(A^{×}) → K_{1}(A) → L_{0}(A) → 0.

We call a ring A Nesterenko-Suslin if it satisfies the hypothesis of Remark 1.13 of their paper [13]. The precise requirement is that for every finite set F, there is a function f_{F} : F → the center of A so that the sum Σ{f_{F} (s) : s ∈ S} is a unit of A for every nonempty S ⊂ F. If k is an infinite field, every associative k-algebra is Nesterenko-Suslin, and so is every commutative ring with many units in the sense of Van der Kallen. Remark 1.13 of Nesterenko-Suslin [13] permits us to ignore unipotent radicals. This is used crucially in the proof of Theorem 1 (see also Proposition 12). In the first draft of the paper, we conjectured that this theorem is true without any hypothesis on A. Sasha Beilinson then brought to our attention Suslin’s paper [22] on the equivalence of Volodin’s K-groups and Quillen’s. From Suslin’s description of Volodin’s spaces, it is possible to show that these spaces are homotopy equivalent to the total space of the N_{n}(A)-torsor on 𝔽𝕃(A^{n}) given in section 2 of this paper. This requires proposition 1 and a little organisation. Once this is done, Corollary 9 can also be obtained from Suslin’s set-up. The statement “X(R) is acyclic” stated and proved by Suslin in [22] now validates Proposition 12 at the infinite level, thus showing that Theorem 1 is true without any hypothesis on A. The details have not been included here. R. Kottwitz informed us that the maximal simplices of 𝔽𝕃(V) are referred to as “regular stars” in the work of Langlands( see [5]). We hope that this paper will eventually connect with mixed Tate motives (see[3],[1]). The arrangement of the paper is a follows. Section 1 has some topological preliminaries used through most of the paper. The proofs of Corollary 7 and Proposition 11 rely on Quillen’s Theorem A. Alternatively, they can both be proved directly by repeated applications of Proposition 1. The definitions of 𝕊ℙ𝕃(A^{n}), 𝔽𝕃(A^{n}), 𝔼𝕋(A^{n}) and first properties are given in section two. The next four sections are devoted to the proof of Theorem 1. The last four sections are concerned with the homology of 𝔼𝕋(A^{n}). The lemmas, corollaries and propositions are labelled sequentially. For instance, corollary 9 is followed by lemma 10 and later by proposition 11; there is no proposition 10 or corollary 10. The other numbered statements are the three theorems. Theorems 2 and 3 are stated and proved in sections 7 and 9 respectively. Section 0 records some assumptions and notation, some perhaps non-standard, that are used in the paper. The reader might find it helpful to glance at this section for notation regarding elementary matrices and the Borel construction and the use of “simplicial complexes”.

0. Assumptions and Notation

Rings, Elementary matrices, Elem(W ↪ V), Elem(V, q), L(V) and L_{p}(V)

We are concerned with the Quillen K-groups of a ring A. We assume that A has the following property: if A^{m} ≅ A^{n} as left A-modules, then m = n. The phrase “A-module” always means left A-module. For a finitely generated free A-module V, the collection of A-submodules L ⊂ V so that (i) V/L is free and (ii) L is free of rank one , is denoted by L(V). L_{p}(V) is the collection of subsets q of cardinality (p + 1) of L(V) so that ⊕{L : L ∈ q} → V is a monomorphism whose cokernel is free. Given an A-submodule W of a A-module V so that the short exact sequence

0 → W → V → V/W → 0

is split, we have the subgroup Elem(W ↪ V) ⊂ Aut_{A}(V), defined as follows. Let H(W) be the group of automorphisms h of V so that (id_{V} − h)V ⊂ W ⊂ ker(id_{V} − h). Let W′ ⊂ V be a submodule that is complementary to W. Define H(W′) in the same manner. The subgroup of Aut_{A}(V) generated by H(W) and H(W′) is Elem(W ↪ V). It does not depend on the choice of W′ because H(W) acts transitively on the collection of such W′. For example, if V = A^{n}, and W is the A-submodule generated by any r members of the given basis of A^{n}, then Elem(W ↪ A^{n}) equals E_{n}(A) , the subgroup of elementary matrices in GL_{n}(A), provided of course that 0 < r < n. If V is finitely generated free and if q ∈ L_{p}(V), the above statement implies that Elem(L ↪ V) does not depend on the choice of L ∈ q. Thus we denote this subgroup by Elem(V, q) ⊂ GL(V).

The Borel Construction

Let X be a topological space equipped with the action of group G. Let EG be the principal G-bundle on BG (as in [14]). The Borel construction, namely the quotient of X × EG by the G-action, is denoted by X//G throughout the paper.

Categories, Geometric realisations, Posets

Every category C gives rise to a simplicial set, namely its nerve (see [14]). Its geometric realisation is denoted by BC. A poset (partially ordered set) P gives rise to a category. The B-construction of this category, by abuse of notation, is denoted by BP. Associated to P is the simplicial complex with P as its set of vertices; the simplices are finite non-empty chains in P. The geometric realisation of this simplicial complex coincides with BP. Simplicial Complexes, Products and Internal Hom, Barycentric subdivision Simplicial complexes crop up throughout this paper. We refer to Chapter 3,[18], for the definition of a simplicial complex and its barycentric subdivision. S(K) and V(K) denote the sets of vertices and simplices respectively of a simplicial complex K. The geometric realisation of K is denoted by |K|. The set S(K) is a partially ordered set (with respect to inclusion of subsets). Note that BS(K) is simply the (geometric realisation of) the barycentric subdivision sd(K). The geometric realisations of K and sd(K) are canonically homeomorphic to each other, but not by a simplicial map. Given simplicial complexes K_{1} and K_{2}, the product |K_{1}| × |K_{2}| (in the compactly generated topology) is canonically homeomorphic to B(S(K_{1})×S(K_{2})).

The category of simplicial complexes and simplicial maps has a categorical product: V(K_{1} ×K_{2}) = V(K_{1})×V(K_{2}). A non-empty subset of V(K_{1} ×K_{2}) is a simplex of K_{1} × K_{2} if and only if it is contained in S_{1} × S_{2} for some S_{i} ∈ S(K_{i}) for i = 1, 2. The geometric realisation of the product is not homeomorphic to the product of the geometric realisations, but they do have the same homotopy type. In fact Proposition 1 of section 1 provides a contractible collection of homotopy equivalences |K_{1}| × |K_{2}| → |K_{1} × K_{2}|. For most purposes, it suffices to note that there is a canonical map P (K_{1}, K_{2}) : |K_{1}| × |K_{2}| → |K_{1} × K_{2}|. This is obtained in the following manner. Let C(K) denote the ℝ-vector space with basis V(K) for a simplicial complex K. Recall that |K| is a subset of C(K). For simplicial complexes K_{1} and K_{2}, we have the evident isomorphism

j : C(K_{1}) ⊗_{ℝ} C(K_{2}) → C(K_{1} × K_{2}).

For c_{i} ∈ |K_{i}| for i = 1, 2 we put P (K_{1}, K_{2})(c_{1}, c_{2}) = j(c_{1} ⊗ c_{2}) ∈ C(K_{1} × K_{2}). We note that j(c_{1} ⊗ c_{2}) belongs to the subset |K_{1} × K_{2}|. This gives the canonical P (K_{1}, K_{2}).

Given simplicial complexes K, L there is a simplicial complex Hom(K, L) with the following property: if M is a simplicial complex, then the set of simplicial maps K × M → L is naturally identified with the set of simplicial maps M → Hom(K, L). This simple verification is left to the reader.

Simplicial maps f : K_{1} × K_{2} → K_{3} occur in sections 2 and 5 of this paper.

|f| ∘ P (K_{1}, K_{2}) : |K_{1}| × |K_{2}| → |K_{3}|

is the map we employ on geometric realisations. Maps |K_{1}| × |K_{2}| → |K_{3}| associated to simplicial maps f_{1} and f_{2} are seen (by contiguity) to be homotopic to each other if {f_{1}, f_{2}} is a simplex of Hom(K_{1}×K_{2}, K_{3}). This fact is employed in Lemma 8. Simplicial maps f : K_{1} ×K_{2} → K_{3} are in reality maps V(f) : V(K_{1})×V(K_{2}) → V(K_{3}) with the property that V(f)(S_{1} × S_{2}) is a simplex of K_{3} whenever S_{1} and S_{2} are simplices of K_{1} and K_{2} respectively. One should note that such an f induces a map of posets S(K_{1}) × S(K_{2}) → S(K_{3}), which in turn induces a continuous map B(S(K_{1}) × S(K_{2})) → BS(K_{3}). In view of the natural identifications, this is the same as giving a map |K_{1}| × |K_{2}| → |K_{3}|. This map coincides with the |f| ∘ P (K_{1}, K_{2}) considered above. The homotopy assertion of maps |K_{1}| × |K_{2}| → |K_{3}| associated to f_{1}, f_{2} where {f_{1}, f_{2}} is an edge of Hom(K_{1} × K_{2}, K_{3}) cannot be proved by the quick poset definition of the maps (for |K_{3}| has been subdivided and contiguity is not available any more). This explains our preference for the longwinded |f| ∘ P (K_{1}, K_{2}) definition.

1. Some preliminaries from topology

We work with the category of compactly generated weakly Hausdorff spaces. A good source is Chapter 5 of [11]. This category possesses products. It also possesses an internal Hom in the following sense: for compactly generated Hausdorff X, Y, Z, continuous maps Z → Hom(X, Y) are the same as continuous maps Z × X → Y, where Z × X denotes the product in this category. This internal Hom property is required in the proof of Proposition 1 stated below.

Hom(X, Y) is the space of continuous maps from X to Y. This space of maps has the compact-open topology, which is then replaced by the inherited compactly generated topology. This space Hom(X, Y) is referred to frequently as Map(X, Y), and some times even as Maps(X, Y) , in the text. Now consider the following set-up. Let Λ be a partially ordered set assumed to be Artinian: (i) every non-empty subset in Λ has a minimal element with respect to the partial order, or equivalently (ii) there are no infinite strictly descending chains λ_{1} > λ_{2} > ⋯ in Λ. The poset Λ will remain fixed throughout the discussion below. We consider topological spaces X equipped with a family of closed subsets X_{λ}, λ ∈ Λ with the property that X_{μ} ⊂ X_{λ} whenever μ ≤ λ. Given another Y, Y_{λ}, λ ∈ Λ as above, the collection of Λ-compatible continuous f : X → Y (i.e. satisfying f(X_{λ}) ⊂ Y_{λ}, ∀λ ∈ Λ) will be denoted by Map_{Λ}(X, Y). Map_{Λ}(X, Y) is a closed subset of Hom(X, Y), and this topologises Map_{Λ}(X, Y). We say that {X_{λ}} is a weakly admissible covering of X if the three conditions listed below are satisfied. It is an admissible covering if in addition, each X_{λ} is contractible.

(1) For each pair of indices λ, μ ∈ Λ, we have

X_{λ} ∩ X_{μ} = ∪_{ν≤λ,ν≤μ} X_{ν}

(2) If

∂X_{λ} = ∪_{ν<λ} X_{ν},

then ∂X_{λ} ↪ X_{λ} is a cofibration

(3) The topology on X is coherent with respect to the family of subsets {X_{λ}}_{λ∈Λ}, that is, X = ∪_{λ}X_{λ}, and a subset Z ⊂ X is closed precisely when Z ∩ X_{λ} is closed in X_{λ} in the relative topology, for all λ.

Proposition 1. Assume that {X_{λ}} is a weakly admissible covering of X. Assume also that each Y_{λ} is contractible. Then the space Map_{Λ}(X, Y) of Λ-compatible maps f : X → Y is contractible. In particular, it is non-empty and path-connected.

Corollary 2. If both {X_{λ}} and {Y_{λ}} are admissible, then X and Y are homotopy equivalent.

With assumptions as in the above corollary, the proposition yields the existence of Λ-compatible maps f : X → Y and g : Y → X. Because g ∘ f and f ∘ g are also Λ-compatible, that they are homotopic to id_{X} and id_{Y} respectively is deduced from the path-connectivity of Map_{Λ}(X, X) and Map_{Λ}(Y, Y).

Corollary 3. If {X_{λ}} is admissible, then there is a homotopy equivalence X → BΛ.

Here, recall that BΛ is the geometric realization of the simplicial complex associated to the set of nonempty finite chains (totally ordered subsets) in Λ; equivalently, regarding Λ as a category, BΛ is the geometric realization of its nerve. We put Y = BΛ and Y_{λ} = B{μ ∈ Λ : μ ≤ λ} in Corollary 2 to deduce Corollary 3. The proof of Proposition 1 is easily reduced to the following extension lemma.

Lemma 4. Let {X_{λ}}, {Y_{λ}} etc. be as in the above proposition. Let Λ′ ⊂ Λ be a subset, with induced partial order, so that for any λ ∈ Λ′, μ ∈ Λ with μ ≤ λ, we have μ ∈ Λ′. Let X′ = ∪_{λ∈Λ′}X_{λ}, Y′ = ∪_{λ∈Λ′}Y_{λ}. Assume given a continuous map f′ : X′ → Y′ with f′(X_{λ}) ⊂ Y_{λ} for all λ ∈ Λ′. Then f′ extends to a continuous map f : X → Y with f(X_{λ}) ⊂ Y_{λ} for all λ ∈ Λ.

Proof. Consider the collection of pairs (Λ′′, f′′) satisfying:

(a) Λ′ ⊂ Λ′′ ⊂ Λ

(b) μ ∈ Λ, λ ∈ Λ′′, μ ≤ λ implies μ ∈ Λ′′

(c) f′′ : ∪{X_{μ}|μ ∈ Λ′′} → Y is a continuous map

(d) f′′(X_{μ}) ⊂ Y_{μ} for all μ ∈ Λ′′

(e) f′|X_{μ} = f′′|X_{μ} for all μ ∈ Λ′ This collection is partially ordered in a natural manner. The coherence condition on the topology of X ensures that every chain in this collection has an upper bound. The presence of (Λ′, f′) shows that it is non-empty. By Zorn’s lemma, there is a maximal element (Λ′′, f′′) in this collection. The Artinian hypothesis on Λ shows that if Λ′′ ≠ Λ, then its complement possesses a minimal element μ. Let D′′ be the domain of f′′. The minimality of μ shows that D′′ ∩ X_{μ} = ∂X_{μ}. By condition (d) above, we see that f′′(∂X_{μ}) is contained in the contractible space Y_{μ}. Because ∂X_{μ} ↪ X_{μ} is a cofibration, it follows that f′′|∂X_{μ} extends to a map g : X_{μ} → Y_{μ}. The f′′ and g patch together to give a continuous map h : D′′ ∪ X_{μ} → Y. Since the pair (Λ′′ ∪ {μ}, h) evidently belongs to this collection, the maximality of (Λ′′, f′′) is contradicted. Thus Λ′′ = Λ and this completes the proof.

□

The proof of the Proposition follows in three standard steps. Step 1: Taking Λ′ = ∅ in Lemma 4 we deduce that Map_{Λ}(X, Y) is nonempty. Step 2: For the path-connectivity of Map_{Λ}(X, Y) , we replace X by X × [0, 1] and replace the original poset Λ by the product Λ × {{0}, {1}, {0, 1}}, with the product partial order, where the second factor is partially ordered by inclusion. The subsets of X × I (resp.Y) indexed by (λ, 0), (λ, 1), (λ, {0, 1}) are X_{λ} × {0}, X_{λ} × {1} and X_{λ} × [0, 1] (resp. Y_{λ} in all three cases). We then apply the lemma to the sub-poset Λ × {{0}, {1}}. Step 3: Finally, for the contractibility of Map_{Λ}(X, Y), we first choose f_{0} ∈ Map_{Λ}(X, Y) and then consider the two maps Map_{Λ}(X, Y) × X → Y given by (f, x) ↦ f(x) and (f, x) ↦ f_{0}(x). Putting (Map_{Λ}(X, Y) × X)_{λ} = Map_{Λ}(X, Y) × X_{λ} for all λ ∈ Λ, we see that both the above maps are Λcompatible. The path-connectivity assertion in Step 2 now gives a homotopy between the identity map of Map_{Λ}(X, Y) and the constant map f ↦ f_{0}. This completes the proof of Proposition 1. We now want to make some remarks about equivariant versions of the above statements.

Given X, {X_{λ}; λ ∈ Λ} as above, an action of a group G on X is called Λcompatible if G also acts on the poset Λ so that for all g ∈ G, λ ∈ Λ, we have g(X_{λ}) = X_{gλ}. Under the conditions of Proposition 1, suppose {X_{λ}}, {Y_{λ}} admit Λ-compatible G-actions. There is no G-equivariant f ∈ Map_{Λ}(X, Y) in general. However, if f ∈ Map_{Λ}(X, Y) and g_{X}, g_{Y} denote the actions of g ∈ G on X and Y respectively, we see that , then g_{Y} ∘ f ∘ g_{X}^{−1} is also a Λ-compatible map. By Proposition 1, we see that this map is homotopic to f. Thus g_{Y} ∘ f and f ∘ g_{X} are homotopic to each other. In particular, H_{n}(f) : H_{n}(X) → H_{n}(Y) is a homomorphism of G-modules. In the sequel a better version of this involving the Borel construction is needed. We recall the Borel construction of equivariant homotopy quotient spaces. Let EG denote a contractible CW complex on which G has a proper free cellular action; for our purposes, it suffices to fix a choice of this space EG to be the geometric realization of the nerve of the translation category of G (the category with vertices [g] indexed by the elements of G, and unique morphisms between ordered pairs of vertices ([g], [h]), thought of as given by the left action of hg^{−1}). The classifying space BG is the quotient space EG/G. If X is any G-space, let X//G denote the homotopy quotient of X by G, obtained using the Borel construction, i.e.,

(1)  X//G = (X × EG)/G,

where EG is as above, and G acts diagonally. Note that the natural quotient map

q_{X} : X × EG → X//G

is a Galois covering space, with covering group G. If X and Y are G-spaces, then considering G-equivariant maps f̃: X × EG → Y ×EG compatible with the projections to EG, giving a commutative diagram

\xymatrix{X × EG \ar[dr] \ar[rr]^{f̃} &  & Y × EG \ar[dl] \\  & EG}

is equivalent to considering maps \overline{f} : X//G → Y//G compatible with the maps q_{X} : X//G → BG, q_{Y} : Y//G → BG, giving a commutative diagram

\xymatrix{X//G \ar[dr]_{q_{X}} \ar[rr]^{\overline{f}} &  & Y//G \ar[dl]^{q_{Y}} \\  & BG}

Proposition 5. Assume that, in the situation of proposition 1, there are Λcompatible G-actions on X and Y. Let EG be as above, and consider the Λcompatible families {X_{λ} × EG}, which is a weakly admissible covering family for X×EG, and {Y_{λ}×EG}, which is an admissible covering family for Y ×EG.

Then there is a G-equivariant map f̃: X × EG → Y × EG, compatible with the projections to EG, such that

(i) f̃(X_{λ} × EG) ⊂ (Y_{λ} × EG) for all λ ∈ Λ (ii) if g̃ : X × EG → Y × EG is another such equivariant map, then there is a G-equivariant homotopy between f̃ and g̃, compatible with the projections to EG

(iii) The space of such equivariant maps X × EG → Y × EG, as in (i), is contractible.

Proof. We show the existence of the desired map, and leave the proof of other properties, by similar arguments, to the reader. Let Map_{Λ}(X, Y) be the contractible space of Λ-compatible maps from X to Y; note that it comes equipped with a natural G-action, so that the canonical evaluation map X × Map_{Λ}(X, Y) → Y is equivariant. This induces X × Map_{Λ}(X, Y) × EG → Y × EG. There is also a natural G-equivariant map π : X × Map_{Λ}(X, Y) × EG → X × EG. This map π has equivariant sections, since the projection Map_{Λ}(X, Y)×EG → EG is a G-equivariant map between weakly contractible spaces, so that the map on quotients modulo G is a weak homotopy equivalence (i.e., (Map_{Λ}(X, Y) × EG)/G is another “model” for the classifying space BG = EG/G). However BG is a CW complex, so the map has a section. □

As another preliminary, we note some facts (see lemma 6 below) which are essentially corollaries of Quillen’s Theorem A (these are presumably well-known to experts, though we do not have a specific reference). If P is any poset, let C(P) be the poset consisting of non-empty finite chains (totally ordered subsets) of P. If f : P → Q is a morphism between posets (an order preserving map) there is an induced morphism C(f) : C(P) → C(Q). If S is a simplicial complex (literally, a collection of finite non-empty subsets of the vertex set), we may regard S as a poset, partially ordered with respect to inclusion; then the classifying space BS is naturally homeomorphic to the geometric realisation |S| (and gives the barycentric subdivision of |S|). A simplicial map f : S → T between simplicial complexes (that is, a map on vertex sets which sends simplices to simplices, not necessarily preserving dimension) is also then a morphism of posets. We say that a poset P is contractible if its classifying space BP is contractible.

Lemma 6. (i) Let f : P → Q be a morphism between posets. Suppose that for each X ∈ C(Q), the fiber poset C(f)^{−1}(X) is contractible. Then Bf : BP → BQ is a homotopy equivalence. (ii) Let f : S → T be a simplicial map between simplicial complexes. Suppose that for any simplex σ ∈ T, the fiber f^{−1}(σ), considered as a poset, is contractible. Then |f| : |S| → |T | is a homotopy equivalence.

Proof. We first prove (i). For any poset P, there is morphism of posets φ_{P} : C(P) → P, sending a chain to its first (smallest) element. If a, b ∈ P with a ≤ b, and C is a chain in φ_{P}^{−1}(b), then {a} ∪ C is a chain in φ_{P}^{−1}(a). This gives an order preserving map of posets φ_{P}^{−1}(b) → φ_{P}^{−1}(a) (i.e., a “base-change” functor). This makes C(P) prefibred over P, in the sense of Quillen (see page 96 in [19], for example). Also, φ_{P}^{−1}(a) has the minimal element (initial object) {a}, and so its classifying space is contractible. Hence Quillen’s Theorem A (see [19], page 96) implies that B(φ_{P}) is a homotopy equivalence, for any P. Now let f : P → Q be a morphism between posets. Let C(f) : C(P) → C(Q) be the corresponding morphism on the posets of (finite, nonempty) chains. If A ⊂ B are two chains in C(Q), there is an obvious order preserving map C(f)^{−1}(B) → C(f)^{−1}(A). Again, this makes C(f) : C(P) → C(Q) prefibred. Since we assumed that BC(f)^{−1}(A) is contractible, for all A ∈ C(Q), Quillen’s Theorem A implies that BC(f) is a homotopy equivalence. We thus have a commutative diagram of posets and order preserving maps

C(P) →^{C(f)} C(Q)
φ_{P} ↓  ↓ φ_{Q}
P  →^{f}  Q

where three of the four sides yield homotopy equivalences on passing to classifying spaces. Hence Bf : BP → BQ is a homotopy equivalence, proving (i). The proof of (ii) is similar. This is equivalent to showing that Bf : BS → BT is a homotopy equivalence. Since f : S → T, regarded as a morphism of posets, is naturally prefibered, and by assumption, Bf^{−1}(σ) is contractible for each σ ∈ T, Quillen’s Theorem A implies that Bf is a homotopy equivalence. □

We make use of Propositions 1 and 5 in the following way. Let A, B be sets, Z ⊂ A × B a subset such that the projections p : Z → A, q : Z → B are both surjective. Consider simplicial complexes S_{Z}(A), S_{Z}(B) on vertex sets A, B respectively, with simplices in S_{Z}(A) being finite nonempty subsets of fibers q^{−1}(b), for any b ∈ B, and simplices in S_{Z}(B) being finite, nonempty subsets of fibers p^{−1}(a), for any a ∈ A. Consider also a third simplicial complex S_{Z}(A, B) with vertex set Z, where a finite non-empty subset Z′ ⊂ Z is a simplex if and only it satisfies the following condition:

(a_{1}, b_{1}), (a_{2}, b_{2}) ∈ Z′ ⇒ (a_{1}, b_{2}) ∈ Z.

Note that the natural maps on vertex sets p : Z → A, q : Z → B induce canonical simplicial maps on geometric realizations

|p| : |S_{Z}(A, B)| → |S_{Z}(A)|, |q| : |S_{Z}(A, B)| → |S_{Z}(B)|.

Corollary 7. (1) With the above notation, the simplicial maps

|p| : |S_{Z}(A, B)| → |S_{Z}(A)|, |q| : |S_{Z}(A, B)| → |S_{Z}(B)|

are homotopy equivalences. In particular, |S_{Z}(A)|, |S_{Z}(B)| are homotopy equivalent.

(2) If a group G acts on A and on B, so that Z is stable under the diagonal G action on A × B, then the homotopy equivalences |p|, |q| are Gequivariant homotopy equivalences. Hence there exists a natural Gequivariant homotopy equivalence between |S_{Z}(A)|×EG and |S_{Z}(B)|× EG.

Proof. Since the situation is symmetric with respect to the sets A, B, it suffices to show |p| is a homotopy equivalence. Note that in the context of a G-action as stated, the G-equivariance of |p| is clear. Let Λ be the poset of all simplices of S_{Z}(A), thought of as subsets of A, and ordered by inclusion. Clearly Λ is Artinian. Apply Corollary 2 with X = |S_{Z}(A, B)|, Y = |S_{Z}(A)|, Λ as above, and the following Λ-admissible coverings: for σ ∈ Λ, let Y_{σ} be the (closed) simplex in Y = |S_{Z}(A)| determined by σ (clearly {Y_{σ}} is admissible); take X_{σ} = |p|^{−1}(Y_{σ}) (this is evidently weakly admissible). For admissibility of {X_{σ}}, we need to show that each X_{σ} is contractible. In fact, regarding the sets of simplices S_{Z}(A, B) and S_{Z}(A) as posets, and S_{Z}(A, B) → S_{Z}(A) as a morphism of posets, X_{σ} is the geometric realization of the simplicial complex determined by ∪_{τ≤σ}p^{−1}(τ). The corresponding map of posets

p^{−1}({τ|τ ≤ σ}) → {τ|τ ≤ σ}

has contractible fiber posets – if we fix an element x ∈ p^{−1}(τ), and p^{−1}(τ)(≥ x) is the sub-poset of elements bounded below by x, then y ↦ y ∪x is a morphism of posets r_{x} : p^{−1}(τ) → p^{−1}(τ)(≥ x) which gives a homotopy equivalence on geometric realizations (it is left adjoint to the inclusion of the sub-poset). But the sub-poset has a minimal element, and so its realization is contractible. The poset {τ|τ ≤ σ} is obviously contractible, since it has a maximal element. Hence, applying lemma 6(ii), X_{σ} is contractible. Since the map |p| : X → Y is Λ-compatible, it is the unique such map upto Λ-compatible homotopy, and is a homotopy equivalence. □

We note that the argument with Quillen’s Theorem A in fact implies directly that S_{Z}(A, B) → S_{Z}(A) is a homotopy equivalence; the uniqueness assertion is not, apparently, a formal consequence of Quillen’s Theorem A.

2. Flag Spaces

In this section, we discuss various constructions of spaces (generally simplicial complexes) defined using flags of free modules, and various maps, and homotopy equivalences, between these. These are used as building blocks in the proof of Theorem 1. Let A be a ring, and let V be a free (left) A-module of rank n. Define a simplicial complex 𝔽𝕃(V) as follows. Its vertex set is

F L(V) = = {F = (F_{0}, F_{1}, …, F_{n}) | 0 = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{n} = V are A-submodules,

and each quotient F_{i}/F_{i−1} is a free A-module of rank 1}.

We think of this vertex set as the set of “full flags” in V. To describe the simplices in 𝔽𝕃(V), we need another definition. Let

SP L(V) = {{L_{1}, …, L_{n}} | L_{i} ⊂ V is a free A-submodule of rank 1, and the induced map ⊕_{i=1}^{n}L_{i} → V is an isomorphism}.

Note that {L_{1}, …, L_{n}} is regarded as an unordered set of free A-submodules of rank 1 of L (i.e., as a subset of cardinality n in the set of all free A-submodules of rank 1 of V). We think of SP L(V) as the “set of unordered splittings of V into direct sums of free rank 1 modules”. Given α ∈ SP L(V), say α = {L_{1}, …, L_{n}}, we may choose some ordering (L_{1}, …, L_{n}) of its elements, and thus obtain a full flag in V (i.e., an element in F L(V)), given by

(0, L_{1}, L_{1} ⊕ L_{2}, ⋯, L_{1} ⊕ ⋯ ⊕ L_{n} = V) ∈ F L(V).

Let

[α] ⊂ F L(V)

be the set of n! such full flags obtained from α. We now define a simplex in 𝔽𝕃(V) to be any subset of such a set [α] of vertices, for any α ∈ SP L(V). Thus, 𝔽𝕃(V) becomes a simplicial complex of dimension n! − 1, with the sets [α] as above corresponding to maximal dimensional simplices. Clearly Aut (V) ≅ GL_{n}(A) acts on the simplical complex 𝔽𝕃(V) through simplicial automorphisms, and thus acts on the homology groups H_{∗}(𝔽𝕃(V), ℤ) (and other similar invariants of 𝔽𝕃(V)). Next, remark that if F ∈ F L(V) is any vertex of 𝔽𝕃(V), we may associate to it the free A-module gr_{F} (V) = ⊕_{i=1}^{n}F_{i}/F_{i−1}. If (F, F′) is an ordered pair of distinct vertices, which are joined by an edge in 𝔽𝕃(V), then we obtain a canonical isomorphism (determined by the edge)

φ_{F,F′} : gr_{F} (V) ⟶^{≅} gr_{F′}(V).

One way to describe it is by considering the edge as lying in a simplex [α], for some α = {L_{1}, …, L_{n}} ∈ SP L(V); this determines an identification of gr_{F} (V) with ⊕_{i}L_{i}, and a similar identification of gr_{F′}(V), and thereby an identification between gr_{F} (V) and gr_{F′}(V). Note that from this description of the maps φ_{F.F′}, it follows that if F, F′, F′′ form vertices of a 2-simplex in 𝔽𝕃(V), i.e., there exists some α ∈ SP L(V) such that F, F′, F′′ ∈ [α], then we also have

φ_{F,F′′} = φ_{F′,F′′} ∘ φ_{F,F′}.

The isomorphism φ_{F,F′} depends only on the (oriented) edge in 𝔽𝕃(V) determined by (F, F′), and not on the choice of the simplex [α] in which it lies. One way to see this is to use that, for any two such filtrations F, F′ of V there is a canonical isomorphism gr _{F}^{p} gr _{F′}^{q}(V) ≅ gr _{F′}^{q}gr _{F}^{p} (V) (Schur-Zassenhaus lemma) for each p, q. But in case F, F′ are flags which are connected by an edge, then there is also a canonical isomorphism gr_{F} gr_{F′}(V) ≅ gr_{F′}(V) (in fact the F -filtration induced on gr _{F′}^{p}(V) has only 1 non-trivial step, for each p), and similarly there is a canonical isomorphism gr_{F′}gr_{F} (V) ≅ gr_{F} (V). These three canonical isomorphisms combine to give the isomorphism φ_{F,F′}. Hence there is a well-defined local system gr(V) of A-modules on the geometric realization |𝔽𝕃(V)| of the simplicial complex 𝔽𝕃(V), whose fibre over a vertex F is gr_{F} (V). Notice further that this local system gr(V) comes equipped with a natural Aut(V) action, compatible with the natural actions on F L(V) and 𝔽𝕃(V). Indeed, any element g ∈ Aut(V) gives a bijection on the set of full flags F L(V), with

F = (0 = F_{0}, F_{1}, …, F_{n} = V) ∈ F L(V))

mapping to

gF = (0 = gF_{0}, gF_{1}, …, gF_{n} = V).

This clearly gives an induced isomorphism ⊕_{i}F_{i}/F_{i−1} ≅ ⊕_{i}gF_{i}/gF_{i−1}, identifying the fibers of the local system over F and gF in a specific way. It is easy to see that if α = {L_{1}, …, L_{n}} ∈ SP L(V), then gα = {gL_{1}, …, gL_{n}} ∈ SP L(V), giving the action of Aut (V) on SP L(V), so that if a pair of vertices F, F′ of F L(V) lie on an edge contained in [α], then gF, gF′ lie on an edge contained in [gα], and so the induced identification φ_{F,F′} is compatible with φ_{gF,gF′}. This induces the desired action of Aut(V) on the local system. Further, note that if F, F′ ∈ F L(V) are connected by an edge in 𝔽𝕃(V), then we may realize φ_{F,F′} by the action of a suitable element of Aut (V), which preserves a simplex [α] in which the edge lies, and permutes the lines in the splitting α ∈ SP L(V). Thus, given any edge-path joining vertices F, F′ in 𝔽𝕃(V), the induced composite isomorphism gr_{F} (V) → gr_{F′}(V) is again realized by the action of an element of Aut (V). In particular, given an edge-path loop based as F ∈ F L(V), the induced automorphism of gr_{F} (V) is induced by the action on gr_{F} (V) of an element of the isotropy group of F in Aut (V), which is the “Borel subgroup” corresponding to the flag F. Hence, the monodromy group of the local system gr(V) is clearly contained in N_{n}(A), defined as a semidirect product

(2)  N_{n}(A) = (A^{×} × ⋯ × A^{×}) ⋉ S_{n}

where S_{n} is the permutation group; we regard N_{n}(A) as a subgroup of Aut (⊕_{i}L_{i}) in an obvious way. Now we make infinite versions of the above constructions. Let A^{∞} be the set of sequences (a_{1}, a_{2}, …, a_{n}, …) of elements of A, all but finitely many of which are 0, considered as a free A-module of countable rank. There is a standard inclusion i_{n} : A^{n} ↪ A^{∞} of the standard free A-module of rank n as the submodule of sequences with a_{m} = 0 for all m > n. The induced inclusion i : A^{n} → A^{n+1} is the usual one, given by i(a_{1}, …, a_{n}) = (a_{1}, …, a_{n}, 0).

We may thus view A^{∞} as being given with a tautological flag, consisting of the A-submodules i_{n}(A^{n}). We define a simplicial complex 𝔽𝕃(A^{∞}), with vertex set F L(A^{∞}) equal to the set of flags 0 = V_{0} ⊂ V_{1} ⊂ V_{2} ⊂ ⋯ ⊂ A^{∞} where V_{i}/V_{i−1} is a free A-module of rank 1, for each i ≥ 1, and with V_{n} = i(A^{n}) for all sufficiently large n. Thus F L(A^{∞}) is naturally the union of subsets bijective with F L(A^{n}). To make 𝔽𝕃(A^{∞}) into a simplicial complex, we define a simplex to be a finite set of vertices in some subset F L(A^{n}) which determines a simplex in the simplicial complex 𝔽𝕃(A^{n}); this property does not depend on the choice of n, since the natural inclusion F L(A^{n}) ↪ F L(A^{n+1}), regarded as a map on vertex sets, identifies 𝔽𝕃(A^{n}) with a subcomplex of 𝔽𝕃(A^{n+1}), such that any simplex of 𝔽𝕃(A^{n+1}) with vertices in F L(A^{n}) is already in the subcomplex 𝔽𝕃(A^{n}). We consider GL(A) ⊂ Aut(A^{∞}) as the union of the images of the obvious maps i_{n} : GL_{n}(A) ↪ Aut (A^{∞}), obtained by automorphisms which fix all the basis elements of A^{∞} beyond the first n. We clearly have an induced action of GL(A) on the simplicial complex 𝔽𝕃(A^{∞}), and hence on its geometric realisation |𝔽𝕃(A^{∞})| through homeomorphisms preserving the simplicial structure. The inclusion 𝔽𝕃(A^{n}) ↪ 𝔽𝕃(A^{∞}) as a subcomplex is clearly GL_{n}(A)- equivariant. Next, observe that there is a local system gr(A^{∞}) on 𝔽𝕃(A^{∞}) whose fiber over a vertex F = (F_{0} = 0, F_{1}, …, F_{n}, …) is gr_{F} (V) = ⊕_{i}F_{i}/F_{i−1}. This has monodromy contained in

N(A) = ∪_{n}N_{n}(A) ⊂ GL(A),

where we may also view N(A) as the semidirect product of

(A^{×})^{∞} = diagonal matrices in GL(A)

by the infinite permutation group S_{∞}. This local system also carries a natural GL(A)-action, compatible with the GL(A)-action on 𝔽𝕃(A^{∞}). Next, we prove a property (Corollary 9) about the action of elementary matrices on homology, which is needed later. The corollary follows immediately from the lemma below. For the statement and proof of the lemma, we suggest that the reader browse the remarks on Hom(K_{1} × K_{2}, K_{3}) in section 0, given simplicial complexes K_{i} for i = 1, 2, 3. The notation Elem(V′ ↪ V′ ⊕ V′′) that appears in the lemma has also been introduced in section 0 under the heading “elementary matrices”.

Lemma 8. Let V′, V′′ be two free A-modules of finite rank, i′ : V′ → V′ ⊕ V′′, i′′ : V′′ → V′ ⊕ V′′ the inclusions of the direct summands. Consider the two natural maps

(3)  α, β : F L(V′) × F L(V′′) → F L(V′ ⊕ V′′)

given by

α : ((F′_{1}, …, F′_{r} = V′), (F′′_{1}, …, F′′_{s})) ↦ (i′(F′_{1}), …, i′(F′_{r}) = i′(V′), i′(V′) + i′′(F′′_{1}), …, i′(V′) + i′′(F′′_{s}) = V′ ⊕ V′′), β : ((F′_{1}, …, F′_{r} = V′), (F′′_{1}, …, F′′_{s})) ↦ (i′′(F′′_{1}), …, i′′(F′′_{s}) = i′′(V′′), i′′(V′′) + i′(F′_{1}), …, i′′(V′′) + i′(F′_{r}) = V′ ⊕ V′′).

(A) α and β are vertices of a one-simplex of Hom(𝔽𝕃(V′) × 𝔽𝕃(V′′), 𝔽𝕃(V′ ⊕ V′′)). (B) The maps |𝔽𝕃(V′)| × |𝔽𝕃(V′′)| → |𝔽𝕃(V′ ⊕ V′′)| induced by α, β are homotopic to each other. (C) Let c : |𝔽𝕃(V′)| × |𝔽𝕃(V′′)| → |𝔽𝕃(V′ ⊕ V′′)| denote the map produced by α. Denote the action of g ∈ GL(V′ ⊕ V′′) on |𝔽𝕃(V′ ⊕ V′′)| by |𝔽𝕃(g)|. Then c and |𝔽𝕃(g)| ∘ c are homotopic to each other, if g ∈ Elem(V′ ↪ V′ ⊕ V′′) ⊂ GL(V′ ⊕ V′′).

Proof. Part (A). By the definition of Hom(K_{1} × K_{2}, K_{3}) in section 0, we only have to check that α(σ′ × σ′′) ∪ β(σ′ × σ′′) is a simplex of F L(V′ ⊕ V′′) for all simplices σ′ of 𝔽𝕃(V′) and all simplices σ′′ of 𝔽𝕃(V′′). Clearly it suffices to prove this for maximal simplices, so we assume that both σ′ and σ′′ are maximal. Note that if we consider any maximal simplex σ′ in 𝔽𝕃(V′), it corresponds to a splitting {L′_{1}, …, L′_{r}} ∈ SP L(V′). Similarly any maximal simplex σ′′ of 𝔽𝕃(V′′) corresponds to a splitting {L′′_{1}, …, L′′_{s}} ∈ SP L(V′′). This determines the splitting {i′(L′_{1}), …, i′(L′_{r}), i′′(L′′_{1}), …, i′′(L′′_{s})} of V′ ⊕ V′′, giving rise to a maximal simplex τ of 𝔽𝕃(V′ ⊕ V′′), and clearly α(σ′ × σ′′) and β(σ′ × σ′′) are both contained in τ. Thus their union is a simplex. (B) follows from (A). We now address (C). We note that c = g ∘ c for all g ∈ id + Hom_{A}(V′′, V′). Denoting by d the map produced by β we see that d = g ∘ d for all g ∈ id + Hom_{A}(V′, V′′). Because c, d are homotopic to each other, we see that c and g ∘c are in the same homotopy class when g is in either of the two groups above. These groups generate Elem(V′ ↪ V′ ⊕ V′′), and so this proves (C). □

Corollary 9. (i) The group E_{n+1}(A) of elementary matrices acts trivially on the image of the natural map

i_{∗} : H_{∗}(𝔽𝕃(A^{⊕n}), ℤ) → H_{∗}(𝔽𝕃(A^{⊕n+1}), ℤ).

(ii) The action of the group E(A) of elementary matrices on H_{∗}(𝔽𝕃(A^{∞}), ℤ) is trivial.

Proof. We put V′ = A^{n} and V′′ = A in the previous lemma. The c in part (B) of the lemma is precisely the i being considered here. By (C) of the lemma, g ∘ i is homotopic to i for all g ∈ Elem(A^{n} ↪ A^{n+1}) = E_{n+1}(A). This proves (i). The direct limit of the r-homology of |𝔽𝕃(A^{n})|, taken over all n, is the r-th homology of |𝔽𝕃(A^{∞}|. Thus (i) implies (ii).

□

We will find it useful below to have other “equivalent models” of the spaces 𝔽𝕃(V), 𝔽𝕃(A^{∞}), by which we mean other simplicial complexes, also defined using collections of appropriate A-submodules, such that there are natural homotopy equivalences between the different models of the same homotopy type, compatible with the appropriate group actions, etc. We apply corollary 7 as follows. Let V ≅ A^{n}. We put A = SP L(V), B = F L(V) and Z = {α, F) : F ∈ [α]}. The simplicial complex S_{Z}(B) of corollary 7 is 𝔽𝕃(V) by its definition. The simplicial complex S_{Z}(A) is our definition of 𝕊ℙ𝕃(V). The homotopy equivalence of 𝕊ℙ𝕃(V) and 𝔽𝕃(V) follows from this corollary. We define SP L(A^{∞}) to be the collection of sets S satisfying

(a) L ∈ S implies that L is a free rank one A-submodule of A^{∞}, (b)⊕{L : L ∈ S} → A^{∞} is an isomorphism, and

(c) the symmetric difference of S and the standard collection: {A(1, 0, 0, ...), A(0, 1, 0, ...), ⋯ } is a finite set. Corollary 7 is then applied to the subset Z ⊂ SP L(A^{∞}) × F L(A^{∞}) consisting of the pairs (S, F) so that there is a bijection h : S → ℕ so that for every L ∈ S, L ⊂ F_{h(L)} and L → gr_{h(L)}^{F} is an isomorphism. The above Z defines 𝕊ℙ𝕃(A^{∞}). The desired homotopy equivalence of the geometric realisations of 𝕊ℙ𝕃(A^{∞}) and 𝔽𝕃(A^{∞}) comes from the same corollary. We also find it useful to introduce a third model of the homotopy types of 𝔽𝕃(V) and 𝔽𝕃(A^{∞}), the “enriched Tits buildings” 𝔼𝕋(V) and 𝔼𝕋(A^{∞}). The latter is defined in the last remark of this section. Let V ≅ A^{n} as a left A-module. Let E(V) be the set consisting of ordered pairs

(F, S) = ((0 = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{r} = V), (S_{1}, S_{2}, …, S_{r})) ,

where F is a partial flag in V, which means that F_{i} ⊂ V is an A-submodule, such that F_{i}/F_{i−1} is a nonzero free module for each i, and S_{i} ∈ SP L(F_{i}/F_{i−1}) is an unordered collection of free A-submodules of F_{i}/F_{i−1} giving rise to a direct sum decomposition ⊕_{L∈S_{i}}L ≅ F_{i}/F_{i−1}. Thus S is a collection of splittings of the quotients F_{i}/F_{i−1} for each i. We may put a partial order on the set E(V) in the following way: (F, S) ≤ (F′, T) if the filtration F is a refinement of F′, and the data S, T of direct sum decompositions of quotients are compatible, in the following natural sense — if F′_{i−1} = F_{j−1} ⊂ F_{j} ⊂ ⋯ F_{l} = F′_{i}, then T_{i} must be partitioned into subsets, which map to the sets S_{j}, S_{j+1}, …, S_{l} under the appropriate quotient maps. In particular, (F′, T) has only finitely many possible predecessors (F, S) in the partial order. We have a simplicial complex ET (V) := NE(V), the nerve of the partially ordered set E(V) considered as a category, so that simplices are just nonempty finite chains of elements of the vertex (po)set E(V). Note that maximal elements of E(V) are naturally identified with elements of SP L(V), while minimal elements are naturally identified with elements of F L(V). Simplices in 𝔽𝕃(V) are nonempty finite subsets of F L(V) which have a common upper bound in E(V), and similarly simplices in 𝕊ℙ𝕃(V) are nonempty finite subsets of SP L(V) which have a common lower bound in E(V).

We now show that 𝔼𝕋(V) = BE(V), the classifying space of the poset E(V), is another model of the homotopy type of |𝔽𝕃(V)|. In a similar fashion, we may define a poset E(A^{∞}), and a space 𝔼𝕋(A^{∞}), giving another model of the homotopy type of |𝔽𝕃(A^{∞})|. We first have a lemma on classifying spaces of certain posets. For any poset (P, ≤), and any S ⊂ P, let

L(S) = {x ∈ P | x ≤ s ∀ s ∈ S}, U(S) = {x ∈ P | s ≤ x ∀ s ∈ S}

be the upper and lower sets of S in P, respectively. Let P_{min} denote the simplicial complex with vertex set P_{min} given by minimal elements of P, and where a nonempty finite subset S ⊂ P_{min} is a simplex if U(S) ≠ ∅. Let |P_{min}| denote the geometric realisation of P_{min}.

Lemma 10. Let (P, ≤) be a poset such that

(a) ∀ s ∈ P, the set L({s}) is finite

(b) if ∅ ≠ S ⊂ P with L(S) ≠ ∅, then the classifying space BL(S) of L(S) (as a subposet) is contractible.

Then |P_{min}| is naturally homotopy equivalent to BP.

Proof. We apply Proposition 1. Take

Λ = {L(S)| ∅ ≠ S ⊂ P and L(S) ≠ ∅}.

This is a poset with respect to inclusion. All λ ∈ Λ are finite subsets of P, so Λ is Artinian. By assumption, the subsets B(λ) ⊂ BP, for λ ∈ Λ, are contractible. On the other hand, the sets λ ∩ P_{min} give simplices in |P_{min}|. Thus, both the spaces BP and |P_{min}| have Λ-admissible coverings, and are thus homotopy equivalent. □

Remark. If a poset P has g.c.d. in the sense that ∅ ≠ S ⊂ P and ∅ ≠ L(S) implies L(S) = L(t) for some t ∈ P, then condition (b) of the lemma is immediately satisfied. However E(V) does not enjoy the latter property. For example, if V = A^{3} with basis e_{1}, e_{2}, e_{3}, let s = {Ae_{1}, Ae_{2}, Ae_{3}} and t = {Ae_{1}, A(e_{1} + e_{2}), Ae_{3}} and let S = {s, t} ⊂ SP L(V) ⊂ E(V). Then L(S) has three minimal elements and two maximal elements. In particular, g.c.d. (s, t) does not exist. In this example, B(L(S)) is an oriented graph in the shape of the letter M.

Proposition 11. If V is a free A module of finite rank, the poset E(V) satisfies the hypotheses of lemma 10. Thus, |𝔽𝕃(V)| is naturally homotopy equivalent to 𝔼𝕋(V) = B(E(V)).

Proof. Clearly the condition (a) of lemma 10 holds, so it suffices to prove (b). We now make a series of observations.

(i) Regard SP L(V) as the set of maximal elements of the poset E(V). We observe that for any s ∈ E(V), if H(s) = SLP (V) ∩ U({s}), then we have that L({s}) = L(H(s)). This is easy to see, once one has unravelled the definitions.

Thus, it suffices to show that for sets S of the type ∅ ≠ S ⊂ SP L(V) ⊂ E(V), we have that B(L(S) is contractible. We assume henceforth that S ⊂ SP L(V).

(ii) Given a submodule W ⊂ V which determines a partial flag 0 ⊂ W ⊂ V, we have a natural inclusion of posets

E(W) × E(V/W) ⊂ E(V),

where on the product, we take the partial order

(a_{1}, a_{2}) ≤ (b_{1}, b_{2}) ⇔ a_{1} ≤ b_{1} ∈ E(W) and a_{2} ≤ b_{2} ∈ E(V/W).

Note that α ∈ E(V) lies in the sub-poset E(W) × E(V/W) precisely when W is one of the terms in the partial flag associated to α. Hence, if α lies in the sub-poset, so does the entire set L({α}).

(iii) With notation as above, if ∅ ≠ S ⊂ SP L(V) ⊂ E(V), and L(S) has nonempty intersection with the image of E(W)×E(V/W) ⊂ E(V), then clearly there exist nonempty subsets S′(W) ⊂ SP L(W), S′′(W) ⊂ 𝕊ℙ𝕃(V/W) such that

(4)  L(S) ∩ (E(W) × E(V/W)) = L(S′(W)) × L(S′′(W))

(iv) If ∅ ≠ S ⊂ SP L(V), then each s ∈ S is a subset of

L(V) = {L ⊂ V |L is a free direct summand of rank 1 of V },

the set of lines in V. Let

T (S) = ∩_{s∈S}s = lines common to all members of S,

so that T (S) ⊂ L(V). Let M(S) denote the direct sum of the elements of T (S), so that M(S) is a free A-module of finite rank, and 0 ⊂ M(S) ⊂ V is a partial flag, in the sense explained earlier; further, T (S) may be regarded also as an element of SP L(M(S)) ⊂ E(M(S)).

(v) We now claim the following: if ∅ ≠ S ⊂ SP L(V) and b ∈ L(S), then there exists a unique subset f(b) ⊂ T (S) such that if M(b) is the (direct) sum of the lines in f(b), then

b = (f(b), b′) ∈ SP L(M(b)) × E(V/M(b)) ⊂ E(M(b)) × E(V/M(b)) ⊂ E(V).

Indeed, if

b = ((0 = W_{0} ⊂ W_{1} ⊂ ⋯ ⊂ W_{h} = V), (t_{1}, t_{2}, …, t_{h})}

where t_{i} ∈ SP L(W_{i}/W_{i−1}), then since b ∈ L(S), we must have that t_{1} ⊂ s for all s ∈ S, which implies that t_{1} ⊂ T (S). Take M(b) = W_{1}, t_{1} = f(b) ∈ SP L(M(b)).

Let P(T (S)) be the poset of nonempty subsets of T (S), with respect to inclusion. Then b ↦ f(b) gives an order-preserving map f : L(S) → P(T (S)).

(vi) For any b ∈ L(S), we have

L(b) = L({f(b)}) × L(b′) ⊂ E(M(b)) × E(V/M(b)),

so that if b_{1} ∈ L(b), then ∅ ≠ f(b_{1}) ⊂ f(b).

We will now complete the proof of Proposition 11. We proceed by induction on the rank of V. Suppose S ⊂ SP L(V) is nonempty, and L(S) ≠ ∅. If M(S) = V, then S = {s} for some s, and L(S) = L({s}) is a cone, hence contractible. So assume M(S) ≠ V. If T ⊂ T (S) is non-empty, and M(T) ⊂ V the (direct) sum of the lines in T, then in the notation of (4) above, with W = M(T), we have S′(W) = {T }, and so f^{−1}(T) = {T } × L(S′′(W))) for some S′′(W) ⊂ SP L(V/W). Now by induction, we have that L(S′′(W)) is contractible, provided it is nonempty. Hence the non-empty fiber posets of f are contractible. If ∅ ≠ T ⊂ T′ ⊂ T (S), then there is a morphism of posets f^{−1}(T) → f^{−1}(T′) given as follows: if b ∈ f^{−1}(T), and

b = ((0 = W_{0} ⊂ W_{1} ⊂ ⋯ ⊂ W_{h} = V), (t_{1}, t_{2}, …, t_{h})}

where t_{i} ∈ SP L(W_{i}/W_{i−1}), then since b ∈ f^{−1}(T), we must have t_{1} = T, W_{1} = M(T). Now define b′ ∈ f^{−1}(T′) using the partial flag

0 = W′ ⊂ W_{1} + M(T′) ⊂ W_{2} + M(T′) ⊂ ⋯ ⊂ W_{h} + M(T′) = V

and elements t′_{i} ∈ SP L(W_{i} + M(T′)/W_{i−1} + M(T′)) induced by the t_{i}. This is easily seen to be well-defined, and gives a morphism of posets f^{−1}(T) → f^{−1}(T′). In particular, if T ⊂ T′ ⊂ T (S) and f^{−1}(T) is non-empty, then so is f^{−1}(T′). Now take any b ∈ L(S) and put T = f(b), T′ = T (S) in the above to deduce that f^{−1}(T (S)) ≠ ∅. By (iii) above, we see that every f^{−1}X is nonempty (and therefore contractible as well) for every nonempty X ⊂ T (S). We see that all the fiber posets f^{−1}(T) considered above are nonempty. This makes f pre-cofibered, in the sense of Quillen (see [19], page 96), with contractible fibers. Hence by Quillen’s Theorem A, f induces a homotopy equivalence on classifying spaces. But P(T (S)) is contractible (for example, since T (S) is the unique maximal element). □

Remark. Proposition 5 and the remarks preceding it apply to the above Proposition. In particular, we obtain homotopy equivalences f : 𝔼𝕋(V) → 𝔽𝕃(V) so that the induced maps on homology are GL(V)-equivariant.

Remark. We now define the poset E(A^{∞}) and show that 𝔼𝕋(A^{∞}) = BE(A^{∞}) is homotopy equivalent to |𝔽𝕃(A^{∞}|. We have already observed that a short exact sequence of free modules of finite rank

0 → V′ → V → V′′ → 0

induces a natural inclusion E(V′) × E(V′′) ↪ E(V) of posets. In particular, when V′′ ≅ A, this yields an inclusion E(V′) ↪ E(V). We have ... ⊂ A^{n} ⊂ A^{n+1} ⊂ ... ⊂ A^{∞} as in the definition of 𝔽𝕃(A^{∞}). From the above, we obtain a direct system of posets

...E(A^{n}) ↪ E(A^{n+1}) ↪ ...

and we define E(A^{∞}) to be the direct limit of this system of posets.

We put P = E(A^{∞}) in lemma 10. We note that α ≤ β, α ∈ E(A^{∞}), β ∈ E(A^{n}) implies that α ∈ E(A^{n}). It follows that P = E(A^{∞}) satisfies the requirements of the lemma because each E(A^{n}) does. It is clear that P_{min} = F L(A^{∞}), and furthermore that P_{min} = 𝔽𝕃(A^{∞}). This yields the homotopy equivalence of |𝔽𝕃(A^{∞})| with 𝔼𝕋(A^{∞}). By Proposition 5, it follows that 𝔼𝕋(A^{∞})//GL(A) and |𝔽𝕃(A^{∞})|//GL(A) are also homotopy equivalent to each other. It has already been remarked that Corollary 7 gives the homotopy equivalence of |𝕊ℙ𝕃(A^{∞}| with |𝔽𝕃(A^{∞})|. Combined with Proposition 5, this gives the homotopy equivalence of |𝕊ℙ𝕃(A^{∞}|//GL(A) with |𝔽𝕃(A^{∞})|//GL(A). The remarks preceding that proposition, combined with corollary 9, show that the action of E(A) on the homology groups of |𝕊ℙ𝕃(A^{∞})| is trivial.

3. homology of the Borel constrcuction

Let V be a free A-module of rank n. Fix β ∈ SP L(V) and let N(β) ⊂ GL(V) be the stabiliser of β (when V = A^{n} and β is the standard splitting, then N(β) is the subgroup N_{n}(A) of the last section). That there is a GL(V)-equivariant N(β)-torsor on |𝔽𝕃(V)| has been observed in the previous section. In a similar manner, one may construct a GL(V)-equivariant N(β)-torsor on 𝔼𝕋(V). This gives rise to a N(β)-torsor on 𝔼𝕋(V)//GL(V). Because BN(β) is a classifying space for such torsors, we obtain a map 𝔼𝕋(V)//GL(V) → BN(β), well defined up to homotopy. On the other hand, the inclusion of β in 𝔼𝕋(V) gives rise to an inclusion BN(β) = {β}//N(β) ↪ 𝔼𝕋(V)//GL(V). It is clear that the composite BN(β) ↪ 𝔼𝕋(V)//GL(V) → BN(β) is homotopic to the identity. Thus BN(β) is a homotopy retract of 𝔼𝕋(V)//GL(V), but not homotopy equivalent to 𝔼𝕋(V)//GL(V). Nevertheless we have the following statement:

Proposition 12. The map BN(β) → 𝔼𝕋(V)//GL(V) induces an isomorphism on integral homology, provided A is as in theorem 1.

Proof. Fix a basis for V, identifying GL(V) with GL_{n}(A). Let β ∈ SP L(V) be the element naturally determined by this basis. Regarded as a vertex of 𝔼𝕋(V), let (β, ∗) ↦ \overline{β} under the natural map

π : 𝔼𝕋(V)//GL(V) → 𝔼𝕋(V)/GL(V)

from the homotopy quotient to the geometric quotient, where ∗ ∈ EGL(V) is the base point (corresponding to the vertex labelled by the identity element of GL(V)). For any x ∈ 𝔼𝕋(V), let H(x) ⊂ GL(V) be the isotropy group of x for the GL(V)-action on 𝔼𝕋(V). Note that since

𝔼𝕋(V)//GL(V) = (𝔼𝕋(V) × EGL(V)) /GL(V),

the fiber π^{−1}(π((x, ∗)) may be identified with EGL(V)/H(x), which has the homotopy type of BH(x).

In particular, the fiber π^{−1}(\overline{β}) has the homotopy type of BN_{n}(A), Further, the principal N_{n}(A) bundle on EGL(V)/H(β) is naturally identified with the universal N_{n}(A)-bundle on BN_{n}(A) – its pullback to {β}×EGL(V) is the trivial N_{n}(A)-bundle, regarded as an N_{n}(A)-equivariant principal bundle, where N_{n}(A) acts on itself (the fiber of the trivial bundle) by translation. This means that the composite

π^{−1}(\overline{β}) → 𝔼𝕋(V)/GL(V) → BN_{n}(A)

is a homotopy equivalence, which is homotopic to the identity, if we identify EGL(V)/H(β) with BN_{n}(A). Thus, the lemma amounts to the assertion that π^{−1}(\overline{β}) → 𝔼𝕋(V)//GL(V) induces an isomorphism in integral homology. Fix α ∈ F L(V) with α ≤ β in the poset E(V). Let

P = {λ ∈ E(V)|α ≤ λ ≤ β}.

One sees easily that (i) BP is contractible, and (ii) the map BP → 𝔼𝕋(V)/GL(V) is a homoemorphism. The first assertion is obvious, since P has a maximal (as well as a minimal) element, so that BP is a cone. For the second assertion, we first note that an element b ∈ P ⊂ E(V) is uniquely determined by the ranks of the modules in the partial flag in V associated to b. Conversely, given any increasing sequence of numbers n_{1} < … < n_{h} = rank V, there does exist an element of P whose partial flag module ranks are these integers. Given any element b ∈ E(V), there exists an element g ∈ GL(V) so that g(b) = b′ ∈ P; the element b′ is the unique one determined by the sequence of ranks associated to b. Finally, one observes that if b ∈ P, and g ∈ GL(V) such that g(b) ∈ P, then in fact g(b) = b: this is a consequence of the uniqueness of the element of P with a given sequence of ranks. These observations imply that BP → 𝔼𝕋(V)/GL(V) is bijective; it is now easy to see that it is a homeomorphism. We may view 𝔼𝕋(V)//GL(V) as the quotient of BP × EGL(V) by the equivalence relation

(5)  (x, y) ∼ (x′, y′) ⇔ x = x′, and y′ = g(y) for some g ∈ H(x).

The earlier map π : 𝔼𝕋(V)//GL(V) → 𝔼𝕋(V)/GL(V) may be viewed now as the map induced by the projection BP × EGL(V) → BP. We may, with this identification, also identify \overline{β} with β. Next, we construct a “good” fundamental system of open neighbourhoods of an arbitrary point x ∈ BP, which we need below. Such a point x lies in the relative interior of a unique simplex σ(x) (called the carrier of x) corresponding to a chain λ_{0} < λ_{1} < ⋯ < λ_{r}. Then one sees that the stabiliser H(x) ⊂ GL(V) is given by

H(x) = ⋂_{i=0}^{r} H(λ_{i}),

since any element of GL(V) which stabilizes the simplex σ(x) must stabilize each of the vertices (for example, since the GL(V) action preserves the partial order). Let star (x) be the union of the relative interiors of all simplices in BP containing σ(x) (this includes the relative interior of σ(x) as well, so it contains x). It is a standard property of simplicial complexes that star (x) is an open neighbourhood of x in BP. Then if z ∈ star (x), clearly σ(z) contains σ(x), and so H(z) ⊂ H(x). Next, for such a point z, and any y ∈ EGL(V), it makes sense to consider the path

t ↦ (tz + (1 − t)x, y) ∈ σ(z) × EGL(V) ⊂ BP × EGL(V)

(where we view the expression tz+(1−t)x as a point of σ(z), using the standard barycentric coordinates). In fact this path is contained in star (x) × {y}, and gives a continuous map

H(x) : star (x) × EGL(V) × I → star (x) × EGL(V)

which exhibits {x} × EGL(V) as a strong deformation retract of star (x) × EGL(V). Further, this is compatible with the equivalence relation ∼ in (5) above, so that we obtain a strong deformation retraction

\overline{H(x)} : π^{−1}(star (x)) × I → π^{−1}(star (x)).

In a similar fashion, we can construct a fundamental sequence of open neighbourhoods U_{n}(x) of x in BP, with U_{1}(x) = star (x), and set

U_{n}(x) = H(x)(star (x) × EGL(V) × [0, 1/n)).

The same deformation retraction H determines, by reparametrization, a deformation retraction

H_{n}(x) : π^{−1}(U_{n}(x)) × I → π^{−1}(U_{n}(x))

of π^{−1}(U_{n}(x) onto π^{−1}(x). Thus, if P′ = P \ β, then

π^{−1}(star (β)) = 𝔼𝕋(V)//GL(V) \ π^{−1}(BP′),

and from what we have just shown above, the inclusion

π^{−1}(β) → π^{−1}(star (β)) = 𝔼𝕋(V)//GL(V) \ π^{−1}(BP′)

is a homotopy equivalence. To simplify notation, we let X = 𝔼𝕋(V)//GL(V), so that we have the map π : X → BP, and X^{0} = X \ π^{−1}(BP′). Let π^{0} = π |_{X^{0}}: X^{0} → BP. We are reduced to showing, with this notation, that the inclusion of the (dense) open subset

X^{0} → X

induces an isomorphism in integral homology. Equivalently, it suffices to show that this inclusion induces an isomorphism on cohomology with arbitrary constant coefficients M. By the Leray spectral sequence, this is a consequence of showing that the maps of sheaves

R^{i}π_{∗}M_{X} → R^{i}π_{∗}^{0}M_{X^{0}}

is an isomorphism, which is clear on stalks x ∈ BP \ BP′. Now consider stalks at a point x ∈ BP′. For any point x′ ∈ star (x), note that x lies in some face of σ(x′) (the carrier of x′). We had defined a fundamental system of neighbourhoods U_{n}(x) of x in BP; explicitly we have

U_{n}(x) = {tx′ + (1 − t)x|0 ≤ t < 1/n and x′ ∈ star (x)}.

Here, as before, we make sense of the above expression tx′ + (1 − t)x using barycentric coordinates in σ(x′). Define

z_{n}(x) = \frac{1}{2n}β + (1 − \frac{1}{2n})x.

Note that z ∈ BP \ BP′ = star (β). Further, observe that U_{n}(x) ∩ BP \ BP′ is contractible, contains the point z, and for any w ∈ U_{n}(x) ∩ BP \ BP′, contains the line segment joining z and w (this makes sense, in terms of barycentric coordinates of any simplex containing both z_{n}(x) and w; this simplex is either the carrier of w, or the cone over it with vertex β, of which σ(w) is a face). This implies H(w) ⊂ H(z_{n}(x)) = H(x) ∩ H(β), for all w ∈ U_{n}(x). A minor modification of the proof (indicated above) that π^{−1}(x) ⊂ π^{−1}(U_{n}(x)) is a strong deforamtion retract, yields the statement that

π^{−1}(z_{n}(x)) → π^{−1}(U_{n}(x) \ BP′)

is a strong deformation retract. Hence, the desired isomorphism on stalks follows from:

(6) B(H(x) ∩ H(β)) → B(H(x)) induces isomorphisms in integral homology.

We now show how this statement, for the appropriate rings A, is reduced to results of [13]. First, we discuss the structure of the isotropy groups H(x) encountered above. Let λ ∈ P, given by

λ = (F, S) = ((0 = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{r} = V), (S_{1}, S_{2}, …, S_{r})) ,

where we also have α ≤ λ ≤ β for our chosen elements α ∈ F L(V) and β ∈ SP L(V). We may choose a basis for each of the lines in the splitting β; then α ∈ F L(V) uniquely determines an order among these basis elements, and thus a basis for the underlying free A-module V, such that the i-th submodule in the full flag α is the submodule generated by the first i elements in β. Now the stabilizer H(α) may be viewed as the group of upper triangular matrices in GL_{n}(A), while H(β) is the group generated by the diagonal subgroup in GL_{n}(A) and the group of permutation matrices, identified with the permutation group S_{n}. In these terms, H(λ) has the following structure. The filtration F = (0 = F_{0} ⊂ F_{1} ⊂ ⋯ ⊂ F_{r} = V) is a sub-filtration of the full flag α, and so determines a “unipotent subgroup” U(λ) of elements fixing the elements of this partial flag, and acting trivially on the graded quotients F_{i}/F_{i−1}. These are represented as matrices of the form

[ \begin{matrix}I_{n_{1}} & ∗ & ∗ & ⋯ & ∗ \\ 0 & I_{n_{2}} & ∗ & ⋯ & ∗ \\ 0 & 0 & I_{n_{3}} & ⋯ & ∗ \\ & ⋮ & ... & & ⋮ \\ 0 & ⋯ & & & I_{n_{r}} \end{matrix} ]

where n_{i} = rank (W_{i}/W_{i−1}), I_{n_{i}} is the identity matrix of size n_{i}; these are the matrices which are strictly upper triangular with respect to a certain “ladder”. Next, we may consider the group S(λ) ⊂ S_{n} of permutation matrices, supported within the corresponding diagonal blocks, of the form

[ \begin{matrix}A_{1} & 0 & 0 & ⋯ & 0 \\ 0 & A_{2} & 0 & ⋯ & 0 \\ 0 & 0 & A_{3} & ⋯ & 0 \\ & ⋮ & ... & & ⋮ \\ 0 & ⋯ & & & A_{r} \end{matrix} ]

where each A_{j} is a permutation matrix. Finally, we have the diagonal matrices T_{n}(A) ⊂ GL_{n}(A), which are contained in H(λ) for any such λ. In fact H(λ) = U(λ)T_{n}(A)S(λ), where the group T_{n}(A)S(λ) normalizes the subgroup U(λ), making H(λ) a semidirect product of U(λ) and T_{n}(A)S(λ). We also have that S(λ) normalizes U(λ)T_{n}(A). In particular, H(α) has trivial associated permutation group S(α) = {I_{n}}, while H(β) has trivial unipotent group U(β) = {I_{n}} associated to it. Now if x ∈ BP, and σ(x) is the simplex associated to the chain λ_{0} < ⋯ < λ_{r} in the poset P, then it is easy to see that H(x) is the semidirect product of U(x) := U(λ_{r}) and T_{n}(A)S(x), with S(x) := S(λ_{0}), since as seen earlier, H(x) is the intersection of the H(λ_{i}). In other words, the “unipotent part” and the “permutation group” associated to H(x) are each the smallest possible ones from among the corresponding groups attached to the vertices of the carrier of x. Again we have that S(x) normalizes U(x)T_{n}(A). We return now to the situation in (6). We see that the groups H(x) = U(x)T_{n}(A)S(x) and H(x) ∩ H(β) = T_{n}(A)S(x) both have the same associated permutation group S(x), which normalizes U(x)T_{n}(A) as well as T_{n}(A). By comparing the spectral sequences

E_{p,q}^{2} = H_{p}(S(x), H_{q}(U(x)T_{n}(A), ℤ)) ⇒ H_{p+q}(H(x), ℤ),
E_{p,q}^{2} = H_{p}(S(x), H_{q}(T_{n}(A), ℤ)) ⇒ H_{p+q}(H(x) ∩ H(β), ℤ)

we see that it thus suffices to show that the inclusion

(7)  T_{n}(A) ⊂ U(x)T_{n}(A)

induces an isomorphism on integral homology. Now lemma 13 below finishes the proof. □

To state lemma 13 we use the following notation. Let I = {i_{0} = 0 < i_{1} < i_{2} < ⋯ < i_{r} = n} be a subsequence of {0, 1, …, n}, so that I determines a partial flag

0 ⊂ A^{i_{1}} ⊂ A^{i_{2}} ⊂ ⋯ ⊂ A^{i_{r}} = A^{n},

where A^{j} ⊂ A^{n} as the submodule generated by the first j basis vectors. Let U(I) be the “unipotent” subgroup of GL_{n}(A) stabilising this flag, and acting trivially on the associated graded A-module, and let G(I) ⊂ GL_{n}(A) be the subgroup generated by U(I) and T_{n}(A) = (A^{×})^{n}, the subgroup of diagonal matrices. Then T_{n}(A) normalises U(I), and G(I) is the semidirect product of U(I) and T_{n}(A).

Lemma 13. Let A be a Nesterenko-Suslin ring. For any I as above, the homomorphism G(I) → G(I)/U(I) ≅ T_{n}(A) induces an isomorphism on integral homology H_{∗}(G(I), ℤ) → H_{∗}(T_{n}(A), ℤ).

Proof. We work by induction on n, where there is nothing to prove when n = 1, since we must have G(I) = T_{1}(A) = A^{×} = GL_{1}(A). Next, if n > 1, and I = {0 < n}, then U(I) is the trivial group, so there is nothing to prove. Hence we may assume n > 1, r ≥ 2, and thus 0 < i_{1} < n. There is then a natural homomorphism G(I) → G(I′), where I′ = {0 < i_{2}−i_{1} < ⋯ < i_{r} −i_{1} = n−i_{1}}, and G(I′) ⊂ GL_{n−i_{1}}(A). Let n′ = n−i_{1}. The induced homomorphism T_{n}(A) → T_{n′}(A) is naturally split, with kernel T_{i_{1}}(A) ⊂ GL_{i_{1}}(A) ⊂ GL_{n}(A). Let

U_{1}(I) = ker (U(I) → U(I′)) = ker (G(I) → GL_{i_{1}}(A) × GL_{n′}(A)) .

Then U_{1}(I) is a normal subgroup of G(I), from the last description, and

G(I)/U_{1}(I) ≅ U(I′) · T_{n}(A) = T_{i_{1}}(A) × G(I′).

Now U_{1}(I) may be identified with M_{i_{1},n′}(A), the additive group of matrices of size i_{1} × n′ over A; this matrix group has a natural action of GL_{i_{1}}(A), and thus of the diagonal matrix group T_{i_{1}}(A), and the resulting semidirect product of T_{i_{1}}(A) with U_{1}(A) is a subgroup of G(I) (in fact, it is the kernel of G(I) → G(I′)). This matrix group M_{i_{1},n′}(A) is isomorphic, as T_{i_{1}}(A)-modules, to the direct sum

⊕_{i=1}^{i_{1}}A^{n′}(i),

where A^{n′}(i) is the free A-module of rank n′, with a T_{i_{1}}(A)-action given by the the “i-th diagonal entry” character T_{i_{1}}(A) → A^{×}. Thus, the semidirect product T_{i_{1}}(A)U_{1}(I) has a description as a direct product

T_{i_{1}}(A)U_{1}(i) ≅ H × H × ⋯ × H = H^{n_{i}}

with H = A^{n′} ·A^{×} equal to the naturally defined semidirect product of the free A module A^{n′} with A^{×}, where A^{×} operates by scalar multiplication. Proposition 1.10 and Remark 1.13 in the paper [13] of Nesterenko and Suslin implies immediately that H → H/A^{n′} ≅ A^{×} induces an isomorphism on integral homology. We now use the following facts.

(i) If H ⊂ K ⊂ G are groups, with H, K normal in G, and if K → K/H induces an isomorphism in integral homology, so does G → G/H; this follows at once from a comparison of the two spectral sequences

E_{r,s}^{2} = H_{r}(G/K, H_{s}(K, ℤ)) ⇒ H_{r+s}(G, ℤ),
E_{r,s}^{2} = H_{r}(G/K, H_{s}(K/H, ℤ)) ⇒ H_{r+s}(G/H, ℤ).

(ii) If H_{i} ⊂ G_{i} are normal subgroups, for i = 1, …, n, such that G_{i} → G_{i}/H_{i} induce isomorphisms on integral homology, then for G = ∏_{i=1}^{n} G_{i}, H = ∏_{i=1}^{n} H_{i}, the map G → G/H induces an isomorphism on integral homology. This follows from the Kunneth formula.

The fact (ii) implies that T_{i_{1}}(A)U_{1}(I) → T_{i_{1}}(A) induces an isomorphism on integral homology. Then (i) implies that G(I) → T_{i_{1}}(A) × G(I′) induces an isomorphism on integral homology. By induction, we have that G(I′) → G(I′)/U(I′) induces an isomorphism on integral homology. Hence T_{i_{1}}(A) × G(I′) → T_{i_{1}} × G(I′)/U(I′) also induces an isomorphism on integral homology. Thus, we have shown that the composition G(I) → G(I)/U(I) = T_{n}(A) induces an isomorphism on integral homology. □

4. 𝕊ℙ𝕃(A^{∞})^{+} and the groups L_{n}(A)

We first note that there is a small variation of Quillen’s plus construction. Let (X, x) be a pointed CW complex, (X_{0}, x) a contractible pointed subcomplex, G a group of homeomorphisms of X which acts transitively on the path components of X, and let H be a perfect subgroup of G, such that H stabilizes X_{0}. Then X//G is clearly path connected, and comes equipped with

(i) a natural map θ : X//G → BG = EG/G, induced by the projection X × EG → EG (ii) a map (X_{0} × EG)/H → X//G, induced by the H-stable contractible set X_{0} ⊂ X (iii) a homotopy equivalence BH → (X_{0} × EG)/H, such that the composition BH → X//G →^{θ} BG is homotopic to the natural map BH → BG (iv) a natural map (X, x) ↪ (X//G, x_{0}) determined by the base point of EG. Note that, in particular, there is a natural inclusion H ↪ π_{1}(X//G, x_{0}), which gives a section over H ⊂ G of the surjection θ_{∗} : π_{1}(X//G, x_{0}) → π_{1}(BG, ∗) = G.

Lemma 14. In the above situation, there is a pointed CW complex (Y, y), together with a map f : (X//G, x_{0}) → (Y, y) such that

(i) the natural composite map

H ↪ π_{1}(X//G, x_{0}) →^{f} π_{1}(Y, y)

is trivial (ii) if g : (X//G, x_{0}) → (Z, z) such that H is in the kernel of

π_{1}(X//G, x_{0}) → π_{1}(Z, z)

then g factors through f, uniquely upto a pointed homotopy (iii) f induces isomorphisms on integral homology; more generally, if L is any local system on Y, the map on homology with coefficients H_{∗}(X//G, f^{∗}L) → H_{∗}(Y, L) is an isomorphism (iv) h : (X, x) → (X′, x′) is a pointed map of such CW complexes with Gactions, such that h is G-equivariant, then there is a map (Y, y) → (Y′, y′), making (X, x) ↦ (Y, y) is functorial (on the category of pointed CW complexes with suitable G actions, and equivariant maps), and f yields a natural transformation of functors.

The pair (Y, y) is obtained by applying Quillen’s plus construction to (X//G, x_{0}) with respect to the perfect normal subgroup H̃ of π_{1}(X//G, x_{0}) which is generated by H. Part (ii) of the lemma is in fact the universal property of the plus construction. As is well-known, this may be done in a functorial way. We sometimes write (Y, y) = (X//G, x_{0})^{+} to denote the above relationship. In what follows, the pair (G, H) is invariably (GL_{n}(A), A_{n}) for 5 ≤ n ≤ ∞. Here A_{n} is the alternating group contained in N_{n}(A). The normal subgroups of GL_{n}(A) generated by A_{n} and E_{n}(A) coincide with each other. It follows that if we take X = X_{0} to be a point, the Y given by the above lemma is just the “original” BGL_{n}(A)^{+}. Recall that there is a natural action of GL(A) on the simplicial complex 𝕊ℙ𝕃(A^{∞}), and hence on its geometric realization |𝕊ℙ𝕃(A^{∞})|. We apply lemma 14 with G = GL(A), H = A_{∞} the infinite alternating group, X = |𝕊ℙ𝕃(A^{∞})|, and X_{0} = {x_{0}} is the vertex of X fixed by N(A) and obtain the pointed space

(Y (A), y) = (|𝕊ℙ𝕃(A^{∞})|//G), x_{0})^{+}.

Taking X′ to be a singleton in (iii) of the above lemma, we get a canonical map

φ : (Y (A), y) → (BGL(A)^{+}, ∗)

of pointed spaces. Let (𝕊ℙ𝕃(A^{∞})^{+}, z) denote the homotopy fibre of φ. We define

L_{n}(A) = π_{n}(𝕊ℙ𝕃(A^{∞})^{+}, z) ∀ n ≥ 0.

The homotopy sequence of the fibration 𝕊ℙ𝕃(A^{∞})^{+} → Y (A) → BGL(A)^{+} combined with the path-connectedness of Y (A) yields:

Corollary 15. There is an exact sequence

⋯ → K_{n+1}(A) → L_{n}(A) → π_{n}(Y (A), y) → K_{n}(A) ⋯ ⋯ → L_{1}(A) → π_{1}(Y (A), y) → K_{1}(A) → L_{0}(A) → 0

where L_{0}(A) is regarded as a pointed set.

Lemma 16. The natural map |𝕊ℙ𝕃(A^{∞})| → 𝕊ℙ𝕃(A^{∞})^{+} induces an isomorphism on integral homology.

Proof. We may identify the universal covering of BGL(A)^{+} with BE(A)^{+}, where BE(A)^{+} is the plus construction (see lemma 14) applied to BE(A) with respect to the infinite alternating group (or, what is the same thing, with respect to E(A) itself). Let φ̃ : Ỹ → BE(A)^{+} be the corresponding pullback map obtained from φ. We first note that 𝕊ℙ𝕃(A^{∞})^{+} is also naturally identified with the homotopy fiber of φ̃. There is then a homotopy pullback φ̂ : Ŷ → BE(A) of φ̃ with respect to BE(A) → BE(A)^{+}. Thus, our map 𝕊ℙ𝕃(A^{∞}) → 𝕊ℙ𝕃(A^{∞})^{+} may be viewed as the natural map on fibers associated to a map

(8)  𝕊ℙ𝕃(A^{∞})//E(A) → Ŷ

of Serre fibrations over BE(A). From a Leray-Serre spectral sequence argument, we see that since (from lemma 14) BE(A) → BE(A)^{+} induces a isomorphism on integral homology, so does Ŷ → Ỹ. Since also 𝕊ℙ𝕃(A^{∞})//E(A) → Ỹ is a homology isomorphism (from lemma 14 again), we see that 𝕊ℙ𝕃(A^{∞})//E(A) → Ŷ induces an isomorphism on integral homology. Now we use that the map (8) is a map between two total spaces of Serre fibrations over a common base, inducing a homology isomorphism on these total spaces. We also know that the monodromy representation of π_{1}(BE(A)) = E(A) on the homology of the fibers is trivial, in both cases: for Ŷ this is because it is a pullback from a Serre fibration over a simply connected base, while for 𝕊ℙ𝕃(A^{∞}), this is one of the key properties we have already established (see the finishing sentence of section 2). The proof is now complete modulo the remark below, which is a straightforward consequence of the Leray-Serre spectral sequence of a fibration. □

Remark. Let p : E → B and p′ : E′ → B be fibrations with fibers F and F′ respectively over the base-point b ∈ B. Let v : E → E′ be a map so that p′ ∘ v = p. Assume that B is path-connected. Then E → E′ is a homology isomorphism implies F → F′ is a homology isomorphism under the following additional assumption: M ≠ 0 implies H_{0}(π_{1}(B, b), M) ≠ 0 for every π_{1}(B, b)-subquotient M of H_{i}(F), H_{j}(F′) for all i, j.

5. The H-space structure

Recall that BGL(A)^{+} has an H-space structure in a standard way, obtained from the direct sum operation on free modules of finte rank; this was constructed in [6]. The aim of this section is to prove the proposition below.

Proposition 17. The space Y (A) has an H-space structure, such that Y (A) → BGL(A)^{+} is homotopic to an H-map, for the standard H-space structure on BGL(A)^{+}.

We first remark that if V is a free A-module of finite rank, then |𝕊ℙ𝕃(V)|//GL(V) is homeomorphic to the classifying space of the following category SPL(V): its objects are simplices in 𝕊ℙ𝕃(V) (thus, certain finite nonempty subsets of SP L(V)), and morphisms σ → τ are defined to be elements g ∈ GL(V) such that g(σ) ⊂ τ, that is, such that g(σ) is a face of the simplex τ of 𝕊ℙ𝕃(V). Let Aut(V) be the category with a single object ∗, with morphisms given by elements of GL(V), so that the classifying space BAut(V) is the standard model for BGL(V). There is a functor F_{V} : SPL(V) → Aut(V), mapping every obeject σ to ∗, and mappng an arrow σ → τ in SPL(V) to the corresponding element g ∈ GLV (). The fiber F_{V}^{−1}(∗) is the poset of simplices of 𝕊ℙ𝕃(V), whose classifying space is thus homeomorphic to |𝕊ℙ𝕃(V)|. It is fairly straightforward to verify that BSPL(V) is homeomorphic to |𝕊ℙ𝕃(V)|//GL(V) (where we have used the classifying space of the translation category of GL(V) as the model for the contractible space E(GL(V))). One way to think of this is to consider the category { \atop SPL}(V), whose objects are pairs (σ, h) with σ a simplex of 𝕊ℙ𝕃(V), and h ∈ GL(V), with a unique morphism (σ, h) → (τ, g) precisely when g^{−1}h(σ) ⊂ τ. It is clear that by considering the full subcategories of objects of the form (σ, g), where g ∈ GL(V) is a fixed element, each of which is naturally equivalent to the poset of simplices in 𝕊ℙ𝕃(V), that the classifying space of { \atop SPL}(V) is homeomorphic to |𝕊ℙ𝕃(V)|×E(GL(V))). Now it is a simple matter to see (e.g., use the criterion of Quillen, given in [19], lemma 6.1, page 89) that B{ \atop SPL}(V) → BSPL(V), given by (σ, h) ↦ h^{−1}(σ), is a covering space which is a principal GL(V)- bundle, where the deck transformations are given by the natural action of GL(V) on |𝕊ℙ𝕃(V)| × E(GL(V)). L(V) denotes the collection of A-submodules L ⊂ V so that L is free of rank one and V/L is a free module. Now we note that if V′, V′′ are free A-modules of finite rank, we note that there is a natural inclusion L(V′) ⊔ L(V′′) ↪ L(V′ ⊕ V′′). This in turn yields a natural map

φ_{V′,V′′} : SP L(V′) × SP L(V′′) → SP L(V′ ⊕ V′′),

given by φ_{V′,V′′}(s, t) = s ⊔ t. It follows easily from the definition of 𝕊ℙ𝕃 that the above map on vertices induces a simplicial map

Φ_{V′,V′′} : 𝕊ℙ𝕃(V′) × 𝕊ℙ𝕃(V′′) → 𝕊ℙ𝕃(V′ ⊕ V′′).

As explained in section 0, at the level of geometric realisations, this has two descriptions. The first description may be used to show that the counterpart of lemma 8(C) is valid for 𝕊ℙ𝕃, namely the homotopy class of the inclusion

|𝕊ℙ𝕃(V′)| × |𝕊ℙ𝕃(V′′)| → |𝕊ℙ𝕃(V′ ⊕ V′′)|

remains unaffected by composition with the action of g ∈ Elem(V′ ↪ V′ ⊕V′′) on |𝕊ℙ𝕃(V′ ⊕ V′′)|. The second description however is more useful in this context. Let us abbreviate notation and denote the (partially ordered) set of simplices of 𝕊ℙ𝕃(V) simply by S(V). The desired map S(V′) × S(V′′) → S(V′ ⊕ V′′) is given simply by (σ, τ) ↦ φ_{V′,V′′}(σ × τ). The resulting map B(S(V′) × S(V′′)) → BS(V′ ⊕ V′′) is the second description of

|𝕊ℙ𝕃(V′)| × |𝕊ℙ𝕃(V′′)| → |𝕊ℙ𝕃(V′ ⊕ V′′)|

for (a) BC′ × BC′′ ≅ B(C′ × C′′) and (b) BS(V) is simply the barycentric subdivision of |𝕊ℙ𝕃(V)|. This latter description also allows us to go a step further and define the functor SPL(V′) × SPL(V′′) → SPL(V′ ⊕ V′′), given on objects by (σ, τ) ↦ φ_{V′,V′′}(σ × τ) as before; on morphisms, it is given by the natural map GL (V′) × GL (V′′) → GL(V′ ⊕ V′′). Hence on classifying spaces, it induces a product

|𝕊ℙ𝕃(V′)|//GL(V′) × |𝕊ℙ𝕃(V′′)|//GL(V′′) → |𝕊ℙ𝕃(V′ ⊕ V′′)|//GL(V′ ⊕ V′′)|.

This is clearly compatible with the product

BGL(V′) × BGL(V′′) → BGL(V′ ⊕ V′′)

under the natural maps induced by the functors SPL → Aut for the three free modules. One verifies that SPL(A) = ∐_{V} SPL(V), with respect to the bifunctor

+ : SPL(A) × SPL(A) → SPL(A)

induced by direct sums on free modules, and the functors Φ_{V′,V′′}, form a symmetric monoidal category. An equivalent category, also denoted SPL(A) by abuse of notation, is that whose objects are pairs (V, σ), where V is a free A-module of finite rank, and σ ∈ 𝕊ℙ𝕃(V) a simplex, and where morphisms (V, σ) → (W, τ) are isomorphisms f : V → W of A-modules such that f(σ) is a face of τ. For the purposes of stabilization, we slightly modify the above to consider the related maps

φ_{m,n} : SP L(A^{m}) × SP L(A^{n}) → SP L(A^{∞})

given by mapping the basis vector e_{i} ∈ A^{m} in the first factor to the basis vector e_{2i−1} ∈ A^{∞}, for each 1 ≤ i ≤ m, and the basis vector e_{j} ∈ A^{n} in the second factor to the basis vector e_{2j} ∈ A^{∞}. A pair of splittings of A^{m}, A^{n} determine one for the free module spanned by the images of the two sets of basis vectors; now one extends this to a splitting of A^{∞} by adjoining the remaining basis vectors of A^{∞} (that is, adjoining those vectors not in the span of the earlier images). If our first two splittings are those given by the basis vectors, which correspond to the base points in |𝕊ℙ𝕃(A^{m})| and |𝕊ℙ𝕃(A^{n})|, the resulting point in SP L(A^{∞}) is again the base point of |𝕊ℙ𝕃(A)|. The corresponding functors

Φ_{m,n} : SPL(A^{m}) × SPL(A^{n}) → SPL(A^{∞})

are compatible with similar functors

Aut(A^{m}) × Aut(A^{n}) → Aut(A^{∞})

which, on classifying spaces, yield the diagram of product maps, preserving base points,

|𝕊ℙ𝕃(A^{m})//GL_{n}(A)| × |𝕊ℙ𝕃(A^{n})|//GL_{n}(A) → |SP L(A^{∞}|//GL(A)
↓  ↓
BGL_{m}(A) × BGL_{n}(A)  →  BGL(A)

where the bottom arrow is the one used in [6] to define the H-space structure on BGL(A)^{+}. As we increase m, n, the corresponding diagrams are compatible with respect to the obvious stabilization maps |𝕊ℙ𝕃(A^{m})| ↪ |𝕊ℙ𝕃(A^{m+1})|, |𝕊ℙ𝕃(A^{n}) ↪ |𝕊ℙ𝕃(A^{n+1})|. Hence we obtain on the direct limits a diagram

|𝕊ℙ𝕃(A^{∞})//GL(A)| × |𝕊ℙ𝕃(A^{∞})|//GL(A) → |SP L(A^{∞}|//GL(A)
↓  ↓
BGL(A) × BGL(A)  →  BGL(A)

From lemma 14, it follows that there is an induced diagram at the level of plus constructions

Y (A) × Y (A) → Y (A)
↓  ↓
BGL(A)^{+} × BGL(A)^{+} → BGL(A)^{+}

Here we have, as remarked above, taken the homotopy equivalent model |𝕊ℙ𝕃(A^{∞}|//GL(A) for the homotopy type earlier denoted Y (A). We abuse notation and use the same symbol to denote this model as well. It is shown in [6] that the botton arrow defines an H-space structure on BGL(A). We claim that, by analogous arguments, the top arrow also defines an H-space structure on Y (A). Granting this, the map Y (A) → BGL(A)^{+} is then an H-map between path connected H-spaces, and so the homotopy fiber Z(A) has the homotopy type of an H-group as well (and this was what we set out to prove here). To show that the product Y (A) × Y (A) → Y (A) defines an H-space structure, we need to show that left or right translation on Y (A) (with respect to this product) by the base point is homotopic to the identity. This is also the main point in [6], for the case of BGL(A)^{+}. We first show:

Lemma 18. An arbitrary inclusion j : {1, 2, …, n} ↪ ℕ determines an inclusion of A-modules A^{n} → A^{∞}, given on basis vectors by e_{i} ↦ e_{j(i)}, which induces a map

|𝕊ℙ𝕃(A^{n})|//GL_{n}(A) → Y (A)

which is homotopic (preserving the base point) to the map induced by standard inclusion i_{n} : A^{n} → A^{∞}.

Proof. We can find an automorphism g of A^{∞} contained in the infinite alternating group A_{∞}, such that g ∘ j = i_{n}, where g acts on A^{∞} by permuting the basis vectors (note that the induced self-map of |SP L(A^{∞})| × EGL(A) fixes the base point). Regarding g as an element of π_{1}(|𝕊ℙ𝕃(A^{∞})|//GL(A)), this implies that the maps (i_{n})_{∗} and j_{∗}, considered as elements of the set of pointed homotopy classes of maps

[|𝕊ℙ𝕃(A^{n})|//GL_{n}(A), |𝕊ℙ𝕃(A^{∞})|//GL(A)] ,

are related by g_{∗}(j_{∗}) = (i_{n})_{∗}, where g_{∗} denotes the action of the fundamental group of the target on the set of pointed homotopy classes of maps. However, g is in the kernel of the map on fundamental groups associated to the map

|𝕊ℙ𝕃(A^{∞})|//GL(A) → (|𝕊ℙ𝕃(A^{∞}|//GL(A))^{+}.

Hence the induced maps

|𝕊ℙ𝕃(A^{n})|//GL(A) → (|𝕊ℙ𝕃(A^{∞})|//GL(A))^{+}

determined by i_{n} and j are homotopic. □

Corollary 19. The map Y (A) → Y (A) defined by an arbitrary injective map α : ℕ ↪ ℕ is homotopic, preserving the base point, to the identity.

Proof. We first note that if for n ≥ 5, we let Y_{n}(A) = (|𝕊ℙ𝕃(A^{n})|//GL_{n}(A))^{+} be the result of applying lemma 14 to |𝕊ℙ𝕃(A^{n})|//GL_{n}(A) for the alternating group A_{n}, then there are natural maps Y_{n}(A) → Y (A), preserving base points, and inducing an isomorphism \underrightarrow{lim}_{n} π_{∗}(Y_{n}(A)) = π_{∗}(Y (A)).

We claim that if α_{n} : {1, 2 …, n} ↪ ℕ is the inclusion induced by restricting α, then the induced map (α_{n})_{∗} : Y_{n}(A) → Y (A) is homotopic, preserving the base points, to the natural map Y_{n}(A) → Y (A). This follows from lemma 18, combined with the defining universal property of the plus construction, given in lemma 14. This implies that the map α : Y (A) → Y (A) must then induce isomorphisms on homotopy groups, and hence is a homotopy equivalence, by Whitehead’s theorem. Thus, we have a map from the set of such injective maps α to the group of base-point preserving homotopy classes of self-maps of Y (A). This is in fact a homomorphism of monoids, where the operation on the injective self-maps of ℕ is given by composition of maps. Now we use a trick from [6]: any homomorphism of monoids from the monoid of injective self-maps of ℕ to a group is a trivial homomorphism, mapping all elements of the monoid to the identity. This is left to the reader to verify (or see [6]). □

We note that the above monoidal category SPL(A) can be used to give another, perhaps more insightful construction of the homotopy type Y (A), analogous to Quillen’s S^{−1}S construction for BGL(A)^{+}. We sketch the argument below. We first take SPL_{0}(A) to be the full subcategory of SPL(A) consisting of pairs (V, σ) where σ ∈ SP L(V), i.e.,σ is a 0-simplex in 𝕊ℙ𝕃(V). This full subcategory is in fact a monoidal subcategory, which is a groupoid (all arrow are isomorphisms). Also, SPL(A) is a symmetric monoidal category, in that the sum operation is commutative upto coherent natural isomorphisms.

Then, using Quillen’s results (see Chapter 7 in [19], particularly Theorem 7.2), one can see that SPL_{0}(A)^{−1}SPL(A) is a monoidal category whose classifying space is a connected H-space, which is naturally homology equivalent to |𝕊ℙ𝕃(A^{∞})|//GL(A). This then forces this classifying space to be homotopy equivalent to Y (A), such that the H-space operations are compatible upto homotopy. This is analogous to the identification made in Theorem 7.4 in [19] of S^{−1}S with K_{0}(R) × BGL(R)^{+} for a ring R, and appropriate S. (We do not get the factor K_{0} appearing in our situation since we work only with free modules).

6. Theorem 1 and the groups H_{n}(A^{×})

Proof of Theorem 1. In view of Proposition 17, we see that 𝕊ℙ𝕃(A^{∞})^{+}, the homotopy fiber of the H-map Y (A) → BGL(A)^{+}, is a H-space as well. It follows that L_{0}(A) = π_{0}(𝕊ℙ𝕃(A^{∞})^{+}) is a monoid. Furthermore, the arrow K_{1}(A) → L_{0}(A) in Corollary 15 is a monoid homomorphism. Thus this corollary produces an exact sequence of Abelian groups. 𝕊ℙ𝕃(A^{n}) has a canonical base point fixed under the action of N_{n}(A). As in sections 3 and 4, this gives a natural inclusion BN_{n}(A) → 𝕊ℙ𝕃_{n}(A)//GL_{n}(A). This is a homology isomorphism by lemma 12. Taking direct limits over all n ∈ ℕ, we see that BN(A) → 𝕊ℙ𝕃(A^{∞})//GL(A) is a homology isomorphism. Applying Quillen’s plus construction with respect to the normal subgroup of N(A) generated by the infinite alternating group, we obtain a space BN(A)^{+}. That BN(A)^{+} has a canonical H-space structure follows easily by the method of the previous section. Now the map BN(A)^{+} → Y (A) obtained by lemma 14(ii) is a homology isomorphism of simple path-connected CW complexes and is therefore a homotopy equivalence (see [4]) Theorem 4.37, page 371 and Theorem 4.5, page 346). This gives the isomorphism H_{n}(A^{×}) → L_{n}(A). The theorem now follows from corollary 15. We now turn to the description of the groups. Let X = B(A^{×}). Let X_{+} = X ⊔ {∗} be the pointed space with ∗ as its base-point. Let QX_{+} be the direct limit of Ω^{n}Σ^{n}X_{+} where Σ denotes reduced suspension.

Proposition 20. H_{n}(A^{×}) ≅ π_{n}(QX_{+}).

This statement was suggested to us by Proposition 3.6 of [17]. A complete proof of the proposition was shown us by Peter May. A condensed version of what we learnt from him is given below. Theorem 2.2, page 67 of [8] asserts that α_{∞} : C_{∞}X_{+} → QX_{+} is a group completion. This is proved in pages 50-59, [10]. The C_{∞} here is a particular case of the construction 2.4, page 13 of [9], given for any operad. For C_{∞}(Y), where Y is a pointed space, the easiest definition to work with is found in May’s review of [16]. It runs as follows. Let V = ∪_{n=0}^{∞}ℝ^{n}. Let C_{k}(Y) be the collection of ordered pairs (c, f) where c ⊂ V has cardinality k and f : c → Y is any function. We identify (c, f) with (c′, f′) if

(i) c′ ⊂ c, (ii) f|c′ = f′, and (iii) f(a) = ∗ for all a ∈ c, a ∉ c′. Here ∗ stands for the base-point of Y. Then C_{∞}Y is the space obtained from the disjoint union of the C_{k}(Y), k ≥ 0 by performing these identifications. This is a H-space. In our case, when Y = X ⊔ {∗}, it is clear that C_{∞}Y is the disjoint union of all the C_{k}(X) as a topological space. Thanks to “infinite codimension” one gets easily the homotopy equivalence of C_{k}(X) with X^{k}//S_{k} where S_{k} is the permutation group of {1, 2, .., k}. Now assume that X is any path-connected space equipped with a nondegenerate base-point x ∈ X. This x gives an inclusion of X^{n} ↪ X^{n+1}. Denote by X^{∞} the direct limit of the X^{n}. Thus X^{∞} is a pointed space equipped with the action of the infinite permutation group S_{∞} = ∪nS_{n}. Put Z = X^{∞}//S_{∞}. As in section 4, we obtain Z^{+} by the use of the infinite alternating group. As in section 5, we see that this is a H-space. It is an easy matter to check that the group completion of ⊔_{k}C_{k}X is homotopy equivalent to ℤ × Z^{+}. This shows that π_{n}(QX_{+}) ≅ π_{n}(Z^{+}) for all n > 0. The proposition is the particular case: X = B(A^{×}).

7. Polyhedral structure of the enriched Tits building

From what has been shown so far, we see that it is of interest to determine the stable rational homology of the flag complexes |𝔽𝕃(A^{n})| (or equivalently, of |𝕊ℙ𝕃(A^{n})|, or 𝔼𝕋(A^{n})). We will construct a spectral sequence that, in principle, gives an inductive procedure to do so. But first we introduce some notation and a definition for posets. Let P be a poset. For p ∈ P, we put e(p) = BL(p) where L(p) = {q ∈ P |q ≤ p} and ∂e(p) = BL′(p) where L′(p) = L(p) \ {p}. If ∂e(p) is homeomorphic to a sphere for every p ∈ P, we say the poset P is polyhedral. We denote by d(p) the dimension of e(p). When P is polyhedral, the space BP gets the structure of a CW complex with {e(p) : p ∈ P } as the closed cells. Its r-skeleton is BP_{r} where P_{r} = {p ∈ P : d(p) ≤ r}. The homology of BP is then computed by the associated complex of cellular chains Cell_{•}(BP), where

Cell_{r}(BP) = ⨁_{{p| dim e(p)=r}} H_{r}(e(p), ∂e(p), ℤ).

Lemma 21. E(A^{n}) is a polyhedral poset in the above sense. Its dimension is n − 1.

Proof. First consider the case when p ∈ SP L(A^{n}) is a maximal element in E(A^{n}). Then p is an unordered collection of n lines in A^{n} (here, as in §2, a “line” denotes a free A-submodule of rank 1 which is a direct summand, and the set of lines in A^{n} is denoted by L(A^{n})). Note that the subset p ⊂ L(A^{n}) of cardinality n determines a poset p̃, whose elements are chains q_{•} = {q_{1} ⊂ q_{2} ⊂ ⋯ ⊂ q_{r} = p} of nonempty subsets, where r_{•} ≤ q_{•} if each q_{i} is an r_{j} for some j, i.e., the “filtration” r_{•} “refines” q_{•}. We claim that, from the definition of the partial order on E(A^{n}), the poset p̃ is naturally isomorphic to the poset L(p). Indeed, an element q ∈ E(A^{n}) consists of a pair, consisting of a partial flag

0 = W_{0} ⊂ W_{1} ⊂ ⋯ ⊂ W_{r} = A^{n}

such that W_{i}/W_{i−1} is free, and a sequence t_{1}, …, t_{r} with t_{i} ∈ SP L(W_{i}/W_{i−1}). The condition that this element of E(A^{n}) lies in L(p) is that each W_{i} is a direct sum of a subset of the lines in p, say q_{i} ⊂ p, giving the chain of subsets q_{1} ⊂ q_{2}... ⊂ q_{r} = p; the splitting t_{i} is uniquely determined by the lines in q_{i} \ q_{i−1}. Let Δ(p) be the (n − 1)-simplex with p as its set of vertices. Now the chains of non-empty subsets of p correspond to simplices in the barycentric subdivison sdΔ(p), where the barycentre b corresponds to the chain {p}. Hence, from the definition of p̃, it is clear that it is isomorphic to the poset whose elements are simplices in the barycentric subdivision of Δ_{n} with b as a vertex, with partial order given by reverse inclusion. Hence Bp̃ is naturally identified with the subcomplex of the second barycentric subdivision sd^{2}Δ(p) which is the union of all simplices containing the barycentre. This explicit description implies in particular that BL′(p) is homeomorphic to S^{n−2} (with a specific triangulation). Before proceeding to the general case, we set up the relevant notation for orientations. For a set q of cardinality r, we put det(q) = ∧^{r}ℤ[q] where ℤ[q] denotes the free Abelian group with q as basis. we observe that there is a natural isomorphism:

H_{n−1}(e(p), ∂e(p)) ≅ H_{n−1}(Δ(p), ∂Δ(p)) = det(p).

Now let p ∈ E(A^{n}) be arbitrary, corresponding to a partial flag

0 = W_{0} ⊂ W_{1} ⊂ ⋯ ⊂ W_{r} = A^{n}.

and splittings t_{i} ∈ SP L(W_{i}/W_{i−1}). Then the natural map

∏_{i=1}^{r} E(W_{i}/W_{i−1}) → E(A^{n})

is an embedding of posets, where the product has the ordering given by (a_{1}, …, a_{r}) ≤ (b_{1}, …, b_{r}) precisely when a_{i} ≤ b_{i} in E(W_{i}/W_{i−1}) for each i. One sees that, by the definition of the partial order in E(A^{n}), the induced map

∏_{i=1}^{r} L(t_{i}) → L(p)

is bijective. Hence there is a homeomorphism of pairs

(BL(p), BL′(p)) = ∏_{i=1}^{r}(BL(t_{i}), BL′(t_{i})),

and so BL′(p) ≅ S^{n−r−1}, and BL(p) is an n − r-cell. □

We now proceed to construct the desired spectral sequence. We use the following notation: if p ∈ E(V), where V is a free A-module of finite rank, and W_{1} ⊂ V is the smallest non-zero submodule in the partial flag associated to p, define t(p) = rank W_{1} − 1. Clearly t : E(V) → ℤ is monotonic. Hence F_{r}E(V) = {p ∈ E(V)|t(p) ≤ r} is a sub-poset. Define

F_{r}𝔼𝕋(V) = BF_{r}E(V) = ∪{e(p)|t(p) ≤ r},

so that

F_{0}𝔼𝕋(V) ⊂ F_{1}𝔼𝕋(V) ⊂ ⋯ F_{n−1}𝔼𝕋(V) = 𝔼𝕋(V)

is an increasing finite filtration of the CW complex 𝔼𝕋(V) by subcomplexes. Hence there is an associated spectral sequence

E_{r,s}^{1} = H_{r+s}(F_{r}𝔼𝕋(V), F_{r−1}𝔼𝕋(V), ℤ) ⇒ H_{r+s}(𝔼𝕋(V), ℤ).

Our objective now is to recognise the above E^{1} terms. It is convenient to use the complexes of cellular chains for these sub CWcomplexes, which are thus sub-chain complexes of Cell_{•}(𝔼𝕋(V)). For simplicity of notation, we write Cell_{•}(V) for Cell_{•}(𝔼𝕋(V)). We have the description

E_{r,s}^{1} = H_{r+s}(gr_{r}^{F} Cell_{•}(V)).

We will now exhibit gr_{r}^{F} Cell_{•}(V) as a direct sum of complexes. Let W ⊂ V be a submodule such that W, V/W are both free, and rank W = r + 1. Let q ∈ SP L(W). The we have an inclusion of chain complexes

Cell_{•}(e(q)) ⊗ Cell_{•}(V/W) ⊂ Cell_{•}(W) ⊗ Cell_{•}(V/W) ⊂ F_{r}Cell_{•}(V).

It is clear that

image Cell_{•}(∂e(q)) ⊗ Cell_{•}(V/W) ⊂ F_{r−1}Cell_{•}(V),

so that we have an induced homomorphism of complexes

(Cell_{•}(e(q))/Cell_{•}(∂e(q))) ⊗ Cell_{•}(V/W) → gr_{r}^{F} Cell_{•}(V).

Composing with the natural chain homomorphism

H_{r}(e(q), ∂e(q), ℤ)[r] → (Cell_{•}(e(q))/Cell_{•}(∂e(q)))

for each q, we finally obtain a chain map

I : ⨁_{(W,q∈SP L(W))} H_{r}(e(q), ∂e(q), ℤ)[r] ⊗ Cell_{•}(V/W) → gr_{r}^{F} Cell_{•}(V).

Finally, it is fairly straightforward to verify that I is an isomorphism of complexes. We deduce that the E^{1} terms have the following description:

E_{r,s}^{1} = ⨁ ; \begin{matrix}rank W = r + 1 \\ q ∈ SP L(W) \end{matrix} det(q) ⊗ H_{s}(Cell_{•}(V/W), ℤ).

We define L_{r}(V) to be the collection of q ⊂ L(V) of cardinality (r + 1) for which (a) and (b) below hold:

(a) ⊕{L : L ∈ q} → V is injective. Its image will be denoted by W (q)

(b) V/W (q) is free of rank (n − r − 1). Summarising the above, we obtain:

Theorem 2. There is a spectral sequence with E^{1} terms

E_{r,s}^{1} = \begin{matrix}⨁ \\ q ∈ L_{r}(V) \end{matrix} det(q) ⊗ H_{s}(𝔼𝕋(V/W (q)), ℤ).

that converges to H_{r+s}(𝔼𝕋(V)). We note that E_{r,s}^{1} = 0 whenever (r + s) ≥ (n − 1) with one exception: (r, s) = (n − 1, 0). Here V ≅ A^{n}.

8. Compatible homotopy

It is true ^{3} that i : 𝔼𝕋(W) × 𝔼𝕋(V/W) ↪ 𝔼𝕋(V) has the property that g ∘ i is freely homotopic (not preserving base points) to i whenever g ∈ Elem(W ↪ V). There are several closed subsets of 𝔼𝕋(A^{n}) with the property that homotopy class of the inclusion morphism into 𝔼𝕋(A^{n}) remains unaffected by composition with the action of g ∈ E_{n}(A). To prove that the union of a finite collection of such closed subsets has the same property, one would require the homotopies provided for any two members of the collection to agree on their intersection. This is the problem we are concerned with in this section. We proceed to set up the notation for the problem. With q ∈ L_{r}(V) as in theorem 2, we shall define the subspaces U(q) ⊂ 𝔼𝕋(V) as follows. Let W (q) = ⊕{L|L ∈ q}. We regard q as an element of SP L(W (q)) and thus obtain the cell e(q) = BL(q) ⊂ 𝔼𝕋(W (q)). This gives the inclusion 𝔼𝕋′(q) = e(q) × 𝔼𝕋(V/W (q)) ⊂ 𝔼𝕋(W (q)) × 𝔼𝕋(V/W (q)) ⊂ 𝔼𝕋(V). We put U(q) = ∪{𝔼𝕋′(t)|∅ ≠ t ⊂ q}.

Main Question: Let i : U(q) ↪ 𝔼𝕋(V) denote the inclusion. Is it true that g ∘ i is homotopic to i for every g ∈ Elem(V, q)? We focus on the apparently weaker question below. Compatible Homotopy Question: Let M ⊂ V be a submodule complementary to W (q). Let g′ ∈ GL((W (q)) be elementary, i.e. g′ ∈ Elem(W (q), q). Define g ∈ GL(V) by gm = m for all m ∈ M and gw = g′w for all w ∈ W (q). Is it true that g ∘ i is homotopic to i? Assume that the second question has an affirmative answer in all cases. In particular, this holds when M = 0. Here V = W (q) and g = g′ is an arbitrary element of Elem(V, q). Let t be a non-empty subset of q. Then U(t) ⊂ U(q). We deduce that j : U(t) ↪ 𝔼𝕋(V) is homotopic to g ∘ j for all g ∈ Elem(V, q). But Elem(V, t) = Elem(V, q). Thus the Main Question has an affirmative answer for (q, i) replaced by (t, j), which of course, up to a change of notation, covers the general case.

Proposition 22. The compatible homotopy question has an affirmative answer if q ∈ L_{r}(V) and r ≤ 2.

The rest of this section is devoted to the proof of this proposition. To proceed, we will require to introduce the class C.

3this only requires the analogue of lemma 8(A) for the enriched Tits building. More general statements are contained in lemmas 23 and 24 .

Documenta Mathematica · Extra Volume Suslin (2010) 459–513

This is our set-up. Let X be a finite set, let V_{x} be a finitely generated free module for each x ∈ X and let V = ⊕{V_{x} : x ∈ X}. Let s = Π_{x∈X}s(x) ∈ Π_{x∈X}SP L(V_{x}). For each x ∈ X, we regard s(x) as a subset of L(V) and put F s = ∪{s(x)|x ∈ X}. Thus F s ∈ SP L(V). The collection of maps f : Π_{x∈X}𝔼𝕋(V_{x}) → 𝔼𝕋(V) with the property that f(Π_{x∈X}BL(s(x))) ⊂ BL(F s) for all s ∈ Π_{x∈X}SP L(V_{x}) is denoted by C. See lemma 10 and proposition 11 and its proof for relevant notation. Every maximal chain C of subsets of X (equivalently every total ordering of X) gives a member i(C) ∈ C. For instance, if X = {1, 2, ..., n} and C consists of the sets {1, 2, ..., k} for 1 ≤ k ≤ n, we put D_{k} = ⊕_{i=1}^{k}V_{i} and E = Π_{i=1}^{n}𝔼𝕋(D_{i}/D_{i−1}), denote by u : Π_{x∈X}𝔼𝕋(V_{x}) → E and v : E → 𝔼𝕋(V) the natural isomorphism and natural inclusion respectively, and put i(C) = v ∘ u.

Lemma 23. The above space C is contractible.

Proof. The aim is to realise C as the space of Λ-compatible maps for a suitable Λ and appeal to Proposition 1. Let Λ(x) = {L(S)|∅ ≠ S ⊂ SP L(V_{x}), ∅ ≠ L(S)}. Proposition 11 asserts that the subspaces {Bλ(x) : λ(x) ∈ Λ(x)} give an admissible cover of 𝔼𝕋(V_{x}). For λ = Π_{x∈X}λ(x) ∈ Λ = Π_{x∈X}Λ(x), we put I(λ) = Π_{x∈X}Bλ(x) and deduce that {I(λ) : λ ∈ Λ} gives an admissible cover of Π_{x∈X}𝔼𝕋(V_{x}). We define next a closed J(λ) ⊂ 𝔼𝕋(V) for every λ ∈ Λ with the properties: (A): J(λ) ⊂ J(μ) whenever λ ≤ μ and (B): J(λ) is contractible for every λ ∈ Λ. For each λ(x) ∈ Λ(x), let Uλ(x) be its set of upper bounds in SP L(V_{x}). It follows that that LUλ(x)) = λ(x). As observed before, we have F : Π_{x∈X}SP L(V_{x}) → SP L(V). Thus given λ = Π_{x}λ(x) ∈ Λ, we set H(λ) = F (Π_{x∈X}Uλ(x)) ⊂ SP L(V) and then put J(λ) = BLH(λ) ⊂ 𝔼𝕋(V). The space of Λ-compatible maps Π_{x∈X}𝔼𝕋(V_{x}) → 𝔼𝕋(V) is seen to coincide with C. That the J(λ) satisfy property (A) stated above is straightforward. The contractibility of J(λ) for all λ ∈ Λ is guaranteed by proposition 11 once it is checked that these sets are nonempty. But we have already noted that C is nonempty. Let f ∈ C. Now I(λ) ≠ ∅ and f(I(λ)) ⊂ J(λ) implies J(λ) ≠ ∅. Thus the J(λ) are contractible, and as said earlier, an application of Proposition 1 completes the proof of the lemma.

□

We remark that the class C of maps Π_{i=1}^{n}𝔼𝕋(W_{i}) → 𝔼𝕋(⊕_{i=1}^{n}W_{i}) has been defined in general. We will continue to employ the notation: V = ⊕{V_{x} : x ∈ X} all through this section. Let P be a partition of X. Each p ∈ P is a subset of X and we put

V_{p} = ⊕{V_{x}|x ∈ p} and 𝔼𝕋(P) = Π{𝔼𝕋(V_{p})|p ∈ P }.

When Q ≤ P is a partition of X (i.e. Q is finer than P), we shall define the contractible collection C(Q, P) of maps f : 𝔼𝕋(Q) → 𝔼𝕋(P) by demanding (a) that f is the product of maps f(p)

f(p) : Π{𝔼𝕋(V_{q}) : q ⊂ p and q ∈ Q} → 𝔼𝕋(V_{p})

and also (b) each f(p) is in the class C. For this one should note that V_{p} = ⊕{V_{q} : q ∈ Q and q ⊂ p}. We observe next that there is a distinguished collection D(Q, P) ⊂ C(Q, P). To see this, recall that we had the embedding i(C) for every maximal chain C of subsets of X (alternatively, for every total ordering of X). Given Q ≤ P, denote the set of total orderings of {q ∈ Q : q ⊂ p} by T (p), for every p ∈ P. The earlier C ↦ i(C) now yields, after taking a product over p ∈ P, i : Π{T (p) : p ∈ P } → C(Q, P), and we denote by D(Q, P) ⊂ C(Q, P) the image of i. The lemma below is immediate from the definitions.

Lemma 24. Given partitions R ≤ Q ≤ P of X, if f is in C(R, Q) (resp. in D(R, Q))and g is in C(Q, P) (resp. in D(Q, P)), then it follows that g ∘ f is in C(R, P) (resp. D(R, P)).

We will soon have to focus on the fixed points of certain unipotent g ∈ GL(V) on 𝔼𝕋(V). For instance, if x, y ∈ X and x ≠ y, we may consider g = id_{V} + h where h(V) ⊂ V_{y} and h(V_{z}) = 0 for all z ∈ X, z ≠ x. Let C be a chain of subsets of X, so that X ∈ C. This chain C gives rise to a partition P (C) of X and also i(C) ∈ D(P (C), {X}) in a natural manner. Let C_{x} = ∩{S ∈ C : x ∈ S}. Then C_{x} ∈ C because C is a chain. Define C_{y} in a similar manner. We say the chain C is (x, y)-compatible if C_{y} ⊂ C_{x} and C_{x} ≠ C_{y} . This condition on C ensures that the embedding i(C) : 𝔼𝕋(P (C)) → 𝔼𝕋(V) has its image within the fixed points of the above g ∈ GL(V). Now let Q be a partition of X so that q ∈ Q, x ∈ q implies y ∉ q. We shall define next the class of (x, y)-compatible C maps 𝔼𝕋(Q) → 𝔼𝕋(V) in the following manner. Let Λ be the set of chains C of subsets of X so that X ∈ C and Q ≤ P (C) (i.e. Q is finer than the partition P (C)). For each C ∈ Λ, let Z(C) be the collection of i(C) ∘ f where f ∈ C(Q, P (C)). Finally, let Z = ∪{Z(C) : C ∈ Λ}. This set Z is defined to be the collection of (x, y)- compatible maps of class C from 𝔼𝕋(Q) to 𝔼𝕋(V). Every z ∈ Z is a map z : 𝔼𝕋(Q) → 𝔼𝕋(V) whose image is contained in the fixed points of the above g on 𝔼𝕋(V). Furthermore, in view of lemma 24, this collection of maps is contained in C(Q, {X}).

Lemma 25. Let Q be a partition of X that separates x and y. Then the collection of (x, y)-compatible class C maps 𝔼𝕋(Q) → 𝔼𝕋(V) is contractible.

Proof. In view of the fact that each C(Q, P) is contractible, by cor 3, it follows that the space of (x, y)-compatible chains is homotopy equivalent to BΛ, where Λ is the poset of chains C in the previous paragraph. It remains to show that BΛ is contractible. We first consider the case where Q is the set of all singletons of X. Let S be the collection of subsets S ⊂ X so that y ∈ S and x ∉ S. For S ∈ S, let F(S) be the collection of chains C of subsets of X so that S ∈ C and X ∈ C. We see that Λ is precisely the union of F(S) taken over all S ∈ S. Let D be a finite subset of S. We see that the intersection of the BF(S), taken over S ∈ D, is nonempty if and only if D is a chain. Furthermore, when D is a chain, this intersection is clearly a cone, and therefore contractible. By cor 3, we see that BΛ has the same homotopy type as the classifying space of the poset of chains of S. But this is simply the barycentric subdivision of BS. But the latter is a cone as well, with {y} as vertex. This completes the proof that BΛ is contractible, when Q is the finest possible partition of X. We now come to the general case, when Q is an arbitrary partition of X that separates x, y. So we have x′, y′ ∈ Q with x ∈ x′, y ∈ y′ and x′ ≠ y′. The set Λ is identified with the collection of chains C′ of subsets of Q so that

(a) Q ∈ C′, and (b) there is some L ∈ C′ so that x′ ∉ L and y′ ∈ L. Thus the general case follows from the case considered first: one replaces (X, x, y) by (Q, x′, y′). □

In a similar manner, we may define, for every ordered r-tuple (x_{1}, x_{2}, ..., x_{r}) of distinct elements of X, the set of (x_{1}, x_{2}, ..., x_{r})-compatible chains C–we demand that for each 0 < i < r, there is a member S of the chain so that x_{i} ∉ S and x_{i+1} ∈ S. Let Q be a partition of X that separates x_{1}, x_{2}, ..., x_{r}. Then the poset of chains C , compatible with respect to this ordered r-tuple, and for which Q ≤ i(P), is also contractible. One may see this through an inductive version of the proof of the above lemma. A corollary is that the collection of (x_{1}, ..., x_{r})-compatible class C maps 𝔼𝕋(Q) → 𝔼𝕋(V) is also contractible. We skip the proof. This result is employed in the proof of Proposition 22 for r = 2 (which has already been verified in the above lemma), and for r = 3, with #(Q) ≤ 4. Here it is a simple verification that the poset of chains that arises as above has its classifying space homeomorphic to a point or a closed interval.

We are now ready to address the proposition. For this purpose, we assume that there is c ∈ X so that V_{x} ≅ A for all x ∈ X \ {c}. To obtain consistency with the notation of the proposition, we set q = X \ {c}. The closed subset U(q) ⊂ 𝔼𝕋(V) in the proposition is the union of 𝔼𝕋′(t) taken over all ∅ ≠ t ⊂ q. For such t, we have W (t) = V (t) = ⊕{V_{x} : x ∈ t}. Recall that 𝔼𝕋′(t) is the product of the cell e(t) ⊂ 𝔼𝕋(V (t)) with 𝔼𝕋(V/V (t)). To proceed, it will be necessary to give a contractible class of maps D → 𝔼𝕋(V) for certain closed subsets D ⊂ U(q). The closed subsets D ⊂ U(q) we consider have the following shape. For each ∅ ≠ t ⊂ q, we first select a closed subset D(t) ⊂ e(t) and then take D to be the union of the D(t) × 𝔼𝕋(V/V (t)), taken over all such t. This D remains unaffected if D(t) is replaced by its saturation sD(t). Here sD(t) is the collection of a ∈ e(t) for which {a} × 𝔼𝕋(V/V (t)) is contained in D. When ∅ ≠ t ⊂ q, we denote by p(t) the partition of X consisting of all the singletons contained in t, and in addition, the complement X \ t. Then there is a canonical identification j(t) : 𝔼𝕋(p(t)) → 𝔼𝕋(V/V (t)).

A map f : D → 𝔼𝕋(V) is said to be in class C if for every ∅ ≠ t ⊂ q and for every a ∈ sD(t), the map 𝔼𝕋(p(t)) → 𝔼𝕋(V) given by b ↦ f(a, j(t)b) belongs to C(p(t), {X}). By lemma 24, we see that it suffices to impose this condition on all a ∈ D(t), rather than all a ∈ sD(t). We observe that for every a ∈ e(t), the map 𝔼𝕋(p(t)) → 𝔼𝕋(V) given by b ↦ (a, j(t)b) belongs to C(p(t), {X}). As a consequence, we see that the inclusion D ↪ 𝔼𝕋(V) is of class C. When concerned with (x, y)-compatible maps, we will assume that D(t) = ∅ whenever t and {x, y} are disjoint. Under this assumption, a map f : D → 𝔼𝕋(V) is said to be (x, y)-compatible of class C if 𝔼𝕋(p(t)) → 𝔼𝕋(V) given by b ↦ f(a, j(t)b) is a (x, y)-compatible map in C(p(t), {X}) for all pairs (a, t) such that a ∈ sD(t). In a similar manner, we define (x, y, z)-compatible maps of class C as well. For this, it is necessary to assume that D(t) is empty whenever the partition p(t) does not separate (x, y, z), equivalently if {x, y, z} \ t has at least two elements.

Lemma 26. Assume furthermore that D(t) is a simplicial subcomplex of e(t). Then the space of maps D → 𝔼𝕋(V) in class C is contractible. The same is true of the space of such maps that are (x, y)-compatible, or (x, y, z)-compatible.

Proof. We denote by d the cardinality of {t : D(t) ≠ ∅}. We proceed by induction on d, beginning with d = 0 where the space of maps is just one point. We choose t_{0} of maximum cardinality so that D(t_{0}) ≠ ∅. Let D′ be the union of D(t) × 𝔼𝕋(V/V (t)) taken over all t ≠ t_{0}. Let C(D′) and C(D) denote the space of class C maps D′ → 𝔼𝕋(V) and D → 𝔼𝕋(V) respectively. By the induction hypothesis, C(D′) is contractible. We observe that the intersection of D′ and e(t_{0}) × 𝔼𝕋(V/V (t_{0})) has the form G × 𝔼𝕋(V/V (t_{0}) where G ⊂ e(t_{0}) is a subcomplex. Furthermore, G ∪ D(t_{0}) is the saturated set sD(t_{0}) described earlier. For a closed subset H ⊂ e(t_{0}), denote the space of C-maps H ×𝔼𝕋(V/V (t_{0})) → 𝔼𝕋(V) by A(H). Note that A(H) = Maps(H, C(p(t_{0}), {X})). By lemma 23, the space C(p(t_{0}), {X}) is itself contractible. It follows that A(H) is contractible. In particular, both A(G) and A(sD(t_{0})) are contractible. The natural map A(sD(t_{0})) → A(G) is a fibration, because the inclusion G ↪ sD(t_{0}) is a cofibration. The fibers of A(sD(t_{0})) → A(G) are thus contractible. It follows that

A(sD(t_{0})) ×_{A(G)} C(D′) → C(D′)

which is simply C(D) → C(D′), enjoys the same properties: it is also a fibration with contractible fibers. Because C(D′) is contractible, we deduce that C(D) is itself contractible. This completes the proof of the first assertion of the lemma. The remaining assertions follow in exactly the same manner by appealing to lemma 25. □

Proof of Proposition 22. Choose x ≠ y with x, y ∈ q. Let g = id_{V} + α where α(V) ⊂ V_{y} and α(V_{k}) = 0 for all k ≠ x ∈ X. To prove the proposition, it suffices to show that g ∘ i is homotopic to i where i : U(q) → 𝔼𝕋(V) is the given inclusion. This notation x, y, α, g will remain fixed throughout the proof.

Case 1. Here q = {x, y}. Now x, y are separated by the partitions p(t) for every non-empty t ⊂ q. By the second assertion of the above lemma, there exists f : U(q) → 𝔼𝕋(V) of class C and (x, y)-compatible. The given inclusion i : U(q) → 𝔼𝕋(V) is also of class C. By the first assertion of the same lemma, f is homotopic to i. Now the image of f is contained in the fixed-points of g and so we get g ∘ f = f. It follows that g ∘ i is homotopic to i. This completes the proof of the proposition when 1 = r = #(q) − 1.

Case 2. Here q = {x, y, z} with x, y, z all distinct. We take Y_{1} to be the union of e(t) × 𝔼𝕋(V/V (t)) taken over all t ⊂ q, t ≠ {z}, t ≠ ∅. We put Y_{2} = 𝔼𝕋(V_{z}) × 𝔼𝕋(V/V_{z}) and Y_{3} = Y_{1} ∩ Y_{2}. We note that U(q) = Y_{1} ∪ Y_{2}. The given inclusion i : U(q) → 𝔼𝕋(V) restricts to i_{k} : Y_{k} → 𝔼𝕋(V) for k = 1, 2, 3. The required homotopy is a path γ : I → Maps(U(q), 𝔼𝕋(V)) so that γ(0) = i and γ(1) = g∘i. Equivalently we require paths γ_{k} in Maps(Y_{k}, 𝔼𝕋(V)) for k = 1, 2 so that

(a) γ_{k}(0) = i_{k} and γ_{k}(1) = g ∘ i_{k} for k = 1, 2 and

(b) both γ_{1} and γ_{2} restrict to the same path in Maps(Y_{3}, 𝔼𝕋(V)). In view of the fact that Y_{3} ↪ Y_{1} is a cofibration, the weaker conditions (a′) and (b′) on fundamental groupoids suffice for the existence of such a γ: (a′): γ_{k} ∈ π_{1}(Maps(Y_{k}, 𝔼𝕋(V)); i_{k}, g ∘ i_{k}) for k = 1, 2 (b′): both γ_{1} and γ_{2} restrict to the same element of π_{1}((Maps(Y_{3}, 𝔼𝕋(V)); i_{3}, g∘ i_{3}) We have the spaces: Z_{k} = Maps(Y_{k}, 𝔼𝕋(V)) for k = 1, 2, 3.These spaces come equipped with the data below: (A)The GL(V)-action on 𝔼𝕋(V) induces a GL(V)-action on Z_{k} (B)The maps of class C give contractible subspaces C_{k} ⊂ Z_{k} for k = 1, 2, 3. (C) We have i_{k} ∈ C_{k} for k = 1, 2, 3. (D) The natural maps Z_{k} → Z_{3} for k = 1, 2 are GL(V)-equivariant, they take i_{k} to i_{3} and restrict to maps C_{k} → C_{3}. Note that the GL(V)-action on Z_{k} turns the disjoint union: G_{k} = ⊔{π_{1}(Z_{k}; i_{k}, hi_{k})|h ∈ GL(V)} into a group: given ordered pairs (h_{j}, v_{j}) ∈ G_{k}, i.e. h_{j} ∈ GL(V) and v_{j} ∈ π_{1}(Z_{k}; i_{k}, h_{j}i_{k}) for j = 1, 2, we get h_{1}v_{2} ∈ π_{1}(Z_{k}; h_{1}i_{k}, h_{1}h_{2}i_{k}) and obtain thereby v = (h_{1}v_{2}).v_{1} ∈ π_{1}(Z_{k}; i_{k}, h_{1}h_{2}i_{k}) and this produces the required binary operation (h_{1}, v_{1}) ∗ (h_{2}, v_{2}) = (h_{1}h_{2}, v). The projection G_{k} → GL(V) is a group homomorphism. The following elementary remark will be used in an essential manner when checking condition (b′). The data (H, F, Δ) where

(i) H ⊂ GL(V) is a subgroup, (ii)F ∈ Z_{k} is a fixed-point of H, and (iii)Δ ∈ π_{1}(Z_{k}; F, i_{k}) produces the lift H → G_{k} of the inclusion H ↪ GL(V) by h ↦ (h, (hΔ).Δ^{−1}).

Finally we observe that there are natural homomorphisms G_{k} → G_{3} induced by Z_{k} → Z_{3} for k = 1, 2. Construction of γ_{1}. The partitions p(t) for t ≠ {z} separate x, y. By lemma 26, we have a (x, y)- compatible class C-map f : Y_{1} → 𝔼𝕋(V). Both i_{1} and f belong to C_{1} and thus we get δ ∈ π_{1}(C_{1}; f, i_{1}). Now f is fixed by our g ∈ GL(V), so we also get gδ ∈ π_{1}(gC_{1}; f, gi_{1}). The path (gδ).δ^{−1} is the desired γ_{1} ∈ π_{1}(Z_{1}; i_{1}, gi_{1}). Construction of γ_{2}. Recall that g = id_{V} + α. We choose m : V_{x} → V_{z} and n : V_{z} → V_{y} so that nm(a) = α(a) for all a ∈ V_{x}. We extend m, n by zero to nilpotent endomorphisms of V, once again denoted by m, n : V → V and put u = id_{V} + n, v = id_{V} + m and note that g = uvu^{−1}v^{−1}. Note that the partition p({z}) separates both the pairs (x, z) and (z, y). We thus obtain f′, f′′ ∈ C_{2} so that f′ is (x, z)-compatible and f′′ is (y, z)- compatible and also δ′ ∈ π_{1}(C_{2}; f′, i_{2}) and δ′′ ∈ π_{1}(C_{2}; f′′, i_{2}). Noting that f′, f′′ are fixed by v, u respectively, we obtain ϵ′ = (vδ′).δ^{′−1} ∈ π_{1}(Z_{2}; i_{2}, vi_{2}) and ϵ′′ = (uδ′′).δ^{′′−1} ∈ π_{1}(Z_{2}; i_{2}, ui_{2}). Thus v′ = (v, ϵ′) and u′ = (u, ϵ′′) both belong to G_{2}. We obtain γ_{2} by u′ ∗ v′ ∗ u^{′−1} ∗ v^{′−1} = (g, γ_{2}) ∈ G_{2} Checking the validity of (b′). Let γ_{13}, γ_{23} ∈ π_{1}(Z_{3}; i_{3}, gi_{3}) be the images of γ_{1} and γ_{2} respectively. We have to show that γ_{13} = γ_{23}. Consider the spaces H, H′, H′′ consisting of ordered pairs (f_{3}, δ_{3}), (f′_{3}, δ′_{3}), (f′′_{3}, δ′′_{3}) respectively, where f_{3}, f′_{3}, f′′_{3} are all in C_{3}, f_{3} is (x, y)-compatible, f′_{3} is (x, z)-compatible, and f′′_{3} is (y, z)-compatible, and δ_{3}, δ′_{3}, δ′′_{3} are all paths in C_{3} that originate at f_{3}, f′_{3}, f′′_{3} respectively, and they all terminate at i_{3}. By lemma 26, we see that the spaces H, H′, H′′ are all contractible. For t = {x, z}, {y, z}, {x, y, z}, the partition p(t) separates (x, y, z). Note that Y_{3} is contained in the union of these three 𝔼𝕋′(t). By lemma 26, there is a (x, y, z)-compatible F ∈ C_{3}. Let Δ be a path in C_{3} that originates at F and terminates at i_{3}. We see that (F, Δ) ∈ H ∩ H′ ∩ H′′. Note that H → π_{1}(Z_{3}; i_{3}, gi_{3}) given by (f_{3}, δ_{3}) ↦ (gδ_{3}).δ_{3}^{−1} is a constant map because H is contractible. The (f, δ) employed in the construction of γ_{1} restricts to an element of H. Also, (F, Δ) belongs to H. It follows that γ_{13} = (gΔ).Δ^{−1}. In a similar manner, we deduce that if ϵ′_{3}, ϵ′′_{3} denote the images of ϵ′, ϵ′′ in the fundamental groupoid of Z_{3}, then ϵ′_{3} = (vΔ).Δ^{−1} ∈ π_{1}(Z_{3}; i_{3}, vi_{3}) and ϵ′′_{3} = (uΔ).Δ^{−1} ∈ π_{1}(Z_{3}; i_{3}, ui_{3}) Thus G_{2} → G_{3} takes v′, u′ ∈ G_{2} to (v, (vΔ).Δ^{−1}), (u, (uΔ).Δ^{−1}) ∈ G_{3} respectively. It follows that their commutator [u′, v′] maps to (g, γ_{23}) ∈ G_{3} under this homomorphism. We apply the remark preceeding the construction of γ_{1} to the subgroup H generated by u, v and F and Δ as above. We conclude that γ_{23} equals (gΔ).Δ^{−1}.

That the latter equals γ_{13} has already been shown. Thus γ_{13} = γ_{23} and this completes the proof of the Proposition.

9. Low dimensional stabilisation of homology

This section contains applications of corollary 9, proposition 22 and Theorem 2 to obtain some mild information on the homology groups of 𝔼𝕋(V). The notation L(V), L_{r}(V), W (q), det(q) introduced to state Theorem 2 will be freely used throughout. The spectral sequence in theorem 2 with coefficients in an Abelian group M will be denoted by SS(V; M). When V = A^{n}, this is further abbreviated to SS(n; M), or even to SS(n) when it is clear from the context what M is. The concept of a commutative ring with many units is due to Van der Kallen. An exposition of the definition and consequences of this term is given in [12]. We note that this class of rings includes semilocal rings with infinite residue fields. The three consequences of this hypothesis on A are listed as I,II,III below. These statements are followed by some elementary deductions. Throughout this section, we will assume that our ring A has this property.

I: SL_{n}(A) = E_{n}(A). This permits a better formulation of Lemma 8 in many instances. Ia: Let 0 → W → P → Q → 0 be an exact sequence of free A-modules with of ranks a, a + b, b. Let d = g.c.d.(a, b). The group H of automorphisms of this exact sequence that induce homotheties on both W and Q may be regarded as a subgroup of GL(P). This group acts trivially on the image of the embedding i : 𝔼𝕋(W) × 𝔼𝕋(Q) → 𝔼𝕋(P). Furthermore {det(g)|g ∈ H} equals (A^{×})^{d}. Thus, if a, b are relatively prime, by lemma 8, we see that g ∘ i is freely homotopic to i for all g ∈ GL(V). We shall take rank(W) = 1 in what follows. Here 𝔼𝕋(W)×𝔼𝕋(Q) is canonically identified with 𝔼𝕋(Q). The induced 𝔼𝕋(Q) → 𝔼𝕋(P) gives rise on homology to an arrow H_{m}(𝔼𝕋(Q)) → H_{m}(𝔼𝕋(P)) which has a factoring:

H_{m}(𝔼𝕋(Q)) ↠ H_{0}(P GL(Q), H_{m}(𝔼𝕋(Q))) → → H^{0}(P GL(P), H_{m}(𝔼𝕋(P)) ↪ H_{m}(𝔼𝕋(P)).

The kernel of H_{m}(𝔼𝕋(Q)) → H_{m}(𝔼𝕋(P)) does not depend on the choice of the exact sequence. Denoting this kernel by KH_{m}(Q) ⊂ H_{m}(𝔼𝕋(Q)) therefore gives rise to unambiguous notation. We abbreviate H_{m}(𝔼𝕋(A^{n})), KH_{m}(A^{n}) to H_{m}(n), KH_{m}(n) respectively. Ib: In the spectral sequence SS(n), we have:

(1) H_{0}(P GL_{n−1}(A), H_{m}(n − 1)) ≅ H_{0}(P GL_{n}(A), E_{0,m}^{1}).

(2) E_{0,m}^{∞} is the image of H_{m}(n − 1) → H_{m}(n).
(3) E_{0,m}^{1} → E_{0,m}^{∞} factors as follows:
E_{0,m}^{1} ↠ E_{0,m}^{2} ↠ H_{0}(P GL_{n}(A), E_{0,m}^{1}) ↠ E_{0,m}^{∞}.
(4) If H_{0}(P GL_{n}(A), E_{p,m+1−p}^{p}) = 0 for all p ≥ 2, then

the given arrow H_{0}(P GL_{n}(A), E_{0,m}^{1}) → E_{0,m}^{∞} is an isomorphism.

(5) Assume that H_{m}(n − 2) → H_{m}(n − 1) is surjective. Then the arrow E_{0,m}^{2} → H_{0}(P GL_{n}(A), E_{0,m}^{1}) in (3) above is an isomorphism.

The factoring in part (3) above is a consequence of the factoring of H_{m}(𝔼𝕋(Q)) → H_{m}(𝔼𝕋(P)) in part I(a). For part (4), one notes that the composite E_{2,m−1}^{2} → E_{0,m}^{2} → H_{0}(P GL_{n}(A), E_{0,m}^{1}) vanishes because H_{0}(P GL_{n}(A), E_{2,m−1}^{2}) itself vanishes. Thus we obtain a factoring: E_{0,m}^{2} → E_{0,m}^{3} → H_{0}(P GL_{n}(A), E_{0,m}^{1}). Proceeding inductively, we obtain the factoring: E_{0,m}^{2} → E_{0,m}^{∞} → H_{0}(P GL_{n}(A), E_{0,m}^{1}). In view of (3), we see that part (4) follows. For part (5), it suffices to note that for every L_{0}, L_{1} ∈ L(A^{n}) with n > 1, there is some L_{2} ∈ L(A^{n}) with the property that both {L_{0}, L_{2}} and {L_{1}, L_{2}} belong to L_{1}(A^{n}). This fact is contained in consequence III of many units.

II: A is a Nesterenko-Suslin ring. Let r > 0, p ≥ 0. Put N = (p + 1)! and n = r + p + 1. Let F_{r} be the category with free rank A-modules of rank r as objects; the morphisms in F_{r} are A-module isomorphisms. Let F be a functor from F_{r} to the category of ℤ[\frac{1}{N}] -modules. Assume that F (a.id_{D}) = id_{F D} for every a ∈ A^{×} and for every object D of F_{r}. In other words, the natural action of GL_{r}(A) on F (A^{r}) factors through the action of P GL_{r}(A). For a free A-module V of rank n, define Ind′F (V) by

Ind′F (V) = ⊕{det(q) ⊗ F (V/W (q)) : q ∈ L_{p}(V)}.

An alternative description of Ind′F (V) is as follows. Fix some q ∈ L_{p}(V). Let G(q) be the stabiliser of q in GL(V). Then det(q) and F (V/W (q)) are G(q)- modules in a natural manner. We have a natural isomorphism of ℤ[GL(V)]- modules:

Ind′F (V) ≅ ℤ[GL(V)] ⊗_{ℤ[G(q)]} [det(q) ⊗_{ℤ} F (V/W (q))].

IIa: If H_{i}(P GL_{r}(A), F (A^{r})) = 0 for all i < m, then H_{i}(P GL_{n}(A), Ind′F (A^{n})) = 0 for all i < p + m. Furthermore,

H_{p+m}(P GL_{n}(A), Ind′F (A^{n})) ≅ H_{m}(P GL_{r}(A), F (A^{r})) ⊗ Sym^{p}(A^{×})) By Shapiro’s lemma, the result of [13] cited earlier, and the fact that the group homology H_{i}(M, C) is isomorphic to C ⊗ Λ^{i}(M) for all commutative groups M and ℤ[1/i!]-modules C given trivial M-action, IIa reduces to the statement: Let Σ(q) denote the group of permutations of a set q of (p + 1) elements. Let M be an Abelian group on which (p + 1)! acts invertibly. Then H_{0}(Σ(q), det(q)⊗Λ^{i}(M^{q})) vanishes when i < p and is isomorphic to Sym^{p}(M) when i = p.

III: The standard application of the many units hypothesis, see ( [20] for instance) is that general position is available in the precise sense given below. Let V ≅ A^{n}. We denote by K(V) the simplicial complex whose set of vertices is L(V). A subset S ⊂ L(V) of cardinality (r + 1) is an r-simplex of K(V) if every T ⊂ S of cardinality t + 1 ≤ n belongs to L_{t}(V). If L ⊂ K(V) is a finite simplicial subcomplex, then there is some e ∈ L(V) with the property that s ∪ {e} is a (r + 1)-simplex of K(V) for every r-simplex s of L. This gives an embedding Cone(L) ↪ K(V) with e as the vertex of the cone. Thus K(V) is contractible. The complex of oriented chains of this simplicial complex will be denoted by C_{•}(V). Thus the reduced homologies H̃_{i}(C_{•}(V)) vanish for all i. The group D(V) = Z_{n−1}C_{•}(V) = B_{n−1}C_{•}(V) comes up frequently. IIIa:

(1) Let p < n. Let M be a ℤ[1/N]-module where N = (p + 1)!. Then H_{j}(P GL(V), C_{p} ⊗ M) vanishes for j < p and is isomorphic to Sym^{p}(A^{×}) ⊗ M when j = p.

(2) H_{j}(P GL(V), C_{n} ⊗ M) vanishes for all j ≥ 0 and for all ℤ[1/(n + 1)!]- modules M.

(3) H_{0}(P GL(V), D(V) ⊗ M) for any Abelian group M is isomorphic to M/2M if n is even, and vanishes if when n is odd

(4) H_{0}(P GL(V), B_{p}C_{•}(V) ⊗ M) = 0 for every ℤ[1/2]-module M and for every 0 ≤ p < n.

(5) H_{1}(P GL(V), Z_{p}C_{•}(V) ⊗ M) = 0 for every ℤ[1/(p + 2)!]- module M and 1 ≤ p ≤ n − 2.

Note that (1) above follows from IIa when F is the constant functor M. For (2), one observes that P GL(V) acts transitively on the set of n-simplices of the simplicial complex K(V). The stabiliser of an n-simplex is the permutation group Σ on (n + 1) letters. The claim now follows from Shapiro’s lemma. The presentation C_{p+2}(V) ⊗ M → C_{p+1}(V) ⊗ M → B_{p}C_{•}(V) ⊗ M → 0 and the observation H_{0}(P GL(V), C_{p+1}(V) ⊗ M) ≅ M/2M whenever p < n suffice to take care of (3) and (4). For assertion (5), one applies the long exact sequence of group homology to the short exact sequence: 0 → Z_{p+1}C_{•}(V) ⊗ M → C_{p+1}(V) ⊗ M → Z_{p}C_{•}(V) ⊗ M → 0. One therefore obtains the exact sequence: H_{1}(P GL(V), C_{p+1}(V) ⊗ M) → H_{1}(P GL(V), Z_{p}C_{•}(V) ⊗ M) → H_{0}(P GL(V), Z_{p+1}C_{•}(V) ⊗ M). The end terms here vanish by (1) and (4). IIIb:

(1) 𝔼𝕋(V) is connected.

(2) E_{0,0}^{2} = ℤ, E_{n−1,0}^{2} = D(V), E_{m,0}^{2} = 0 if m ≠ 0, n − 1 for the spectral sequence SS(V).

(3) H_{1}(𝔼𝕋(V)) ≅ ℤ/2ℤ if rank(V) > 2.

Note that (1) is a consequence of (2). Part (2) is deduced by induction on rank(V) = n. The induction hypothesis enables the identification of the E_{m,0}^{1} terms of the spectral sequence for V (together with differentials) with the C_{m}(V) (together with boundary operators) when m < n. Thus (2) follows.

For part (3), consider the spectral sequence SS(3). Here E_{2,0}^{2} = D(A^{3}) and H_{0}(P GL_{3}(A), D(A^{3})) = 0 by IIIa(3). Thus the hypothesis of Ib(4) holds for SS(3). Consequently, H_{1}(3) ≅ E_{1,0}^{∞} ≅ H_{0}(P GL_{2}(A), H_{1}(2)) = H_{0}(P GL_{2}(A), D(A^{2})) ≅ ℤ/2ℤ, the last isomorphism given by IIIa(3) once again. The isomorphism H_{1}(n) ≅ ℤ/2ℤ for n > 3 is contained in the lemma below for N = 1.

Lemma 27. Let M be an Abelian group. Let N ∈ ℕ. For 0 < r < N, we are given m(r) ≥ 0 so that H_{r}(d; M) → H_{r}(d + 1; M) is a surjection if d = r + m(r) + 1 , and an isomorphism if d > r + m(r) + 1. Let m(N) = max{0, m(1) + 1, m(2) + 1, ..., m(N − 1) + 1}. Then H_{N}(d; M) → H_{N}(d + 1; M) is

(a) an isomorphism if d > N + m(N) + 1,

(b) is a surjection if d = N + m(N) + 1.

(c) The surjection in (b) above factors through an isomorphism H_{0}(P GL_{d}(A), H_{N}(d)) → H_{N}(d + 1; M) if if M is a ℤ[1/2]-module.

Proof. Consider the spectral sequence SS(V; M) that computes the homology of 𝔼𝕋(V) with coefficients in M. Here V is free of rank N + h + 2, where h ≥ m(N). We make the following claim: Claim:If E_{s,r}^{2} ≠ 0 and 0 < s and r < N, then s + r ≥ N + 1 + h − m(N). Furthermore, when equality holds, H_{0}(P GL(V), E_{s,r}^{2} ⊗ ℤ[1/2]) = 0. We assume the claim and prove the lemma. We take h = m(N). All the E_{s,r}^{2} with s + r = N are zero except possibly for (s, r) = (0, N). Part (b) of the lemma now follows from Ib(2). We consider next the E_{s,r}^{2} with s + r = N + 1 and s ≥ 2 (or equivalently with r < N). It follows that E_{s,r}^{s} is a quotient of E_{s,r}^{2}. The second assertion of the claim now show that H_{0}(P GL(V), E_{s,r}^{s}) = 0 if M is a ℤ[1/2]-module. Part (c) of the lemma now follows from Ib(4). We take h > m(N) and prove part (a) by induction on h. The inductive hypothesis implies that H_{N}(N + h; M) → H_{N}(N + h + 1; M) is surjective. By Ib(5), it follows that H_{N}(N + 1 + h; M) → E_{0,N}^{2} is an isomorphism. Now there are no nonzero E_{s,r}^{2} with s + r = N + 1 and s ≥ 2. Thus E_{0,N}^{2} = E_{0,N}^{∞}. It follows that H_{N}(N + 1 + h; M) → H_{N}(N + 2 + h; M) is an isomorphism. It only remains to prove the claim. We address this matter now. For r = 0, both assertions of the claim are valid by IIIb(2)and IIIa(3). So assume now that 0 < r < N. Let SH(r) = H_{r}(d; M) for d = r + m(r) + 2. In view of our hypothesis, the chain complex E_{0,r}^{1} ← E_{1,r}^{1} ← ... ← E_{p−1,r}^{1} ← E_{p,r}^{1} for N + h + 1 = p + r + m(r) + 1 is identified with C_{0}(V) ⊗ SH(r) ← ... ← C_{p−1}(V) ⊗ SH(r) ← ⊕{det(q) ⊗ H_{r}(𝔼𝕋(V/W (q)); M)|q ∈ L_{p}(V)}. As in IIIb(2), it follows that E_{s,r}^{2} = 0 whenever 0 < s < p. Furthermore, we deduce the following exact sequence for E_{p,r}^{2}:

⊕{det(q) ⊗ KH_{r}(V/W (q); M)|q ∈ L_{p}} → E_{p,r}^{2} → Z_{p}C_{•} ⊗ SH(r) → 0. By IIIa(4), we see that H_{0}(P GL(V), E_{p,r}^{2}) = 0 if M is a ℤ[1/2]-module. Note that p + r = N + h − m(r) ≥ N + 1 + h − M(N). This completes the proof of the claim, and therefore, the proof of the lemma as well.

□

The Proposition below is an application of Proposition 22. The notation here is that of Theorem 2. We regard B_{p,q}^{r} and Z_{p,q}^{r} as subgroups of E_{p,q}^{1} for all r > 1. The notation KH_{m}(Q) has been introduced in Ia, the first application of many units.

Proposition 28. Let rank(V) = n. Let M be a ℤ[1/2]-module. In the spectral sequence SS(V; M), we have:

(1) ⊕{det(q) ⊗ KH_{m}(V/W (q))|q ∈ L_{1}(V)} ⊂ B_{1,m}^{∞} if n > 1.

(2) E_{1,m}^{∞} = 0 if n > 2 and M is a ℤ[1/6]-module.

(3) If, in addition, it is assumed that H_{m+1}(n − 2; M) → H_{m+1}(n − 1; M) is surjective, then ⊕{det(q) ⊗ KH_{m}(V/W (q))|q ∈ L_{2}(V)} ⊂ B_{2,m}^{∞}.

Proof. Let q ∈ L_{r}(V). We have U(q) ⊂ 𝔼𝕋(V) as in Proposition 22. The spectral sequence of Theorem 2 was constructed from an increasing filtration of subspaces of 𝔼𝕋(V). Intersecting this filtration with U(q) we obtain a spectral sequence that computes the homology of U(q). Its terms will be denoted by E_{b,c}^{a}(q). One notes that E_{b,m}^{1}(q) is the direct sum of det(u)⊗H_{m}(𝔼𝕋(V/W (u))) taken over all u ⊂ q of cardinality (b + 1). We denote the terms of the spectral sequence in theorem 2 by E_{b,c}^{a}(V). The given data also provides a homomorphism E_{b,c}^{a}(q) → E_{b,c}^{a}(V) of E^{1}-spectral sequences. We assume that M is a ℤ[1/(r + 1)!]-module. We choose a basis e_{1}, e_{2}, ..., e_{n} of V so that q = {Ae_{i} : 1 ≤ i ≤ r + 1}. Let G ⊂ GL(V) be the subgroup of g ∈ GL(V) so that (A) g(q) = q, (B)g(e_{i}) = e_{i} for all i > r + 1, (C), the matrix entries of g are 0, 1, −1 and (D) det(g) = 1. Now G acts on the pair U(q) ⊂ 𝔼𝕋(V). Thus the above homomorphism of spectral sequences is one such in the category of G-modules.We observe:

(a) G is a group of order 2(r + 1)!

(b) there are no nonzero G-invariants in E_{i,m}^{1}(q) for i > 0, and consequently the same holds for all G-subquotients, in particular for E_{i,m}^{a}(q) for all a > 0 as well. Proof of part 1. Take r = 1. Proposition 22 implies that the image of H_{m}(U(q)) → H_{m}(𝔼𝕋(V)) has trivial G-action. In view of (b) above, this shows that E_{1,m}^{∞}(q) → E_{1,m}^{∞}(V) is zero. But E_{1,m}^{∞}(q) = Z_{1,m}^{∞}(q) = det(q) ⊗ KH_{m}(V/W (q)). It follows that det(q) ⊗ KH_{m}(V/W (q)) ⊂ B_{1,m}^{∞}(V). Part (1) follows. Proof of part (2). We take r = 2. Here we have Z_{1,m}^{∞}(q) = Z_{1,m}^{2}(q). Appealing to Proposition 22 and observation (b) once again, we see that the image of the homomorphism Z_{1,m}^{2}(q) → Z_{1,m}^{2}(V) is contained in B_{1,m}^{∞}. Part (2) therefore follows from the claim below. Claim: ⊕{Z_{1,m}^{2}(q)|q ∈ L_{2}(V)} → Z_{1,m}^{2}(V) is surjective. Denote the image of H_{m}(n−2; M) → H_{m}(n−1; M) by I. A simple computation produces the exact sequences: 0 → ⊕{det(u)⊗KH_{m}(V/W (u) : u ∈ L_{1}(V), u ⊂ q} → Z_{1,m}^{2}(q) → det(q)⊗I → 0, and 0 → ⊕{det(u) ⊗ KH_{m}(V/W (u) : u ∈ L_{1}(V)} → Z_{1,m}^{2}(V) → Z_{1}C_{•}(V) ⊗ I → 0. The claim now follows from the above description of Z_{1,m}^{2}(q) and Z_{1,m}^{2}(V). Thus part (2) is proved. Proof of part (3).We take r = 2 once again. The surjectivity of H_{m+1}(n − 2; M) → H_{m+1}(n − 1; M) implies that E_{0,m+1}^{2}(q) has trivial G-action. By observation (b), we see that d_{2,m}^{2} : E_{2,m}^{2}(q) → E_{0,m+1}^{2}(q) is zero. It follows that E_{2,m}^{∞}(q) = Z_{2,m}^{2}(q) here. Proposition 22 and observation (b) once again show that the image of Z_{2,m}^{2}(q) → Z_{2,m}^{2}(V) is contained in B_{2,m}^{∞}(V). Because Z_{2,m}^{2}(q) = det(q) ⊗ KH_{m}(V/W (q)), part (3) follows. This completes the proof of the Proposition. □

Theorem 3. Let H_{m}(n; M) denote H_{m}(𝔼𝕋(A^{n}); M) where M is a ℤ[1/6]- module. We have:

(1) H_{1}(n; M) = 0 for all n > 2,

(2) H_{0}(GL_{3}(A), H_{2}(3; M)) → H_{2}(n; M) is an isomorphism for all n ≥ 4,

(3) H_{0}(GL_{4}(A), H_{3}(4; M)) → H_{3}(n; M) is an isomorphism for all n ≥ 5,

(4) H_{0}(GL_{2m−2}(A), H_{m}(2m − 2; M)) → H_{m}(n; M) is an isomorphism for all n > 2m − 2.

Proof. Part (1) has already been proved. Proof of part 2. For this, we study SS(V; M) where rank(V) = 4. We first note that

(i) E_{3,0}^{2} = D(V) and therefore H_{0}(P GL(V), E_{3,0}^{2}) = 0. (ii) E_{1,1}^{2} = E_{1,1}^{1} = ⊕{det(q) ⊗ D(V/W (q)) : q ∈ L_{1}(V)}, and therefore H_{1}(P GL(V), E_{1,1}^{2}) = 0 by IIa. (iii) E_{u,v}^{2} = 0 except when (u, v) = (0, 0), (0, 2), (1, 1), (3, 0). We have E_{1,1}^{∞} = 0 by proposition 28 and thus obtain the short exact sequence: 0 → E_{3,0}^{3} → E_{3,0}^{2} → E_{1,1}^{2} → 0. By (i) and (ii) above, we see that H_{0}(P GL(V), E_{3,0}^{3}) = 0. By Ib(1,4), we see that H_{0}(P GL_{3}(A), H_{2}(3; M)) → H_{2}(4; M) is an isomorphism. In particular, H_{2}(4; M) receives the trivial P GL_{4}(A)-action. Taking N = 2 and m(1) = 0 in lemma 27, we see that H_{2}(4; M) → H_{2}(n; M) is an isomorphism for all n ≥ 4. This proves part (2). Proof of part 3. We inspect SS(V; M) where V = A^{5}. We note that

(1) E_{1,2}^{∞} and E_{2,1}^{∞} both vanish. This follows from proposition 28, once it is noted that KH_{1}(2; M) = H_{1}(2; M).

(2) E_{0,2}^{2} = H_{2}(4; M) has the trival P GL(V)-action.

(3) H_{i}(P GL(V), E_{2,1}^{2}) = 0 for all i < 3. This follows from IIa and IIIa(3) after observing that H_{1}(2; M) ≅ D(A^{2}) ⊗ M.

(4) From (2) and (3) we see that d_{2,1}^{2} = 0.

(5) We deduce that E_{2,1}^{2} ≅ E_{0,4}^{2}/E_{0,4}^{3} and E_{1,2}^{2} = E_{1,2}^{3} ≅ E_{0,4}^{3}/E_{0,4}^{4} from observations (1) and (4).

(6) H_{1}(P GL(V), E_{1,2}^{2}) = 0. To see this, first note the the short exact sequence: 0 → P → E_{1,2}^{2} → Q → 0, where P = ⊕{det(q) ⊗ KH_{2}(V/W (q) : q ∈ L_{1}(V)} and Q = H_{2}(4; M) ⊗ Z_{1}C_{•}(V). The vanishing of H_{1}(P GL_{3}(A), Q) follows from IIIa(5). By IIa, the vanishing of H_{1}(P GL_{3}(A), P) is reduced to the vanishing of H_{0}(P GL_{3}(A), KH_{2}(A^{3}). Now let I be the augmentation ideal of the group algebra R[P GL_{3}(A)] where R = ℤ[1/6]. In view of the fact that P GL_{3}(A)_{ab} is 3-torsion, we see that I = I^{2}. It follows that for all ℤ[1/6]-modules N equipped with P GL_{3}(A)- action, we have IN = I^{2}N, or equivalently, H_{0}(P GL_{3}(A), IN) = 0. We apply this remark to N = H_{2}(3; M). By part (2) of the proposition, we see that KH_{2}(3; M) = IN. This proves that H_{0}(P GL_{3}(A), KH_{2}(A^{3})) = 0. We have completed the proof of observation 6.

(7) H_{0}(P GL(V), E_{4,0}^{4}) = 0. In view of the filtration of (5), it suffices to check that H_{1}(P GL(V), E_{a,b}^{2}) = 0 for (a, b) = (1, 2) and (2, 1) (which has been seen in observations (3) and (6)) and also that H_{0}(P GL(V), E_{4,0}^{2}) = 0 (and this is clear because E_{4,0}^{2} = D(V)).

(8) H_{0}(P GL_{4}(A), H_{3}(4; M)) → H_{3}(V; M) is an isomorphism. That H_{0}(P GL_{4}(A), H_{3}(4; M)) → E_{0,3}^{∞} is an isomorphism follows from observation (7) and Ib(4). Now E_{a,b}^{∞} = 0 whenever a + b = 3 and (a, b) ≠ (3, 0). This proves (8).

(9) H_{3}(5; M) → H_{3}(n; M) is an isomorphism for all n ≥ 5. This follows from lemma 27 by taking N = 3 and m(1) = m(2) = 0. This finishes the proof of part (3). Part (4) now follows from the same lemma and induction.

□

Remark. It can be checked that parts (1,2,4) of the above theorem are valid for ℤ[1/2]-modules M. In part (3), it is true that H_{3}(n; M) ≅ H_{3}(n + 1; M) for n > 4 and also that H_{0}(P GL_{4}(A), H_{3}(4; M) → H_{3}(5; M) is a surjection.

Proposition 29. Assume that the Compatible Homotopy Question has an affirmative answer. Then, for all ℤ[1/r!]-modules M and for all d > r + 1, H_{0}(P GL_{r+1}(A), H_{r}(r + 1; M)) → H_{r}(d; M) is an isomorphism.

Proof. For r = 1, this statement has been checked in IIIb(3) and lemma 27. Let N > 1. We assume that the above statement has been proved for all r < N. Let M be a ℤ[1/N!]-module. In lemma 27, may may now take m(1) = m(2) = ... = m(N − 1) = 0. From this lemma, we obtain:

H_{0}(P GL_{N+2}(A), H_{N}(N + 2; M)) → H_{N}(N′; M)

is an isomorphism for all N′ > N + 2. So the proposition is proved once it is checked that

H_{0}(P GL_{N+1}(A), H_{N}(N + 1; M)) → H_{N}(N + 2; M)

is an isomorphism. To prove this, we consider the spectral sequence SS(V; M) where V = A^{N+2}. We will prove:

(i) E_{a,b}^{2} = 0 or a = 0 or a + b = N or (a, b) = (N + 1, 0). Furthermore the action of P GL(V) on E_{0,b}^{2} is trivial when b < N. (ii) if a > 0 and b > 0, then H_{i}(P GL(V), E_{a,b}^{2}) = 0 for i = 0, 1. (iii) E_{a,b}^{∞} = 0 when a > 0 and b > 0. (iv) E_{a,b}^{2} ≅ E_{N+1,0}^{b+1}/E_{N+1,0}^{b+2} whenever a > 0, b > 0 and a + b = N. We first observe that (iv) is true for any spectral sequence of P GL(V)- modules where (i),(ii) and (iii) hold. Next note that (ii) and (iv) imply that H_{0}(P GL(V), E_{N+1,0}^{N+1}) is contained in H_{0}(P GL(V), E_{N+1,0}^{2}). And since the latter is zero, we see that the former also vanishes. We deduce that both arrows H_{0}(P GL(V), E_{0,N}^{1}) → E_{0,N}^{∞} → H_{N}(N +2; M) are isomorphisms exactly as in earlier proofs. Thus it only remains to prove (i), (ii) and (iii). Proof of (i). This is contained in the proof of lemma 27. Proof of (ii). 0 → P → E_{a,b}^{2} → Q → 0 is exact, where P = ⊕{det(q) ⊗ KH_{b}(V/W (q)|q ∈ L_{a}(V)}, and Q = Z_{a}C_{•} ⊗ H_{b}(b + 2; M) as in the proof of the lemma 27. The required vanishing of H_{i}(P GL(V), T) for i = 0, 1 holds for T = Q by IIIa(5). For T = P and a > 1, the required vanishing follows from II(a). For T = P and a = 1, this is deduced from the vanishing of H_{0}(P GL_{N}(A), KH_{N−1}) (see the proof of observation (6) in the proof of theorem 3). Proof of (iii). We follow the steps of the proof of Proposition 28. We first choose q ∈ L_{N−1}(V) and consider the inclusion U(q) ↪ 𝔼𝕋(V). As in that proof we get a homomorphism of E^{1} spectral sequences of G-modules with G ⊂ SL(V) as given there. The terms of these spectral sequences are denoted by E_{b,c}^{a}(q) and E_{b,c}^{a}(V) respectively. From the inductive hypothesis, we deduce: (i′) E_{a,b}^{2}(q) = 0 or a = 0 or a + b = N or (a, b) = (N + 1, 0). Furthermore the action of G on E_{0,b}^{2}(q) is trivial when b < N. (ii′) if a > 0, h > 0, then H_{0}(G, E_{a,b}^{h}(q)) = 0. These observations together imply (iii′) Z_{a,b}^{∞}(q) = Z_{a,b}^{2}(q) when a > 0 and b > 0. For a > 0, b > 0, we obtain E_{a,b}^{∞}(q) → E_{a,b}^{∞}(V) is zero, from the affirmative answer to the Compatible Homotopy Question. For such (a, b), the image of x(q) : Z_{a,b}^{2}(q) → Z_{a,b}^{2}(V) is thus contained in B_{a,b}^{∞}(V). As in the proof of proposition 28, we see that the sum of the images of x(q), taken over all q ∈ L_{N−1}(V), is all of Z_{a,b}^{2}(V). It follows that Z_{a,b}^{2}(V) = B_{a,b}^{∞}(V) and thus E_{a,b}^{∞}(V) = 0 whenever a > 0, b > 0. This proves assertion (iii) and this completes the proof of the Proposition.

□

10. a double complex

We will continue to assume that A is a commutative ring with many units. The paper [2] of Beilinson, Macpherson and Schechtman introduces a Grassmann complex, intersection and projection maps, and a torus action. The terms of the double-complex constructed below may be obtained from the quotients by the torus action of the objects of [2]. The arrows of the double-complex are signed sums of their intersection and projection maps. D(V), C_{•}(V) etc. are as in the previous section. When rank(V) = n, we have the resolution: 0 ← D(V) ← C_{n}(V) ← C_{n+1}(V).... We put \overline{C}_{r}(V) = H_{0}(P GL(V), C_{r}(V)) when r ≥ n and define \overline{C}_{r}(V) to be zero otherwise. We put \overline{C}_{r}(A^{n}) = \overline{C}_{r}(n). We observe that the above resolution of D(V) tensored with the rationals is a projective resolution in the category of ℚ[P GL(V)]-modules. It follows that H_{i}(P GL_{n}(A), D(A^{n})) ⊗ ℚ ≅ H_{n+i}(\overline{C}(n)_{•}) ⊗ ℚ. We denote by ∂′ : \overline{C}_{r}(n) → \overline{C}_{r−1}(n) the boundary operator of \overline{C}(n)_{•}. We will now define ∂′′ : \overline{C}_{r}(n) → \overline{C}_{r}(n − 1). Let V ≅ A^{n}. Let (L_{0}, L_{1}, ..., L_{r}) be an ordered (r + 1)-tuple in L(V) that gives rise to a r-simplex of K(V) (see consequence III of many units for notation). We define ∂_{i}(L_{0}, L_{1}, ..., L_{r}) ∈ C_{r−1}(V/L_{i}) by ∂_{i}(L_{0}, L_{1}, ..., L_{r}) = (\overline{L}_{0}, .., \overline{L}_{i−1}, \overline{L}_{i+1}, ..., \overline{L}_{r}) where \overline{L}_{j} = L_{j} + L_{i}/L_{i} ∈ L(V/L_{i}) whenever j ≠ i. Now let

g_{r}(L_{0}, L_{1}, ..., L_{r}) = Σ_{i=0}^{r}(−1)^{i}∂(L_{0}, L_{1}, ..., L_{r}) ∈ ⊕{C_{r−1}(V/L) : L ∈ L(V)}.

The above g_{r} : C_{r}(V) → ⊕{C_{r−1}(V/L) : L ∈ L(V)} anti-commutes with the boundary operator. The functor M → H_{0}(P GL(V), M) takes g_{r} to ∂′′ : \overline{C}_{r}(n) → \overline{C}_{r−1}(n − 1). This defines ∂′′. We put F_{r}(A) = ⊕\{\overline{C}_{r}(n) : n ≥ 1\} and define ∂ : F_{r}(A) → F_{r−1}(A) by ∂ = ∂′ + ∂′′. The exact relation between the homology of F_{•}(A) and groups L_{n}(A) is as yet unclear. However, we do have:

Lemma 30. H_{3}(F_{•}(A)) ⊗ ℚ ≅ L_{2}(A) ⊗ ℚ ≅ H_{3}(\overline{C}_{•}(2)) ⊗ ℚ.

We sketch a proof. In view of the H-space structure, L_{i}(A) ⊗ ℚ is the primitive homology of 𝔼𝕋(A^{n}) with ℚ coefficients for n large. The vanishing of H_{1}(n; ℚ) for n > 2 implies that the primitive homology is all of H_{i}(n; ℚ) for i = 2, 3 and n large. By theorem 3, we get L_{i}(A) ⊗ ℚ ≅ H_{0}(P GL_{i+1}(A), H_{i}(i + 1; ℚ)) for i = 2, 3. For the computation of H_{0}(P GL(V), H_{2}(𝔼𝕋(V); ℚ)) where V = A^{3}, we recall the exact sequence obtained from SS(V; ℚ): 0 → H_{2}(𝔼𝕋(V); ℚ) → D(V) ⊗ ℚ → D_{2}(V) = ⊕{D(V/L) : L ∈ L(V)} → 0. This identifies L_{2}(A) ⊗ ℚ with the cokernel of H_{1}(P GL(V), D(V)) ⊗ ℚ → H_{1}(P GL(V), D_{2}(V)) ⊗ ℚ. In view of IIa and the above remarks, this is readily identified with H_{3}(F_{•}(A)) ⊗ ℚ.That gives the first isomorphism of the lemma. For the second isomorphism, what one needs is: Claim: The arrow H_{4}(\overline{C}_{•}(3)) → H_{3}(\overline{C}_{•}(2)) induced by ∂′′ is zero. The proof of this claim, which we address now, was already known to Spencer Bloch. Let V = A^{3}. Given an ordered 5-tuple (L_{0}, ..., L_{4}) with the L_{i} ∈ L(V) as vertices of a 4-simplex in K(V) (i.e. in general position), they belong to a conic C and the projection from the points L_{i} induces an isomorphism p_{i} : C → ℙ(V/L_{i}). We put (M_{0}, ..., M_{4}) = (p_{0}L_{0}, p_{0}L_{1}, ..., p_{0}L_{4}). Let q_{i} = p_{i} ∘ p_{0}^{−1}. With the ∂_{i} as in the definition of g_{4}, we see that ∂_{i}(L_{0}, ..., L_{4}) ∈ C_{3}(V/L_{i}) and q_{i}∂_{i}(L_{0}, ..., L_{4}) ∈ C_{3}(V/L_{0}) both give rise to the same element of \overline{C}_{3}(2). It follows that ∂(M_{0}, M_{1}, ..., M_{4}) ↦ ∂′′(L_{0}, ..., L_{4}) under the map C_{3}(V/L_{0}) → \overline{C}_{3}(2). Thus ∂′′(L_{0}, ..., L_{4}) ↦ 0 ∈ H_{3}(\overline{C}_{•}(2). This proves the claim and the lemma. Thus we have shown that L_{2}(A) ⊗ ℚ ≅ coker(\overline{C}_{4}(A^{2}) → \overline{C}_{3}(A^{2})). The Bloch group tensored with ℚ is the homology of

\overline{C}_{4}(A^{2}) → \overline{C}_{3}(A^{2}) → Λ^{2}(A^{×}) ⊗ ℚ.

Thus this discussion amounts to a proof of Suslin’s theorem on the Bloch group. It remains to obtain a closed form for L_{3}(A) ⊗ ℚ by this method. Acknowledgements: This paper owes, most of all, to Sasha Beilinson who introduced us to the notion of a homotopy point. The formulation in terms of contractible spaces of maps occurs in many places in this manuscript. Rob de Jeu lent a patient ear while we groped towards the definition of 𝔽𝕃(V). Peter May showed us a proof of Proposition 20. We thank them, Kottwitz and Gopal Prasad for interesting discussions, and also Spencer Bloch for his support throughout the project. Finally we thank the referee who brought our attention to several careless slips in the first draft.

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M. V. Nori Dept. of Mathematics University of Chicago 5734 University Ave. Chicago IL-60637 USA nori@math.uchicago.edu V. Srinivas School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Colaba Mumbai-400005 India srinivas@math.tifr.res.in 514

Documenta Mathematica · Extra Volume Suslin (2010)