K-theory and the Enriched Tits Building
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 $…
Authors: M. V. Nori, V. Srinivas
To A. A. Suslin with admiration, on his sixtieth birthday.
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.
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 1 Gopal Prasad informed us that we should take G reductive or k perfect. 2 John 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] . 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 1 p,q terms and the differentials d 1 p,q are recognisable since they involve only the homology groups of ET(A a ) for a < n. The differentials d r p,q 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
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
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
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:
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".
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
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 ).
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.
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 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 :
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
For most purposes, it suffices to note that there is a canonical map
. 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
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.
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
) 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
) 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.
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
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 M ap Λ (X, Y ). M ap Λ (X, Y ) is a closed subset of Hom(X, Y ), and this topologises M ap 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
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.
In both corollaries, what one obtains is a contractible collection of homotopy equivalences; there is no preferred or 'natural' choice. Naturally, this situation persists in all applications of the above proposition and corollaries. 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
Proof. Consider the collection of pairs (Λ ′′ , f ′′ ) satisfying:
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 Λ-compatible. The pathconnectivity 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 Λ-compatible 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 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 ) :
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
w w w w w w w 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 f : X × EG → Y × EG, compatible with the projections to EG, such that
EG is another such equivariant map, then there is a G-equivariant homotopy between f and g, 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
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 ) :
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.
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 -1 P (b), then {a} ∪ C is a chain in ϕ -1 P (a). This gives an order preserving map of posets ϕ -1 P (b) → ϕ -1 P (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 -1 P (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
We thus have a commutative diagram of posets and order preserving maps
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:
Note that the natural maps on vertex sets p : Z → A, q : Z → B induce canonical simplicial maps on geometric realizations
Corollary 7.
(1) With the above notation, the simplicial maps Proof. We first discuss (1). Since the situation is symmetric with respect to the sets A, B, it suffices to show |p| is a homotopy equivalence.
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:
(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
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, corollary 2 provides a contractible collection of Λ-compatible homotopy inverses of |p|.
In the presence of a G-action, Proposition 5 provides a contractible family of G-
This suffices to give (2).
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
We think of this vertex set as the set of "full flags" in V .
To describe the simplices in FL(V ), we will need another definition. Let
and the induced map ⊕ n i=1 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
[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
) 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)
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
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 p F gr q F ′ (V ) ∼ = gr q F ′ gr p F (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 p F ′ (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
, 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 ϕ F,F ′ is a direct sum of isomorphisms of the form
between free modules of rank 1, where σ is a permutation of {1, . . . , n}. 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 such a direct sum of isomorphisms, upto permuting the factors. In particular, given an edge-path loop based as F ∈ F L(V ), the induced automorphism of gr F (V ) is the composition of a "diagonal" automorphism and a permutation.
Hence, the monodromy group of the local system gr(V ) is clearly contained in N n (A), defined as a semidirect product
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
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
Next, observe that there is a local system gr(A ∞ ) on FL(A ∞ ) whose fiber over a vertex
where we may also view N (A) as the semidirect product of
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".
the inclusions of the direct summands. Consider the two natural maps
given by
(A) α and β are vertices of a one-simplex of Hom(FL
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
) 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).
The group E n+1 (A) of elementary matrices acts trivially on the image of the natural map
(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.
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:
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 F h(L) 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
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
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
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 ) := N E(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
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 . 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
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
where on the product, we take the partial order
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
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
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
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
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.
)) 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
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
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, * ) → β under the natural map
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
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 (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
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 (β) → ET(V )//GL(V ) induces an isomorphism in integral homology.
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 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
Then one sees that the stabiliser H(x) ⊂ GL(V ) is given by
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
(where we view the expression tz + (1t)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
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
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
The same deformation retraction H determines, by reparametrization, a deformation retraction
and from what we have just shown above, the inclusion
is a homotopy equivalence. To simplify notation, we let X = ET(V )//GL(V ), so that we have the map π : X → BP , and
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
Here, as before, we make sense of the above expression tx ′ +(1-t)x using barycentric coordinates in σ(x ′ ). Define
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
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
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
) 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
where n i = rank (W i /W i-1 ), I ni 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
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
, 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
p,q = 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
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
Then U 1 (I) is a normal subgroup of G(I), from the last description, and
Now U 1 (I) may be identified with M i1,n ′ (A), the additive group of matrices of size i 1 × n ′ over A; this matrix group has a natural action of GL i1 (A), and thus of the diagonal matrix group T i1 (A), and the resulting semidirect product of T i1 (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 i1,n ′ (A) is isomorphic, as T i1 (A)-modules, to the direct sum
, where A n ′ (i) is the free A-module of rank n ′ , with a T i1 (A)-action given by the the "i-th diagonal entry" character T i1 (A) → A × . Thus, the semidirect product T i1 (A)U 1 (I) has a description as a direct product
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
H induces an isomorphism on integral homology. This follows from the Kunneth formula. The fact (ii) implies that T i1 (A)U 1 (I) → T i1 (A) induces an isomorphism on integral homology. Then (i) implies that G(I) → T i1 (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 i1 (A) × G(I ′ ) → T i1 × 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
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 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 H 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
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
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
where L 0 (A) is regarded as a pointed set. 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 ϕ : Y → 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 ϕ. There is then a homotopy pullback ϕ : Y → BE(A) of ϕ 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)
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 Y → Y . Since also SPL(A ∞ )//E(A) → Y is a homology isomorphism (from lemma 14 again), we see that SPL(A ∞ )//E(A) → Y 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 Y 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
a homology isomorphism under the following additional assumption:
) for all i, j.
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 -1 V ( * ) 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 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 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 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
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
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
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
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
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
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
which, on classifying spaces, yield the diagram of product maps, preserving base points,
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
Hence we obtain on the direct limits a diagram
From lemma 14, it follows that there is an induced diagram at the level of plus constructions [6] that the bottom 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 Hmap between path connected H-spaces, and so the homotopy fiber Z(A) has the homotopy type of an H-space 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
which is homotopic (preserving the base point) to the map induced by standard inclusion i n :
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
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
Hence the induced maps
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 lim
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 basepoint 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).
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.
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
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 × ).
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
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 p, 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 p is naturally isomorphic to the poset L(p). Indeed, an element q ∈ E(A n ) consists of a pair, consisting of a partial flag
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 ∆(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 sd∆(p), where the barycentre b corresponds to the chain {p}. Hence, from the definition of p, it is clear that it is isomorphic to the poset whose elements are simplices in the barycentric subdivision of ∆ n with b as a vertex, with partial order given by reverse inclusion. Hence B p is naturally identified with the subcomplex of the second barycentric subdivision sd 2 ∆(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:
Now let p ∈ E(A n ) be arbitrary, corresponding to a partial flag
and splittings t i ∈ SP L(W i /W i-1 ). Then the natural map
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
and so BL ′ (p) ∼ = S n-r-1 , and BL(p) is an nr-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
) is an increasing finite filtration of the CW complex ET(V ) by subcomplexes. Hence there is an associated spectral sequence
Our objective now is to recognise the above E 1 terms. It is convenient to use the complexes of cellular chains for these sub CW-complexes, 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
r,s = H r+s (gr F r Cell • (V )). We will now exhibit gr F r 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
so that we have an induced homomorphism of complexes
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 :
H r (e(q), ∂e(q), Z)
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:
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 (nr -1).
Summarising the above, we obtain:
There is a spectral sequence with E 1 terms
that converges to H r+s (ET(V )). We note that E 1 r,s = 0 whenever (r + s) ≥ (n -1) with one exception: (r, s) = (n -1, 0). Here V ∼ = A n .
It is true3 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. 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
), 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 and the proof of lemma 10 combine to show 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 U λ(x) be its set of upper bounds in SP L(V x ). It follows that that LU λ(x)) = λ(x). As observed before, we have
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 n i=1 ET(W i ) → ET(⊕ n i=1 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
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 , 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
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
The required homotopy is a path γ : I → Maps(U (q), ET(V )) so that γ(0) = i and this exact sequence for which the induced automorphisms on W and Q are of the type α.id W and β.id Q respectively where α, β are arbitrary units of A. We may regard H as a subgroup of GL(P ). This group H 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:
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
Ib: In the spectral sequence SS(n), we have:
) If H 0 (P GL n (A), E p p,m+1-p ) = 0 for all p ≥ 2, then the given arrow H 0 (P GL n (A), E 1 0,m ) → E ∞ 0,m is an isomorphism. (5) Assume that H m (n -2) → H m (n -1) is surjective. Then the arrow E 2 0,m → H 0 (P GL n (A), E 1 0,m ) 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
,m-1 ) itself vanishes. Thus we obtain a factoring: E 2 0,m → E 3 0,m → H 0 (P GL n (A), E 1 0,m ). Proceeding inductively, we obtain the factoring:
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.
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 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[ 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
For assertion (5), one applies the long exact sequence of group homology to the short exact sequence: 0
One therefore obtains the exact sequence:
The end terms here vanish by ( 1) and (4). IIIb:
(1) ET(V ) is connected.
(2) E 2 0,0 = Z, E 2 n-1,0 = D(V ), E 2 m,0 = 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 1 m,0 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 2,0 = 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) 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 2 s,r = 0 and 0 < s and r < N , then s + r ≥ N + 1 + hm(N ). Furthermore, when equality holds, H 0 (P GL(V ), E 2 s,r ⊗ Z[1/2]) = 0. We assume the claim and prove the lemma. We take h = m(N ). All the E 2 s,r 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 2 s,r with s + r = N + 1 and s ≥ 2 (or equivalently with r < N ). It follows that E s s,r is a quotient of E 2 s,r . The second assertion of the claim now show that H 0 (P GL(V ), E s s,r ) = 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 2 0,N is an isomorphism. Now there are no nonzero E 2 s,r with s + r = N + 1 and s ≥ 2. Thus E 2 0,N = 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
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 2 s,r = 0 whenever 0 < s < p. Furthermore, we deduce the following exact sequence for E 2 p,r : ⊕{det(q) ⊗ KH r (V /W (q); M )|q ∈ L p } → E 2 p,r → Z p C • ⊗ SH(r) → 0. By IIIa(4), we see that H 0 (P GL(V ), E 2 p,r ) = 0 if M is a Z[1/2]-module. Note that p + r = N + hm(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 r p,q and Z r p,q as subgroups of E 1 p,q 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
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 a b,c (q). One notes that E 1 b,m (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 a b,c (V ). The given data also provides a homomorphism E a b,c (q) → E a b,c (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 Gmodules.We observe: (a) G is a group of order 2(r + 1)! (b) there are no nonzero G-invariants in E 1 i,m (q) for i > 0, and consequently the same holds for all G-subquotients, in particular for E a i,m (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 2 1,m (q). Appealing to Proposition 22 and observation (b) once again, we see that the image of the homomorphism Z 2 1,m (q) → Z 2 1,m (V ) is contained in B ∞ 1,m . Part (2) therefore follows from the claim below. Claim: ⊕{Z 2 1,m (q)|q ∈ L 2 (V )} → Z 2 1,m (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 2 1,m (q) → det(q) ⊗ I → 0, and 0 → ⊕{det(u) ⊗ KH m (V /W (u) : u ∈ L 1 (V )} → Z 2 1,m (V ) → Z 1 C • (V ) ⊗ I → 0. The claim now follows from the above description of Z 2 1,m (q) and Z 2 1,m (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 2 0,m+1 (q) has trivial G-action. By observation (b), we see that d 2 2,m : E 2 2,m (q) → E 2 0,m+1 (q) is zero. It follows that E ∞ 2,m (q) = Z 2 2,m (q) here. Proposition 22 and observation (b) once again show that the image of Z 2 2,m (q) → Z 2 2,m (V ) is contained in B ∞ 2,m (V ). Because Z 2 2,m (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 2 3,0 = D(V ) and therefore H 0 (P GL(V ), E 2 3,0 ) = 0. (ii) E 2 1,1 = E 1 1,1 = ⊕{det(q) ⊗ D(V /W (q)) : q ∈ L 1 (V )}, and therefore H 1 (P GL(V ), E 2 1,1 ) = 0 by IIa. (iii) E 2 u,v = 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: 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 -1 0 . 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 C 3 (2). It follows that ∂(M 0 , M 1 , ..., M 4 ) → ∂ ′′ (L 0 , ..., L 4 ) under the map C 3 (V /L 0 ) → C 3 (2). Thus ∂ ′′ (L 0 , ..., L 4 ) → 0 ∈ H 3 (C • (2). This proves the claim and the lemma. Thus we have shown that L 2 (A) ⊗ Q ∼ = coker(C 4 (A 2 ) → C 3 (A 2 )). The Bloch group tensored with Q the homology of
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.
this only requires the analogue of lemma 8(A) for the enriched Tits building. More general statements are contained in lemmas 23 and 24 .
Original Paper
Loading high-quality paper...
Comments & Academic Discussion
Loading comments...
Leave a Comment