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On Wagoner complexes

Wagoner complexes are simplicial complexes associated to groups of Kac-Moody type. They admit interesting homotopy groups which are related to integral group homology if the root datum is of 2-spherical type. We give a general definition of Wagoner complexes, exhibit some simple properties and calculate low dimensional homotopy groups. In addition, we give a definition of affine Wagoner complexes related to groups admitting a root datum with valuation.


Introduction
Wagoner complexes are certain simplicial complexes first constructed by Wagoner in [Wag73]. In his original construction, they were associated to the group GL n (R) over an arbitrary ring R. Wagoner used the homotopy groups of these complexes to give a tentative definition of higher algebraic K-theory. Later Anderson, Karoubi and Wagoner proved that this definition is equivalent to Quillen's definition of K-theory.
The complexes Wagoner defined are closely related to the building associated to the general linear group. Unlike the building, the Wagoner complex is not homotopy equivalent to a bouquet of spheres, instead it has interesting homotopy and homology groups. Wagoner complexes have already been generalised to reductive F -groups by Petzold in [Pet04]. There, rational homology groups of Wagoner complexes are used to construct rational representations of reductive groups.
In this paper, we define Wagoner complexes in the context of groups of Kac-Moody type (or groups with root data), which we believe to be the appropriate level of generality. Important examples of such groups are semi-simple Lie groups and algebraic groups as well as Kac-Moody groups. All definitions are very natural and the connection to the theory of buildings is clearly visible.
We exhibit interesting substructures in Wagoner complexes analogous to apartments in buildings and present some general properties of Wagoner complexes. The main result is the following characterisation of low dimensional homotopy groups of Wagoner complexes in the case where the associated root datum is of 2-spherical type.
Theorem 0 Let G be a group of Kac-Moody type and assume that its root datum is of 2-spherical type. Let G † be the little projective group and letĜ be the associated Steinberg group in the sense of Tits. Let W(G) be the associated Wagoner complex. Then, in the generic case, we have If G is of non-spherical type or rank(G) ≥ 3, we have By a recent result by Caprace, this means π 1 (|W(G)|) ∼ = H 2 (G † ; Z) except over very small fields.
For the precise statement, see Theorem 14 and Corollary 15.
In addition to this result, we also make the connection to algebraic K-theory explicit. Finally, we give a definition of affine Wagoner complexes associated to groups admitting root data with valuations.
The structure of this text is as follows. In section 1, we define Wagoner complexes. We construct apartments and a projection onto the building for spherical Wagoner complexes in section 2. In section 3, we discuss simple properties of pairs of roots. The definition of the Steinberg group in the sense of Tits is given in section 4. We prove the main result in section 5 and show its relation to algebraic K-theory in section 6. Finally, we give the construction of affine Wagoner complexes in section 7.
We would like to thank Linus Kramer for the introduction to the topic of Wagoner complexes and countless useful discussions. In addition, we are very grateful for suggestions and answers by Pierre-Emmanuel Caprace and Richard Weiss. We are indebted to Guntram Hainke for very good advice on the construction in the non-spherical case.
The author was supported by the Graduiertenkolleg: "Analytische Topologie und Metageometrie" while working on this topic. This work is part of the author's doctoral thesis at the Universität Münster.

The definition of Wagoner complexes
Throughout this text, let (W, S) be a Coxeter system of rank at least 2 whose Coxeter diagram has no isolated nodes. Let Σ be the associated Coxeter complex. Denote by Φ the set of roots of Σ and let G be a group with a twin root datum (G, (U α ) α∈Φ , H) of type Φ. For the precise definition of groups with root data, also known as groups of Kac-Moody type, see the book [Rém02] by Rémy. Denote the subgroup of G generated by all root groups, the little projective group, by G † .
We write R s = {t ∈ Σ : s ⊆ t} for the residue of s in Σ. We will say that a simplex s is co-spherical if the corresponding residue R s is of spherical type. The basic 'building blocks' for Wagoner complexes will be the following groups.

Definition 1 For a co-spherical simplex s ∈ Σ we define
It is well known that there is a twin building of type Φ associated to the group G, see [Rém02,2.6.4]. By the same reference, the positive half of the twin building is a thick Moufang building ∆ on which G acts strongly transitively. There is a standard apartment in the building which we identify with Σ, and there is a fundamental chamber c 0 ∈ Σ. Using the Moufang property of the building, we obtain: For every s ∈ Σ, the parabolic subgroup P s is the stabiliser subgroup of s in G. We have U s ≤ P s .

Remark
The building ∆ is isomorphic to the set of cosets {gP s : s ⊆ c 0 , g ∈ G} of standard parabolics partially ordered by reverse inclusion, see for example [AB08,6.29]. We will henceforth identify the building with this partially ordered set.
Generalising the construction by Wagoner in [Wag73], we define: Definition 2 The Wagoner complex W(G) is the flag complex over the set of cosets {gU s : s ∈ Σ, s co-spherical, g ∈ G} partially ordered by inclusion, which we write as follows: We call a Wagoner complex spherical if the root datum is of spherical type.

Simple Properties of spherical Wagoner complexes
Although we make no direct use of this, let us examine some interesting properties of spherical Wagoner complexes for further reference. Most of these results were already observed in [Pet04] or [Wag73]. We restrict ourselves to spherical Wagoner complexes in this section, because the corresponding results are more interesting. We will comment on the general situation at the end of this section. For the rest of this section, assume that Σ is a spherical Coxeter complex.

Definition 3
We define the standard apartment of W(G) to be Left translates of the standard apartment will be called apartments.
A spherical Wagoner complex admits a natural projection onto the barycentric subdivision of the building, which can be described as the flag complex over the building.
Projection Consider the following map of partially ordered sets This map is well-defined since U s ≤ P s . In addition, it is easily seen to be order-preserving. In particular, it induces a surjective simplicial map p ∆ : W(G) → Flag(∆) onto the barycentric subdivision of the building ∆.
It is easy to see that the projection p ∆ maps the standard apartment A bijectively onto the barycentric subdivision of the standard apartment Flag(Σ) which justifies the terminology. In addition, the map p ∆ induces a surjective map on singular homology: by the theorem of Solomon and Tits, the singular homology of the building ∆ is concentrated in top-dimension and it is generated by the apartments which are all images of apartments in W(G).
Right N -action There is a right action of the group N on the vertices of the Wagoner complex {gU s : g ∈ G, s ∈ Σ} by right multiplication: for any chamber c, this action is free. It induces a free right N -action on W(G) and we have An = nA. As already observed in [Pet04], the projection If we consider a complex analogous to the Wagoner complex but replacing the groups U s by the parabolic subgroups P s , we obtain another simplicial complex with a right N -action. The quotient complex is isomorphic to the barycentric subdivision of the building. The following diagram summarises all these constructions.
The general case In the general case for non-spherical root data, the concept of apartments and the above projection still make sense. The projection is then onto the flag complex over all co-spherical simplices of ∆, which is the simplicial complex underlying the Davis realisation of the building ∆ as in [Dav08,18.2]. The Davis realisation is contractible, so the induced maps on homotopy and homology groups are necessarily trivial.
Remember that a pair of roots (α, β) is called prenilpotent, if the intersections α ∩ β and −α ∩ −β both contain a chamber. For any prenilpotent pair of roots (α, β), we write These sets of roots are always finite. For any of the three sets of roots we write Remark For spherical Coxeter systems, a pair of roots (α, β) is prenilpotent if and only if α = −β. In this case, the root interval can be expressed simply as For any root α ∈ Φ, we denote by s α the reflection at its wall ∂α. In non-spherical Coxeter complexes, there are two possibilities for prenilpotent pairs of roots.

Lemma 4 (Proposition 3.165 and Lemma 8.45 in [AB08]) Let (α, β) be a prenilpotent pair of roots in Φ. Then there are two possibilities
• The product s α s β has infinite order. In this case, the roots α and β are nested, that is, we have α ⊆ β or β ⊆ α.
• The product s α s β has finite order. Then every maximal simplex s of the intersection of the walls ∂α ∩ ∂β has codimension 2 and the link lk(s) is spherical. We writē α = α ∩ lk(s),β = β ∩ lk(s) for the induced roots in the link. The map given by This can be used to prove the following characterisation.
Lemma 5 For any prenilpotent, non-nested pair of roots (α, β), there are either roots α and β such that Proof If the Coxeter complex Σ is of rank two and finite, it is an ordinary 2m-gon for some m ≥ 2. Its roots are paths of length m. We denote the vertices of Σ by (x i ) i∈Z/2m . This leads to an enumeration of the roots (α i ) i∈Z/2m such that every root α i contains the vertices {x i , x i+1 , . . . , x i+m }. Then it is easy to see that Since the numbering of vertices was arbitrary, the Lemma follows for rank two finite Coxeter complexes.
If the rank of the Coxeter complex Σ is greater than two, and if (α, β) is a prenilpotent, non-nested pair of roots, by Lemma 4, we can reduce to the rank 2 case as follows. Fix a maximal simplex s in the intersection of the walls ∂α ∩ ∂β and writeᾱ = α ∩ lk(s), β = β ∩ lk(s) as above. Then consider the bijection Now the lemma follows by applying the finite rank two case to lk(s) and using the bijection to transport the result to the ambient building.
Using this characterisation, we prove another lemma which describes the 'shadows' of root group intervals in the groups U s .
Lemma 6 Let (α, β) be a prenilpotent, non-nested pair of roots. Let s ∈ α ∩ β be a co-spherical simplex. Then one of the following holds.
• There is a prenilpotent, non-nested pair of roots (γ, δ) such that R s ⊂ γ ∩ δ and Proof The set of chambers in R s is finite, and we denote the number of chambers in R s \ (α ∩ β) by n. If n = 0, then (α, β) is already a pair of roots we are looking for. Otherwise, we will replace U [α,β] iteratively by smaller groups as follows, always decreasing n strictly. If n > 0, then R s ⊂ α or R s ⊂ β. We fix a chamber c ∈ R s ∩ α ∩ β, and a chamber d ∈ −α ∩ −β. Denote by e the projection of d onto R s . Since the roots −α and −β are convex and d ∈ −α ∩ −β, we have e ∈ −α ∩ R s or e ∈ −β ∩ R s . By [Rém02, 2.2.6], the minimal gallery from c to d through e crosses one of the walls ∂α or ∂β first among the walls of roots in [α, β]. Assume that the gallery crosses the wall ∂α first. Then there is a chamber c ∈ −α ∩ R s such that c ∈ γ for all γ ∈ ]α, β].
By In the first case, the number of chambers n in R s \ (α ∩ β) is strictly smaller than the number of chambers in R s \ (α ∩ β), since the first one is a subset of the second and does not contain c . If n = 0, we are finished, otherwise we repeat the procedure.
In the second case, we have U ]α,β] = U β . Then either R s ∈ β and we are done, or , in which case we can choose an arbitrary root γ containing R s .

The Steinberg group
From now on, we will often require the following technical condition. Denote by Π the set of simple roots of the root system Φ. For roots α, β ∈ Π, we let Following Caprace in [Cap07], we consider the following condition, which excludes the possibility that X α,β modulo its centre is isomorphic to some small finite groups.
We will require this condition for most of the results later on. Now, following Tits in [Tit87], we introduce the Steinberg group. The definition uses the concept of the direct limit of a system of groups as in [Ser80,§1].

Definition 7
The Steinberg groupĜ associated to the group G is the direct limit of the system of groups U α and U [α,β] with their natural inclusions for all roots α and all prenilpotent pairs of roots (α, β).
There is an improved characterisation of the Steinberg group due to Caprace, where we can restrict ourselves to non-nested pairs of roots.
Theorem 8 (Proposition 3.9 in [Cap07]) Assume that the group G is of 2-spherical type and satisfies condition (Co * ). Then the Steinberg groupĜ is isomorphic to the direct limit of the system of groups U α and U [α,β] with their natural inclusions for all roots α and all prenilpotent, non-nested pairs of roots (α, β).
There is a similar characterisation for the groups U c .

Theorem 9 (Theorem 3.6 in [Cap07])
Assume that the group G is of 2-spherical type and satisfies condition (Co * ). For any chamber c ∈ Σ, the group U c is the direct limit of the system of groups U α and U [α,β] for all roots α containing c and all prenilpotent, non-nested pairs of roots (α, β) containing c, respectively.
We need this statement also for the groups U s .

Lemma 10 Assume again that G is of 2-spherical type and satisfies condition (Co * ).
For any co-spherical simplex s ∈ Σ, the group U s is the direct limit of the system of groups U α and U [α,β] for all roots α where R s ⊂ α and all prenilpotent, non-nested pairs of roots (α, β) with R s ⊂ (α ∩ β), respectively.
Proof If s = c is a chamber, this is Theorem 9. Now let s ∈ Σ be any co-spherical simplex and fix a chamber c that contains s. Consider the following systems of groups with the canonical inclusions. Since U s ⊆ U c , by the universal property of lim − → U s , we obtain a homomorphism which is the identity on all groups U ∈ U s . We have the following diagram: where the bottom row is surjective since U s = U : U ∈ U s . It remains to show that ϕ is injective. Then the diagonal map is an isomorphism onto the image of U s in U c . It is well known that the direct limit of a system of groups can be constructed explicitly as the free product of the involved groups modulo the relations coming from the inclusions in the system of groups, see [Ser80,§1]. Assume that ϕ is not injective, and fix 1 = g ∈ lim − → U s with ϕ(g) = 1. By the above description, the element g can be written as a word g = g 1 · · · g n with g i ∈ U i ∈ U s and with n > 0 minimal.
The group V is either a root group or a root group interval. In both cases, there is a prenilpotent, non-nested pair of roots (α, β) with c ∈ α ∩ β such that ϕ(g i ) and ϕ(g i+1 ) are contained in U [α,β] .
But observe that ϕ(g i ) and ϕ(g i+1 ) are contained in U s , so by Lemma 6, there is a group U ∈ U s such that Hence the product g i g i+1 can already be formed in the group lim − → U s and g 1 · · · g i−1 (g i g i+1 )g i+2 · · · g n is a representation of g as a word of shorter length in lim − → U s , which contradicts the minimality of n. So ϕ is injective.
All of these characterisations allow us to give a different description of the Steinberg group.
Proposition 11 Assume that G is 2-spherical and satisfies condition (Co * ). The direct limitG of the system of groups U s for all co-spherical simplices s ∈ Σ with respect to their natural inclusions is isomorphic to the Steinberg groupĜ.
Proof The proof will be divided into two steps. In the first step, we construct a map G →G, the inverse map is constructed in the second step.
Step 1: Of course, U α ≤ U s for any co-spherical simplex s such that R s ⊂ α. So we have an inclusion U α → U s for any such simplex s. Roots are connected and chambers and panels are always co-spherical. So for any two co-spherical simplices s, s with R s , R s ⊂ α, there is a sequence of co-spherical simplices s 1 , . . . , s n such that and the group U α is contained in all of these. Hence the inclusion U α →G, induced by any inclusion U α → U s for any co-spherical simplex s with R s ⊂ α, does not depend on the choice of s. A similar argument works for the inclusions U [α,β] → U s for any co-spherical simplex s such that R s ⊂ (α ∩ β). By the universal property of the colimit, there is hence a canonical homomorphismĜ →G induced by these inclusions.
Step 2: By Lemma 10, every group U s can be written as a direct limit of a subsystem of the direct system forĜ, hence there is a canonical homomorphism U s →Ĝ. Since these homomorphisms are all induced by the inclusions of root groups, they are compatible with the direct system forG. So there is a canonical epimorphismG →Ĝ.
A closer inspection yields that the composition of these maps is by construction trivial on the root groups U α , hence they invert each other.
The importance of the Steinberg group for this paper comes from the following theorem by Caprace, for which we require some additional notation. For every root α write . For every pair of roots (α, β) let X α,β := X α ∩ X β and let Φ α,β be the rank 2 root system generated by α and β. Set Theorem 12 (Caprace, Theorem 3.11 in [Cap07]) Assume that the group G is of 2-spherical type and that its diagram has no isolated nodes. Suppose also that 2. For each pair of roots (α, β) in Π such that the corresponding Coxeter group elements s α and s β do not commute, the Steinberg groupX α,β is the universal central extension of X α,β .
ThenĜ is the universal central extension of the little projective group G † .
Condition (Co * ) is implied by the first condition, see the proof of Theorem 3.11 in [Cap07]. Furthermore, note that the second condition is essentially a condition on Moufang polygons. There is a complete classification of Moufang polygons satisfying this condition, see [dMT09], stating that it is almost always valid except in the case of small fields. Using [TW02, 33.10-33.17], one can calculate that the first condition is also always fulfilled except for small fields.

Homotopy groups of Wagoner complexes
There are various results to calculate the fundamental groups of complexes formed out of group cosets, for instance by Soulé, Brown and Abels-Holz. An equivalent point of view is to consider a group acting on a simplicial complex and to construct a presentation using stabilisers and the fundamental group of the quotient complex. We require a very simple form of a result of this type. Unfortunately, we could not find such a simple formulation in the literature, so we include a precise statement of this result here. We then use the modern approach of complexes of groups as in [BH99] to give a simple proof.
Theorem 13 (II.12.20 in [BH99]) Let X be a finite-dimensional simplicial complex and let a group G act on X simplicially. Let Y ≤ X be a subcomplex which is a strict fundamental domain for the action, that is, the complex Y intersects each G-orbit in exactly one point.
We assume that Y is connected and simply connected, and we write G 0 = G s : s ∈ Y for the subgroup generated by the stabilisers of simplices of Y . Then where the direct limit is taken over the system of stabilisers (G s ) s∈Y with the natural inclusions.
We will use the language of simple complexes of groups and of stratified spaces in the proof. Since the definition of these objects would enlarge this paper unnecessarily, we will try to treat these concepts in a 'black box' fashion. All definitions and all the results we use can be found in [BH99, Chapter II.12].
Proof All references given in this proof and all missing definitions are contained in Chapter II.12 in [BH99]. The main result we need for this proof is Proposition 12.20. We have adopted the notation of this proposition here. In the following, we will construct the objects appearing there and we will relate our situation to the situation in the proposition.
The geometric realisation |X| is a stratified simplicial complex in the sense of Definition 12.1 and the discussion thereafter, where the strata are given by the G-images of closed simplices of Y . The set of strata Q for the subcomplex |Y | is given by the closed simplices in Y .
The group G then acts by strata preserving morphisms on X, and we construct the associated complex of groups G(Q) as in 12.15. By Proposition 12.20(2), we then have π 0 (|X|) ∼ = G/G 0 . If we consider the G 0 -action on the connected component G 0 · Y ⊆ X, we obtain by Proposition 12.20(4) that This can be used to calculate the number of connected components and the fundamental group of the geometric realisation of the Wagoner complex |W(G)|.

Theorem 14
Assume that G is a group of Kac-Moody type satisfying condition (Co * ) and such that the associated root system is 2-spherical. Then we have If G is of non-spherical type or of rank at least 3, we have Proof Consider the action of G on the Wagoner complex W(G). The stabiliser of a vertex (U s ) is obviously the group U s itself, the stabiliser of a flag is given by the smallest group U s 1 . The group G 0 = U s : s ∈ Σ is equal to the little projective group G † . The direct limit of all stabilisers of simplices in A with the natural inclusions is hence the groupG. A strict fundamental domain for the action on |W(G)| is given by the geometric realisation of the standard apartment |A|. If G is of spherical type of rank n, then |A| is homotopy equivalent to a (n − 1)-sphere, and hence connected and simply connected if n ≥ 3.
If G is of non-spherical type, then |A| is homeomorphic to the Davis realisation of the Coxeter system associated to G as defined in [Dav08, Chapter 7], which is contractible by Theorem 8.2.13 in [Dav08].
So we can apply Theorem 13 and use that we haveG ∼ =Ĝ by Proposition 11.

The connection to algebraic K-theory
Wagoner complexes were originally introduced by Wagoner in [Wag73] to provide a definition of higher algebraic K-theory. He then proved the following theorem, which is a special case of our result only if R is a division ring.
Theorem 16 (Wagoner, Proposition 2 in [Wag73]) Let R be a ring. If n ≥ 2 then If n ≥ 3, then In subsequent papers [AKW73] and [AKW77], Anderson, Karoubi and Wagoner and in a final paper [Wag78] Wagoner proved that Wagoner's definition of higher algebraic K-theory coincides with Quillen's definition of K-theory in the sense that This leads naturally to the question whether for any group with rank(G) i. This question would be interesting for hermitian K-theory, for example, by choosing the group G to be a symplectic or orthogonal group. Unfortunately, we do not know whether these groups are isomorphic.

Affine Wagoner complexes
For completeness, we also include the following construction, imitating the corresponding construction in [Wag75]. Let G be a group with a spherical root datum with discrete valuation, compare [Wei09, Chapter 3]. Let ∆ be the associated affine building and let Σ be a fixed apartment of ∆. We will denote the set of spherical roots by Φ and the half-apartments of Σ by H α,k for α ∈ Φ, k ∈ Z with associated walls ∂H α,k .
Definition 17 Let s be a simplex in Σ and let n ≥ 1. The n-cell around s is defined to be the set C n (s) := α∈Φ, k∈Z Rs⊂H α,nk Definition 18 For any n ≥ 1 and any simplex s ∈ Σ, we write U n s := U α,k : C n (s) ⊂ H α,k , α ∈ Φ, k ∈ Z .
Analogously to Definition 2, for any n ≥ 1, we define the affine Wagoner complex W n (G) to be the flag complex over the partially ordered set of cosets {gU n s : s ∈ Σ, g ∈ G}.
For every fixed n, we have by construction s ⊆ s implying that U n s ≤ U n s . There are also simplicial epimorphisms from W n (G) onto the barycentric subdivisions of certain 'coarsenings' of the affine building ∆. For buildings of typeÃ n , this can be found in [Wag75,§5].
Remark As above, we define the standard apartment of the Wagoner complex by A n = Flag({gU n s : g ∈ G, s ∈ Σ}).
Obviously, the apartment A n is isomorphic to the flag complex over the set of n-cells {C n (s) : s ∈ Σ}. It is not hard to see that this complex is in turn isomorphic to the complex associated to the affine reflection group generated by the reflections at the walls ∂H α,nk for α ∈ Φ and k ∈ Z. In particular, by rescaling by 1 n , this complex is isomorphic to the Coxeter complex Σ, so A n is contractible.
In analogy to the construction above, we denote the direct limit of the system of groups {U n s : s ∈ Σ} with their canonical inclusions byG n . By the same argumentation as in the proof of Theorem 14, using that A n is contractible, we obtain Proposition 19 For any n ≥ 1, we have π 1 (|W n (G)|) ∼ = ker(G n G † ) Construction If n divides m, then C n (s) ⊆ C m (s) and hence U n s ⊇ U m s for any s ∈ Σ. Consequently, the map p m n : W m (G) → W n (G) induced by gU m s → gU n s is well-defined. We obtain an inverse system of the simplicial complexes W n (G) with the maps p m n for n dividing m. This induces an inverse system on the fundamental groups π 1 (|W n (G)|).

Definition 20
The inverse limit of this system is denoted by π aff 1 (G) := lim ← − n π 1 (|W n (G)|) and is called the affine fundamental group of G.
The following theorem was already proved by Wagoner in [Wag75, §2] for G = SL k (F) over a local field F in a very different way.
Theorem 21 Let W(G) be the Wagoner complex associated to the spherical building at infinity. Assume that the rank of the spherical building is at least three. Then, for every n ≥ 1, there is a homomorphism π 1 (|W(G)|) → π 1 (|W n (G)|) that is compatible with the inverse system above. In particular, we obtain a homomorphism π 1 (|W(G)|) → π aff 1 (G).