FUNDAMENTAL GROUPS OF SPLIT REAL KAC-MOODY GROUPS AND GENERALIZED REAL FLAG MANIFOLDS

We determine the fundamental groups of symmetrizable algebraically simply connected split real Kac-Moody groups endowed with the Kac-Peterson topology. In analogy to the finite-dimensional situation, because of the Iwasawa decomposition G = KAU+, the embedding K ,↪ G is a weak homotopy equivalence, in particular π1(G) = π1(K). It thus suffices to determine π1(K), which we achieve by investigating the fundamental groups of generalized ag varieties. Our results apply in all cases in which the Bruhat decomposition of the generalized ag variety is a CW decomposition- in particular, we cover the complete symmetrizable situation; furthermore, the results concerning only the structure of π1(K) actually also hold in the nonsymmetrizable two-spherical case.


Introduction
The structure of maximal compact subgroups in semisimple Lie groups was investigated by Cartan and, later, Mostow. In [30], Mostow gives a new proof of a Cartan's theorem stating that a connected semisimple Lie group G is a topological product of a maximal compact subgroup K and a Euclidean space, implying in particular that G and K have isomorphic fundamental groups. Subsequent case-by-case analysis provided the isomorphism types of these maximal compact subgroups -which in the split real situation turn out to be all classical -and their fundamental groups. Tables of the DOI: 10.1007/S00031-022-09719-7 * Supported by DFG grant KO 4323/11. * * Partially supported by DFG grant KO 4323/11. Isomorphism types of π 1 (G(Π)) for the spherical Dynkin diagrams. 1 Π Π adm coloured by γ π 1 (G(Π)) Dynkin diagram LaTeX styles kindly provided by Max Horn at [18].
Isomorphism types of π 1 (G(Π)) for selected indefinite Dynkin diagrams. 2 Π Π adm coloured by γ π 1 (G(Π)) In [9, Sect. 16], the group Spin(Π, κ) -where κ denotes a so-called admissible colouring of the vertices of Π -is defined as the canonical universal enveloping group of a Spin(2)-amalgam A(Π, Spin(2)) = { G ij , φ i ij | i = j ∈ I} where the isomorphism type of G ij depends on the (i, j)-and (j, i)-entries of the Cartan matrix of Π, as well as the values of κ on the corresponding vertices.
It is shown in [9,Sect. 17] that there exists a finite central extension Spin(Π, κ) → K(Π) which implies that the subspace topology on K(Π) inherited from the Kac-Peterson topology on G(Π) defines a unique topology on Spin(Π, κ) that turns the central extension into a covering map. The resulting group topology on Spin(Π, κ) is called the Kac-Peterson topology on Spin(Π, κ).
In the simply-laced case, there is a unique nontrivial admissible colouring κ and the corresponding group Spin(Π) := Spin(Π, κ) double-covers K as shown in [9]. We prove here that in the simply-laced case Spin(Π) is simply connected which then implies that π 1 (K) ∼ = C 2 .
A key to the proof both in the simply-laced and in the general case is the computation of the fundamental groups of generalized flag varieties -that is, spaces of the form G/P J for a parabolic subgroup P J of G corresponding to an index subset J ⊆ I. It turns out that the aforementioned space Spin(Π)/Spin(3) is a universal covering space of an appropriately chosen generalized flag variety. In general, we prove the following theorem.
Theorem. Let Π be an irreducible Dynkin diagram such that the Bruhat decomposition of G(Π) provides a CW decomposition (i.e., such that the conclusion of Proposition 3.7 holds), let I be the index set of the Dynkin diagram, let J ⊆ I, and let P J be a parabolic of type J. Then a presentation of π 1 (G/P J ) is given by In particular, this statement holds in the 2-spherical and in the symmetrizable case.
We refer to [40] for the analog result in the finite-dimensional situation. In order to determine π 1 (K) in the general case, we compute subgroups of π 1 (K) corresponding to the index sets of connected components of Π adm using the above theorem and covering maps of the type K/K J → K/(K ∩ T )K J where T is a maximal split torus of G(Π) and K J is the subgroup of fixed points of a Levi factor of P J with both T and P J invariant under the Cartan-Chevalley involution. We then show that π 1 (K) is a direct product of appropriately chosen such subgroups.
In a very similar way, the fundamental group of Spin(Π, κ) is determined, establishing the following theorem.
Theorem. Let Π be an irreducible Dynkin diagram such that the Bruhat decomposition of G(Π) provides a CW decomposition (i.e., such that the conclusion of Proposition 3.5 holds). Let n(g) be the number of connected components of Π adm of colour g. Let n(b, κ) be the number of connected components of Π adm on which κ takes the value 1 and which have colour b. Then π 1 (Spin(Π, κ)) ∼ = Z n(g) × C n(b,κ) 2 .
In particular, this statement holds in the 2-spherical and in the symmetrizable case.
Acknowledgements. The research leading to this article has been partially funded by DFG via the project KO 4323/11. The authors thank J. Grüning and two anonymous referees for various helpful remarks on earlier versions of this article.
Since a ij ∈ R, one can analogously define a quadruple (g R (A), h R (A), Ψ,Ψ) where g R (A) is a real Lie algebra that embeds naturally into g C (A) as the real form given by the involution induced by complex conjugation. One refers to g R (A) as the split real Kac-Moody algebra associated with R and to h R (A) as its standard split Cartan subalgebra.
Let Q ⊆ h R (A) * be the group generated by Ψ and Q ± , the subsemigroups generated by ±Ψ, respectively. For k ∈ {C, R} and α ∈ h k (A) * define the root space The set ∆ decomposes as a disjoint union into the subsets ∆ ± := ∆ ∩ Q ± called positive (respectively negative) roots. The restriction of the Lie bracket on g R (A) to u ± := α∈∆± g k α turns u + and u − into Lie subalgebras of g R (A).
For i = {1, . . . , n} define the fundamental root reflection σ i ∈ GL(h R (A) * ) by Then the Weyl group of g R (A) is defined as W := σ 1 , . . . , σn ≤ GL(h R (A) * ) and forms a Coxeter system together with the set of fundamental root reflections. Finally, define the set of real roots Φ := W.Ψ ⊆ ∆ and Φ ± := ∆ ± ∩ Φ, the positive (respectively negative) real roots. The construction in [39] of G A (R) (see Definition 1.1) provides a representation of G A (R) on g R (A) by Lie algebra automorphisms, which is denoted by and referred to as the adjoint representation of G A (R). Since the subgroup Ad(G(Π)) of G(Π) under this representation preserves the commutator subalgebra g R (A), one obtains an adjoint representation Ad : for G(Π). The kernels of the adjoint representations of G A (R) and G(Π) are given by the respective centres.
An element X ∈ g R (A) is ad-locally-finite if for every element Y ∈ g R (A) there exists an ad(X)-invariant finite-dimensional subspace W with Y ∈ W . As pointed out in [27, p. 64], this implies that ad(X)| W is a (finite) matrix in some basis of W , so the exponential exp(ad(X)) can be defined in the ususal way. By [39,(KMG5), p. 545] and the uniqueness properties of G A (R) established in [39,Thm. 1], exp(ad(X)) ∈ Ad(G A (R)). Let F g R (A) and F g R (A) be the subsets of ad-locally-finite elements of the respective algebras. The maps exp : F g R (A) → Ad(G A (R)) and exp : F g R (A) → Ad(G(Π)) given by X → exp(ad(X)) can be lifted to exponential functions exp : The intersection T := G(Π) ∩ T C is called the standard split maximal torus of G(Π); again, A R ∩ T is of finite index in T and T contains the centre of G(Π).
The Lie algebra g R (A) admits a unique involution θ which maps e j to f j for all j = 1, . . . , r and acts as −1 on h R (A). There exists a unique involutive automorphism θ : G A (R) → G A (R) such that θ(exp(X)) = exp(θ(X)) for all X ∈ F g R (A) , and this involutive automorphism is called the Cartan-Chevalley involution of G A (R). We denote by K A (R) := G A (R) θ ⊂ G A (R) the fixed point subgroup of this involution and define K(Π) := K A (R) ∩ G(Π).
Let α ∈ Φ be a real root. Then g R α is one-dimensional and consists of ad-locally-finite elements. One can therefore define the root group Uα := exp(g R α ) ⊆ G A (R). Each root group Uα carries a unique Lie group topology such that Uα ∼ = R as topological groups. Root groups corresponding to positive real roots are called positive root groups, root groups corresponding to negative real roots are called negative root groups.
Define the positive (respectively negative) maximal unipotent subgroup U ± of G A (R) as the group generated respectively by the positive and negative root groups. One has U ± ⊆ G(Π). The groups U ± are normalized by T R and intersect T R trivially. In particular, they intersect the centres of G A (R) and G(Π) trivially and hence embed into both Ad(G A (R) and Ad(G(Π)).
One can show that the pair ((Uα) α∈Φ , T ) defines an RGD system for G(Π). For details concerning RGD systems, we refer the reader to [1,Chap. 8].
Recall that the generalized Cartan matrix A is called 2-spherical, if a ij a ji ≤ 3 for all i = j ∈ I; in other words, if the orders of the products σ i σ j are always finite. The generalized Cartan matrix A is symmetrizable if it is the product of a symmetric and a diagonal matrix. These notions are also applied to any and all objects that are derived from A such as the (extended) Weyl group, the Kac-Moody group, their buildings, etc.
Definition and Remark 2.1. The Kac-Peterson topology on G A (R) equals the finest group topology on G A (R) such that the natural embeddings (Uα → G A (R)) α∈Φ and T R → G A (R) are continuous when T R and the root groups Uα are endowed with their Lie group topologies.
The Kac-Peterson topology is kω by [14,Prop. 7.10] and, in particular, Hausdorff. Moreover, for every α ∈ Φ + , it induces the unique connected Lie group topology on Gα and on T R by [14,Cor. 7.16] For more details on the Kac-Peterson topology, see [14,Chap. 7]. The groups G and K are always endowed with the subspace topologies induced by the Kac-Peterson topology on G A (R) and G/B with the quotient topology.
Unless specified more explicitly, the symbol J will always denote an arbitrary subset of the index set I, the symbol Π J the subdiagram of Π corresponding to J, the symbol G J the subgroup G(Π J ) of G, and the symbols K J and B J the intersections G J ∩ K and G J ∩ B, respectively. This is consistent with the notation for the fundamental rank one subgroups: one has G(Π i ) = G i = Gα i . Remark 2.3. Due to the structure theory of RGD systems (cf. [1,Chap. 8], most notably the fact that restricting an RGD system to a subdiagram again yields an RGD system), for each fundamental rank one subgroup G i there exists an (abstract) isomorphism γ i : SL(2, R) → G i with the following properties. Let B SL(2,R) be the group of upper triangular matrices in SL(2, R) and let U ±β denote the canonical root subgroups of SL(2, R). Then, By [14,Cor. 7.16], the restriction of the Kac-Peterson topology to any spherical subgroup H of G coincides with its Lie topology. That is, the groups G i inherit their Lie group topology from the topological Kac-Moody group G. By the classical theory of Lie groups this yields the existence of a diffeomorphism γ i with the desired properties; in particular, γ i is an open map. be the Weyl distance function on G/B, and let l S be the length function that associates to each element the (unique) length of a corresponding reduced expression in S. Let ≤ be the strong Bruhat order on W . Recall that for w 1 , w 2 ∈ W one has w 1 ≤ w 2 if there exist reduced expressions s i1 · · · s i l S (w 1 ) of w 1 and s j1 · · · s j l S (w 2 ) such that the former is a (not necessarily consecutive) substring of the latter. For w ∈ W and a chamber gB ∈ G/B, define and C<w(gB) := C ≤w (gB) \ Cw(gB).
In particular, one has Cw(B) = BwB/B and C ≤σ (B) = B s B/B for σ ∈ S with representative s ∈ W ⊆ G in the extended Weyl group W . A set C ≤σ (gB) is called a σ-panel.
Moreover, for a subset {σ i } i∈J ⊆ S with representatives {s i } i∈J ⊆ W define P J to be the standard parabolic subgroup corresponding to the index set J: that is, Throughout this paper, Cw(gB) and C ≤w (gB) will always be endowed with the subspace topologies induced by G/B. Lemma 2.5. Let σ i = σ j ∈ S. Then the following hold: Proof. Assertions (a) and (b) follow from [1,Rem. 8.51] and [1, Rem. (2) after Thm. 6.56], respectively, and the Iwasawa decomposition 3. The fundamental group of the generalized flag variety G/P J For a moment, let Π be an irreducible simply-laced diagram distinct from A 1 , and let G = G(Π) and K = K(Π) be as in the preceding section. Moreover, let Spin(Π) be the double cover of K(Π) constructed in [9,Lem. 16.18] (see Definition 4.6 below). By construction, any A 2 -subdiagram of Π yields an embedding Spin(3) → Spin(Π) and, since Spin(3) inherits the Lie topology from the Kac-Peterson topology on Spin(Π) by [14,Cor. 7.16], one obtains a locally trivial fibre bundle by [32] (see Proposition A.13). It will turn out in Section 4 below that Spin(Π)/Spin(3) is a universal covering space of the generalized flag variety G/P J where J ⊂ I equals the set consisting of the two types involved in the chosen A 2 -subdiagram. The fundamental group of Spin(Π) then follows from the homotopy exact sequence This motivates our interest in the fundamental group and covering theory of generalized flag varieties G/P J .
Throughout this section, let J ⊆ I, let W J be the subgroup of W generated by {σ i } i∈J , and let W J ⊆ W be a set of representatives of the cosets in W/W J that have minimal length in the coset they define. Lemma 3.4. Let G be 2-spherical or symmetrizable and let w ∈ W J . Then the canonical map ψw is a homeomorphism. [14,Prop. 5.9] plus Corollary B.8), the preceding observations yield that ψw is an injective quotient map and therefore a homeomorphism.
denotes the subgroup of diagonal matrices and U ±α denote the canonical root subgroups of SL(2, R). This implies that C ≤σi (B) is homeomorphic to the building SL(2, R)/B SL(2,R) P 1 (R) S 1 .
, is a relative homeomorphism, i.e., it is a continuous map and its restriction D k+1 \ S k → e is a homeomorphism.
(c) If e ∈ E, then its closure cl e is contained in a finite union of cells in E.
(d) X has the weak topology determined by {cl e | e ∈ E}, i.e., a subset A of X is closed if and only if A ∩ cl e is closed in cl e for each e ∈ E.
For k ∈ N, let Λ k be an index set for the k-dimensional cells, so that X (k) \ X (k−1) = λ∈Λ k e λ and set χ λ := χe λ . This map is called the characteristic map of e λ .
Proposition 3.7. Let G be 2-spherical or symmetrizable. Then for each w ∈ W , the set Proof. The first statement is immediate by [14, Cor. 3.10 and Prop. 5.9] plus Corollary B.8; see also [24, pp. 170-171]. Furthermore, [14,Prop. 5.9] combined with Corollary B.8 states that the Bruhat decomposition of G/B is a CW decomposition. By Lemma 3.4, G/P J is composed of cells that are homeomorphic to cells in G/B, so composing the characteristic maps of the latter cells with the canonical map ψ : G/B → G/P J yields characteristic maps for the cells in G/P J . For the closure-finiteness, let BwP J /P J be a cell in G/P J . Since ψ is continuous and restricts to a homeomorphism BwB/B → BwP J /P J , it maps cl BwB/B surjectively onto cl BwP J /P J . Now, cl BwB/B = x≤w BxB/B, which implies that where the last equality holds since W J ⊆ P J . This proves that cl BwP J /P J is contained in a finite union of cells.
It remains to show that G/P J has the weak topology determined by the cell closures. For w ∈ W and a representative w ∈ W J of minimal length of wW J , one has BwP J /P J = B wP J /P J . Let ew := BwP J /P J = B wP J /P J and e w := BwB/B. Let ew = cl ew = x≤ w BxP J /P J andē w := cl e w = x≤w BxB/B.
Let A be a closed subset of G/P J and let ew, w ∈ W J , be an arbitrary cell. Then Since ψw is a quotient map by Remark 3.3, this implies that A ∩ēw is closed inēw. Now, let A be a subset of G/P J such that A ∩ēw is closed inēw for all w ∈ W J . Since for each w ∈ W one has ew = e w for any minimal-length representative w ∈ W J of wW J , in fact A ∩ēw is closed inēw for all w ∈ W . Therefore, The preceding result combined with the following lemma (which is a consequence of [29, Chap. 7, Thm. 2.1]) will allow us to efficiently compute the fundamental group of a generalized flag variety in Theorem 3.15 below. Lemma 3.8. Let X be a CW complex with only one 0-cell x 0 . For each λ ∈ Λ 2 , let f λ : [0, 1] → S 1 be a loop whose homotopy class generates π 1 (S 1 ) and whose image is a presentation of π 1 (X, x 0 ), where the brackets denote the respective homotopy classes in X (1) .
Next, we study the characteristic maps of the CW decomposition of a generalized flag variety explicitly. Proof. Let {x 0 } := 1 0 ∈ P 1 where P 1 denotes the real projective line, modelled as the subset of one-dimensional subspaces of R 2 . Since each one-dimensional subspace in P 1 \ {x 0 } contains exactly one element in the upper half circle R([0, 1]) · 1 0 while x 0 contains the two boundary points corresponding to R(0) and R(1), one has a surjection from [0, 1] onto P 1 given by t → R(t) · 1 0 which maps (0, 1) bijectively onto Since SL(2, R) acts transitively on the real projective line P 1 with B SL(2,R) being the stabilizer of x 0 := 1 0 , one has a bijective correspondence gB → gx 0 between SL(2, R)/B SL(2,R) and P 1 . This yields the desired surjectivity and bijectivity properties of R. Continuity is clear, as well as the fact that the restriction to the interior is a homeomorphism.
Lemma 3.12. Let G be 2-spherical or symmetrizable. Then the maps defined above are characteristic maps for the following cells: Proof. (a) One has to show that χ i ([0, 1]) ⊆ C ≤σi (B) and that χ i is a continuous map which maps (0, 1) homeomorphically to Cσ i (B). The first assertion is clear, since by Lemma 2.5 one has C ≤σi = G i B/B.
. Hence, s is the unique preimage of kB under χ i . This yields the desired bijectivity property. The continuity properties are clear.
Theorem 3.15. If the Bruhat decomposition satisfies the conclusion of Proposition 3.7, then a presentation of π 1 (G/P J ) is given by In particular, this statement holds in the 2-spherical and the symmetrizable case.
Proof. By Lemma 3.4 and Proposition 3.7, the Bruhat decomposition is a CW decomposition where each cell BwP J /P J has dimension l(w). The characteristic maps of the 1-cells Bs i P J /P J and 2-cells Bs i s j P J /P J are given by the compositions χ i := ψs i • χ i , respectively χ (i,j) := ψs i sj • χ (i,j) (ψs i and ψs i sj denoting the canonical homeomorphisms from Lemma 3.4). Lemma 3.8 gives a presentation of π 1 (G/P J ). The generating elements are given by the homotopy classes x i := [ χ i ] of the characteristic maps of the 1-cells -namely, the cells . This yields the desired generating set as well as the trivial relation To obtain the set of relators, for k = 1, . . . , 4 let ϕ k : Then the concatenation ϕ := ϕ 1 * ϕ 2 * ϕ 3 * ϕ 4 is a loop in the relative boundary ∂([0, 1] × [0, 1]) S 1 which generates its fundamental group. Moreover, for each characteristic map where P J is the unique 0-cell of the CW complex. Therefore, Lemma 3.8 implies that the set of relators is given by where Moreover, Since This proves the assertion. Proof. For each generator x h in the presentation of Theorem 3.15, one has x 2 h = 1. Recall that λ denotes the labelling map I → V of the vertex set of Π. Since Π is connected, one has a minimal path (i 1 , . . . , im = h) λ in Π such that i 1 ∈ J. If m = 1, one has x h = 1 by the presentation above. Let x i1 , . . . , Multiplying these expressions yields x 2 h = 1. Since each generator has order ≤ 2, the relations show that the group is abelian. One concludes that π 1 (G/P J ) ∼ = C 2 n−|J| .

The fundamental groups of G(Π) and Spin(Π, κ)
The Iwasawa decomposition G = KAU + implies that K acts transitively on the generalized flag varieties G/P J . In this section, we describe the generalized flag varieties and suitable covering spaces as coset spaces of K and its various spin covers defined in [9]. This will then allow us to compute the fundamental group of K and its various spin covers via locally trivial fibre bundles and homotopy exact sequences.
Lemma 4.1. The canonical map ψ : K/(K ∩ P J ) → G/P J is a homeomorphism. In particular, there exists a homeomorphism G/P J → K/(K ∩ T )K J .
Proof. Bijectivity follows from the product formula for subgroups since G = KP J . By Lemma 3.2, the map ψ : G/(K ∩ P J ) → G/P J is continuous, so the same holds for its bijective restriction ψ : In order to show that ψ is closed, let P := P J and let P := P J ∩ K. Consider the commutative diagram where ι denotes the canonical embedding and ϕ denotes the canonical map from G/ P to G/P . Since K is closed in G by [8,Sect. 3F], the map ι is closed. By Lemma 3.2, ϕ is open.
Let X P ⊆ K/ P be a closed subset of K/ P and suppose that ψ(X P ) = XP is not closed in G/P . Then the complement C G/P (XP ) is not open in G/P , hence the complement C G/ P (ϕ −1 (XP )) = ϕ −1 (C G/P (XP )) is not open in G/ P . Therefore, For the second claim, since P J = G J B and θ(P J ) ∩ P J = G J T , one has The claim follows.
The key advantage of the description of a generalized flag variety as a K-coset space lies in the fact that K ∩ T is a finite group. It is therefore straightforward to write down covering spaces of generalized flag varieties via the following well-known basic observation from covering theory. This readily applies in our setting.
Lemma 4.2 now shows that ψ is a covering map. Define c(Π, κ) to be the number of connected components of Π adm on which κ takes the value 2. For a subgraph Π adm J of Π adm that is a union of connected components of Π adm let κ J be the corresponding restriction of κ.
Definition 4.5. Let be the colouring γ : V → {r, g, b} of Π adm that to each connected componentΠ adm of Π adm assigns a colour as follows. LetΠ adm be coloured red (denoted by r) if it contains a vertex i λ such that there exists a vertex j λ ∈ V satisfying ε(i, j) = 1 and ε(j, i) = −1; letΠ adm be coloured green (g) if it is not red and consists only of an isolated vertex; and blue (b) else.
We refer to the introduction for a discussion of various examples.
Definition and Remark 4.6. As recalled in the introduction, in [9, Def. 16.16] the spin group Spin(Π, κ) with respect to Π and κ is defined as the universal enveloping group of a particular Spin (2) where the isomorphism type of G ij depends on the (i, j)-and (j, i)-entries of the Cartan matrix of Π as well as the values of κ on the corresponding vertices. The group K(Π) can be regarded as (being uniquely isomorphic to) the universal enveloping group of an SO(2, R)-amalgam {G ij , φ i ij | i = j ∈ I} where each G ij covers G ij via an epimorphism α ij . By [9, Lem. 16.18] there exists a canonical central extension ρ Π,κ : Spin(Π, κ) → K(Π) that makes the following diagram commute for all i = j ∈ I: Here, τ ij and τ ij denote the respective canonical maps into the universal enveloping groups. By [9, Prop. 3.9], one has ker(ρ Π,κ ) = τ ij (ker(α ij )) | i = j ∈ I Spin(Π,κ) .
In particular, this implies that the subspace topology on K(Π) defines a unique topology on Spin(Π) that turns the extension into a covering map. The resulting group topology on Spin(Π, κ) is called the Kac-Peterson topology on Spin(Π, κ).
Before turning to the general case, we will first consider the simply-laced case and formulate and prove the corresponding simplified versions of the main theorems. Proof. From [9] (exact references below) it follows that the kernel of the covering map Spin(Π) → K coincides with the kernel of the covering map Spin(Π ij ) → K ij and is equal to the group Z := {±1 Spin(Π) } (for the definition of −1 Spin(Π) , see below). This is a consequence of the following facts regarding an irreducible simply-laced diagram Π (all referring to [9]): • There is an epimorphism Spin(2) → SO(2, R) with kernel {±1 Spin(2) } (see [Thm. 6.8]).
The claim now follows from Corollary 3.16 and Corollary 4.1.
The following proposition provides our main result in the simply laced case.
By [15,Chap. 4], this yields the homotopy long exact sequence π 4 (Spin(Π)/Spin(Π ij )) → π 3 (Spin(Π ij )) → π 3 (Spin(Π)) → π 3 (Spin(Π)/Spin(Π ij )) → π 2 (Spin(Π ij )) → π 2 (Spin(Π)) → π 2 (Spin(Π)/Spin(Π ij )) Proof. By Theorem 3.15, π 1 (G/B) ∼ = H I where as defined in 4.10. For J ⊆ I, let the set of relators of H J . Let the set of commutators of pairs of generators from different connected components of Π adm . Then Let π H I and π H be the canonical homomorphisms from the free group x i ; i ∈ I to H I and H, respectively. It suffices to show that k l=1 R J l ∪ R c ⊆ ker π H I and R I ⊆ ker π H . It is clear that a relator x i x ε(i,j) j . This implies that x j has order 2 in H Jm , hence x 2 j ∈ R Jm xi;i∈I , the normal closure of R Jm in the free group.
Since R Jm xi;i∈I ⊆ ker π H , one obtains i λ and j λ in different connected components. As above, we can assume that ε(i, j) = −1 and ε(j, i) = 1. Since In particular, this statement holds in the symmetrizable case.
Proof. By Theorem A.15, π 1 (G) ∼ = π 1 (K), so it suffices to prove that π 1 (K) is of the given isomorphism type; note that Theorem A.15 has only been established in the symmetrizable case. Let J ⊆ I. The diagram with all maps being the respective canonical maps, commutes. Since the maps are continuous by Lemma 3.2, one obtains a commutative diagram of induced homomorphisms π 1 (K) π 1 (K/K J ) where p * and q * are injective, because p and q are covering maps (see Lemma 4.3). By Theorem 3.15 and Lemma 4.1, π 1 (K/(K ∩ T )) and π 1 (K/(K ∩ T )K J ) can be identified with H I = x i ; i ∈ I | R I and x i ; i ∈ I | R I ∪ {x j | j ∈ J} , respectively (R I as in (3) in the above proof), where ψ * corresponds to the canonical homomorphism between these groups as the proof of Theorem 3.15 shows. For the index set Jm of a connected component of Π adm , letJm := I \ Jm. Then by Proposition 4.12, Summing up, one obtains a commutative diagram having replaced p * and q * from above with the corresponding monomorphisms. By Lemma 4.3, the covering K/KJ m → K/KJ m (K ∩ T ) has degree 2 n−|Jm| = 2 |Jm| . This implies that Hm := q * (π 1 (K/KJ m )) is a subgroup of H Jm of index 2 |Jm| . The isomorphism type of Hm is uniquely determined by this index and Lemma 4.11. One has if Π adm Jm has colour r, 2Z ∼ = Z, if Π adm Jm has colour g, C 2 if Π adm Jm has colour b.
In particular, this statement holds in the 2-spherical and the symmetrizable case.
Proof. By Since ρ Π,κ is open as a covering map and ϕ is open by Lemma 3.2, it follows from Lemma 4.2 that ρ J Π,κ is a covering map. From here the proof is analogous to the proof of Theorem 4.13, after extending the commutative diagram at the beginning of the latter proof: One obtains that π 1 (Spin(Π, κ)) ∼ = if Π adm Jm has colour r, 2Z ∼ = Z, if Π adm Jm has colour g, C 2 if Π adm Jm has colour b.
Since Π adm Jm is the union of all connected components except Π adm Jm , one has c(Π, κ) − c(ΠJ m , κJ m ) ∈ {0, 1}, depending on whether κ is constant 1 or 2 on Π adm Jm . This implies if Π adm Jm has colour g, C 2 if Π adm Jm has colour b and κ ≡ 1 on Π adm Jm , {1}, if Π adm Jm has colour b and κ ≡ 2 on Π adm Jm .
This proves the assertion. Now all theorems from the introduction have been proved.

A. Maximal unipotent subgroups of Kac-Moody groups and applications to Kac-Moody symmetric spaces
T. HARTNICK, R. KÖHL Throughout this appendix, we fix a symmetrizable generalized Cartan matrix A with underlying diagram Π. We consider the corresponding algebraically simply-connected semisimple split real Kac-Moody group G := G(Π) = [G A (R), G A (R)] as given by Definition 1.1. As in Section 2, we also denote by K A (R) ≤ G A (R) the fixed point subgroup of the Cartan-Chevalley involution θ and set K := K(Π) = K A (R) ∩ G. We equip all of these groups with the restrictions of the Kac-Peterson topology.
The goal of this appendix is to relate the topology of G to the topology of K. Our main result (see Theorem A.15 below) asserts that the inclusion K → G is a weak homotopy equivalence. This implies in particular that π 1 (G) ∼ = π 1 (K) and thus allows the computation of π 1 (G) by the methods presented in the main part of the article.
In the spherical case, the subgroup K < G is even a deformation retract and hence the inclusion K → G is a homotopy equivalence, as a consequence of the topological Iwasawa decomposition of G. This decomposition also implies that the associated Riemannian symmetric space G/K is contractible.
While real Kac-Moody groups also possess an Iwasawa decomposition, it is currently unknown whether this decomposition is topological. To establish our main result, we thus have to work with a certain central quotient G of G, for which the topological Iwasawa decomposition was established in [8]. We will show that the image K of K in G is a strong deformation retract and that the reduced Kac-Moody symmetric space G/K is contractible. Since the finite-dimensional central extension G → G is a Serre fibration by a classical result of Palais [32], this will allow us to deduce the desired result about G and K.
A.1. The topological Iwasawa decomposition Let us denote by Ad : G A (R) → Aut(g R (A)) and Ad : G(Π) → Aut(g R (A)) the adjoint representations of G A (R) and G = G(Π), respectively. We recall from [8] that the quotient map G → Ad(G) factors as where G is uniquely determined by the fact that T := p 1 (T ) ∼ = (R × ) rk(A) is a torus and p 2 has finite kernel. The group G is referred to as the semisimple adjoint quotient of G, and we equip it with the quotient topology with respect to the Kac-Peterson topology on G. We will denote by U ± the positive, respectively negative maximal unipotent subgroup of G(Π) as introduced in Section 2. Also recall from Section 2 that A R := exp(h R (A)) ≤ G A (R) and set A := A R ∩ G.
A more refined statement has been established in [8] for the semisimple adjoint quotient G of G. To state this result, denote by . Equip these groups with their respective quotient topologies and note that p 1 restricts to a bijection between U + and U + .
Theorem A.2 (Topological Iwasawa decomposition, [8,Thm. 3.23]). Multiplication induces homeomorphisms Since A is contractible, in order to show that K is a deformation retract of G it will suffice to show that U + is contracible. We thus need to understand the topology induced by the Kac-Peterson topology on the standard unipotent subgroups.

A.2. The Kac-Peterson topology on U ±
We now turn to the study of the restriction of the Kac-Peterson topology to the standard maximal unipotent subgroups U − and U + . Recall from Section 2 that the Weyl group W is a Coxeter group, so elements of W can be represented by reduced words in the generators s 1 , . . . , sr. Given such a reduced word w = (s i1 , . . . , s ir ) in W with corresponding simple roots α i1 , . . . α ir we define positive roots β 1 , . . . , βr by We then set Uw := U β1 · · · U βr ⊂ U + and define a map It is established in [4, Sect. 5.5, Lem.] that the map µw is a bijection for every reduced word w, and that its image Uw depends only on the Weyl group element represented by w, but not on the chosen reduced expression. Since G A (R) is a topological group, the bijection µw is continuous. In fact, one can show that µw is a homeomorphism. A proof of this fact was sketched in [14,Lem. 7.25]; since openness of the maps µw is crucial for everything that follows, we fill in the details of this sketch here.
Lemma A.3. For every reduced word w the map µw is a homeomorphism onto its image.
Composing this homeomorphism with the map (6) now provides the desired continuous inverse to µw.
To describe the topology on U + we recall that there exist several distinct but related partial orders on W which in different places in the literature are referred to as the Bruhat order on W . In the sequel we will consider the following version; here denotes the length function with respect to the generating set {s 1 , . . . , sr}.
Definition A.4. The weak right Bruhat order on W is the partial order ≤w defined as According to [4, p. 44] we have w 1 ≤w w 2 if and only if there exists a reduced word (r i1 , . . . , r i (w 2 ) ) for w 2 such that w 1 = r i1 · · · r i (w 1 ) .
Recall that for the strong Bruhat order ≤ one has w 1 ≤ w 2 if there exists a reduced word (r i1 , . . . , r im ) for w 2 and a reduced word (r j1 , . . . , r j l ) for w 1 such that (r j1 , . . . , r j l ) is a substring of (r i1 , . . . , r im ) (not necessarily consecutive). By definition, but the converse is not true. An important difference between the weak right Bruhat order and the strong Bruhat order is that (W, ≤) contains a cofinal chain, i.e., a totally ordered subset T ⊂ W such that for every w ∈ W there exists t ∈ T such that w ≤ t, whereas for the weak right Bruhat order, such a cofinal chain does not exist. In fact, given w 1 , w 2 ∈ W there will in general not exist an element w 3 ∈ W with w 1 ≤w w 3 and w 2 ≤w w 3 .
Note that if w 1 ≤w w 2 , then we can choose a reduced word (r i1 , . . . , r i (w 2 ) ) for w 2 such that w 1 = r i1 · · · r i (w 1 ) . Thus if we define β 1 , . . . , β (w2) as above then we have a commuting diagram where the horizontal maps are inclusions, and the vertical maps are homeomorphisms. In particular, we have a continuous inclusion ι w2 w1 : Uw 1 → Uw 2 , hence we may form the colimit lim in the category of topological spaces. We emphasize that in view of the previous remark the system ((Uw) w∈W , (ι w2 w1 ) w1≤ww2 ) is not directed, hence this colimit is not a direct limit.
Proposition A.5. The kω-space U + is given by the colimit both in the category of topological spaces and in the category of kω-topological spaces.
Proof. The corresponding statement in the category of sets is established in [4,Thm. 5.3].
In view of the applications to Kac-Moody symmetric spaces that we have in mind we recall that U ± are subgroups of the commutator subgroup G of G R (A); in particular we can consider their images U ± := p 1 (U ± ) under the map p 1 : G → G from (4). In this context, we will need the following fact.
Proof. By [14,Prop. 7.27] the map T × U + → T U + is a homeomorphism and the kernel of p 1 is contained in T . The latter implies that p 1 restricts to a continuous bijection U + → U + , and the former implies that this bijection is open.

A.3. Dilation structures on U ±
Definition A.7. Let U be a topological group. By a dilation structure on U , we mean a family of maps (Φ t : U → U ) t∈R with the following properties: (a) Each Φ t is a continuous automorphisms of the topological group U .
Remark A.8. Note that if a topological group U admits a dilation structure, then it is in particular contractible. Indeed, if we define Ψ t := Φ t/(t−1) , then is continuous with Ψ 0 = Φ 0 = Id and Ψ 1 = Φ −∞ , hence a contraction to the identity.
Dilation structures on finite-dimensional simply-connected nilpotent Lie groups play a major role in conducting analysis on such groups, see, e.g., [11]. Not every finite-dimensional simply-connected nilpotent Lie group admits a dilation structure, but if U is the unipotent radical of a minimal parabolic subgroup of a semisimple Lie group, then such a dilation structure always exists. The methods of [25] allow one to extend this result to the Kac-Moody setting.
Following [21, §3.12], we define the fundamental chamber of h R (A) as Since the family (α i ) 1≤i≤n is linearly independent, there exists Indeed, by the linear independence of (α i ) 1≤i≤n the solution space for the system of n−1 linear equations ∀2 ≤ i ≤ n : α 1 (x) − α i (x) = 0 has strictly larger dimension than the solution space for the system of n linear equations ∀1 ≤ i ≤ n : α i (x) = 0. We now define a 1-parameter subgroup of A R by a t := exp(tX 0 ) and denote by ϕ t := Ad(a t ) ∈ Aut(u + ) the associated automorphism of the Lie algebra u + = α∈∆+ g k α . Similarly we denote by the restriction of the conjugation-action of a t on G A (R) to U + . Note that if X ∈ u + is ad-locally finite, then Φ t (exp(X)) = exp(ϕ t (X)).
From (1) and the defining property of X 0 one deduces that for every positive root α with height |α| ∀Y ∈ gα : ϕ t (Y ) = e t|α| Y.
Setting Φ −∞ (u) := e for all u ∈ U + , we deduce that the map is continuous and that Φ 0 = Id Uw . Combining this with Proposition A.5, one deduces that the map Φ : is continuous, hence a dilation structure.
Recall that U + is isomorphic to U − under the Cartan-Chevalley involution of G A (R), which maps a t to a −t . Thus if we define Φ − t := ca −t | U − then we obtain the following.
Corollary A. 10. The family (Φ − t ) t∈R defines a dilation structure on U − .
Combining this with Remark A.8 and Proposition A.6, we can record the following.
Corollary A.11. The topological groups U + and U − are contractible. Consequently, the groups U + and U − are contractible.
A.4. Homotopy groups of real-split semisimple Kac-Moody groups Corollary A.12. The subgroup K < G is a deformation retract. In particular, the inclusion i K : K A (R) → G A (R) is a homotopy equivalence and thus induces isomorphisms (i K ) * : πn(K) → πn(G) for all n ≥ 0.
Proof. We have established in Corollary A.11 that U + is contractible, and A is contractible since it is homeomorphic to R rk(A) . The assertion now follows from Theorem A.2.
Since it is currently unknown whether the Iwasawa decomposition of G is also a topological decomposition, the strategy of the above proof can not be applied to G. However, using the following result of Palais [32, Sect. 4.1, Cor.], one can still obtain an isomorphism between the fundamental groups of G and K.
Recall that the kernel of the quotient map G → G is homeomorphic to (R × ) cork(A) . In particular, it has 2 cork(A) connected components, whereas its higher homotopy groups vanish. Applying Proposition A.13 to the diagram of fibrations we thus obtain the following.
Combining this with Corollary A.12 we deduce the following.

A.5. Kac-Moody symmetric spaces and causal contractions
We conclude this appendix with an application to the results obtained so far to Kac-Moody symmetric spaces. By [8], the homogeneous spaces G A (R)/K A (R) and G/K carry the natural structure of topological reflection spaces, and the same holds for their quotients Ad(G A (R))/Ad(K A (R)) and Ad(G)/Ad(K). The topological reflection space X = G/K is called the unreduced Kac-Moody symmetric space of type A, and the topological reflection space X = Ad(G)/Ad(K) = G/K is called the reduced Kac-Moody symmetric space of type A.
Corollary A. 16. The reduced symmetric space X is contractible.
Proof. In view of the topological Iwasawa decomposition, the orbit map at the basepoint o = eK U + × A → X , (u, a) → ua.o is a homeomorphism. Since U + and A are contractible, this implies contractability of X .
The proof of Theorem A.9 can be used to provide an explicit contraction for X U + × A, using the contraction by conjugation with suitable elements of the torus T R on the group U + and the standard contraction on the finite-dimensional real vector space A. It turns out that this contraction has interesting additional properties. Recall from [8,Sect. 7] that the symmetric space X admits future and past boundaries ∆ + and ∆ − that both carry a simplicial structure which turns them in the geometric realizations of the positive and negative halves of the twin building of G A (R). Following [8,Sect. 7], a causal ray is a geodesic ray of X whose parallelity class equals a point in ∆ + and a piecewise geodesic causal curve is the concatenation of a finite set of segments of causal rays that can be parametrized in such a way that the walking direction always points towards the future boundary. Given x, y ∈ X , we say that x causally preceeds y (in symbols x y) if there exists a piecewise geodesic causal curve from x to y.
Since both conjugation by elements of T R and the standard contraction of the vector space A preserve geodesic rays and the future and past boundaries (cf. [8,Sect. 7]), the set of piecewise geodesic causal curves of X , and hence the causal pre-order , are invariant under the given contraction.
Corollary A.17. The reduced symmetric space X is causally contractible, i.e., it admits a contraction that preserves .
B. The Bruhat decomposition is a CW decomposition J. GRÜNING, R. KÖHL Let G be a Kac-Moody group endowed with the Kac-Peterson topology and let T be the standard maximal torus and U + , U − the standard unipotent subgroups. [22, Thm. 4(a)] asserts without proof that the multiplication map is a homeomorphism with respect to the Kac-Peterson topology. In this note, we provide a proof in the symmetrizable case that makes use of this fact in the two-spherical case ( [14,Prop. 7.31]), of the embedding of Kac-Moody groups constructed in [28,Thm. 3.15(2)], and of the fact that the Kac-Peterson topology is kω. Among the various consequences of this result is that the Bruhat decomposition of a symmetrizable topological Kac-Moody group is a CW decomposition.
Recall that a k-space (alternatively, compactly generated space) is a topological space X in which a set C ⊂ X is closed if and only if its intersection C ∩ K with any compact subset K of X is compact. That is, a k-space is a topological space X whose topology is coherent with the family of all compact subspaces of X. A kω-space is a topological space X whose topology is coherent with respect to a countable ascending family of compact subspaces. By (3) of [7] any kω-space is a k-space.
Proposition B.1 ([33, Cor.]). A continuous proper map f : X → Y from a topological space X to a k-space Y is closed. In particular, a continuous injection ι : X → Y into a kω-space Y = n∈N Ym with compact Ym such that for each m ∈ N the pre-image ι −1 (Ym) is also compact is a topological embedding, i.e., it is a homeomorphism onto its image.
Proof. The first statement is exactly [33,Cor.]. The second statement is an immediate consequence of the first, since a kω-space is a k-space in which any compact subset K of Y is contained in some Ym of the ascending family (Ym) m∈N of compact subsets (statement (3) of [7]). Proposition B.3. Let G be a split real Kac-Moody group. Then the Kac-Peterson topology τ KP on G equals the finest group topology τ MB on G such that the embeddings of the maximal bounded subgroups, each endowed with its Lie group topology, are continuous.
Proof. By [26,Lem. 4.3], the Kac-Peterson topology τ KP on G induces the Lie group topology on its maximal bounded subgroups. A fundamental SL 2 (R) is bounded and, in particular, embeds as a closed subgroup into a maximal bounded subgroup. Therefore, its subspace topology equals its Lie group topology; by [14,Prop. 7.21] the topology τ KP equals the finest group topology on G such that the embeddings of the fundamental SL 2 (R) Lie subgroups is continuous, whence τ KP is finer than or equal to the final group topology τ MB with respect to the embedded maximal bounded subgroups. Again, since by [26,Lem. 4.3] the Kac-Peterson topology on G induces the Lie group topology on its maximal bounded subgroups, the two described topologies actually coincide.
Corollary B.4. Let G be a split real Kac-Moody group endowed with the Kac-Peterson topology and let (G i ) i∈I be a finite family of Lie-subgroups of G such that each fundamental SL 2 (R) is contained in at least one of the G i . Then the Kac-Peterson topology on G equals the finest group topology on G such that the embeddings of the (G i ) i , each endowed with its Lie group topology, are continuous.