Right-angled Coxeter quandles and polyhedral products

RIGHT-ANGLED COXETER QUANDLES AND POLYHEDRAL PRODUCTS

arXiv:1706.06209v1 [math.AT] 19 Jun 2017

DAISUKE KISHIMOTO

Abstract. A Coxeter group W defines a quandle XW which yields a group Ad(XW ), and it was recently proved by Akita that Ad(XW ) is an intermediate object between W and the associated Artin group. It is proved that the right-angularity of W is homotopically inherited by Ad(XW ) in terms of polyhedral products, and applications of this result are given.

1. Introduction A quandle is a set with a binary operation satisfying three conditions which are analogous to Reidemeister moves of all types in knot theory. These three conditions are also thought to axiomatize conjugation in a group since every identity satisfied by conjugation in a group is derived from the three conditions. In particular, there are mutually adjoint constructions of quandles and groups through conjugation: (1) we associate a quandle to a conjugation closed subset of a group; (2) we associate a group to a quandle. Thus by composing these two constructions, we can construct a new group out of a given conjugation closed subset of a group. In this paper, we study a new group Ad(XW ) constructed from a set of reflections of a Coxeter system (W, S) which is closed under conjugation. The group Ad(XW ) has been studied in several directions. For example, Nosaka [N] has studied its cohomological properties in connection with low dimensional topology, and its group structure has been studied in connection with pointed Hopf algebras [AFGV, E] in the special case that W is the symmetric group Σn . In particular, Andruskiewitsch, Fantino, Garc´ıa, and Vendramin [AFGV] proved by a direct calculation that the natural projection Bn → Σn from the braid group Bn factors as the composite of surjections Bn → Ad(XΣn ) → Σn so that Ad(XΣn ) is an intermediate object between Σn and Bn . Recently, Akita [A] proved that Ad(XW ) has the same property for an arbitrary Coxeter system (W, S). Theorem 1.1. For an arbitrary Coxeter system (W, S), the natural projection AW → W factors as the composite of surjections AW → Ad(XW ) → W. Thus Ad(XW ) is an intermediate object between W and AW for a general Coxeter system (W, S), so we here pose the following problem. Problem 1.2. How are properties of a Coxeter system (W, S) and the associated Artin group AW inherited by Ad(XW )? 1

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DAISUKE KISHIMOTO

In this paper, we consider the property of a Coxeter system (W, S) called the right-angularity for this problem. We first characterize the right-angularity of (W, S) homotopically in terms of spaces called polyhedral products, and then prove that Ad(XW ) satisfies the same homotopical condition when the underlying Coxeter system (W, S) is right-angled so that Ad(XW ) inherits the right-angularity of (W, S) homotopically. We give applications of this result and compute the mod 2 cohomology of Ad(XW ) when (W, S) is right-angled. We also discuss further problems on the group Ad(XW ) and related objects. We will assume that spaces have the homotopy types of CW-complexes and non-degenerate basepoints. The author would like to thank Toshiyuki Akita for explaining his work [A] and useful comments. He also would like to thank Takefumi Nosaka for advices. 2. Coxeter quandles and Akita’s results 2.1. Coxeter systems. A pair (W, S) of a group W and a finite set S is called a Coxeter system if it is equipped with a map m : S × S → {1, 2, . . . , ∞} satisfying the following three conditions: (1) m(s, t) = m(t, s); (2) m(s, t) = 1 if and only if s = t; (3) the group W has a presentation W = hs ∈ S | (st)m(s,t) = 1 for m(s, t) < ∞i. Hereafter, (W, S) will denote a Coxeter system. Our reference of Coxeter groups is [D]. The Artin group associated with a Coxeter system (W, S) is defined by AW = has (s ∈ S) | as at as · · · = at as at · · · for 1 < m(s, t) < ∞i. | {z } | {z } m(s,t)

m(t,s)

Then the Coxeter group W is obtained by adding the relations a2s = 1 to AW and replacing as with s for s ∈ S so that there is a natural projection π : AW → W,

as 7→ s.

Example 2.1. The symmetric group Σn will be regarded as a Coxeter group with the generating set s1 , . . . , sn−1 , where si is the transposition of i and i+1 for i = 1, . . . , n−1. Then its associated Artin group is the braid group Bn . 2.2. Coxeter quandles. Recall that a quandle X is a set equipped with a binary operation ∗ : X × X → X satisfying the following three conditions: (1) x ∗ x = x; (2) (x ∗ y) ∗ z = (x ∗ z) ∗ (y ∗ z);


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(3) the map X → X, x 7→ x ∗ w is bijective for any w ∈ X. We consider the following two mutually adjoint constructions producing groups from quandles and vice versa. To a quandle X there is associated a group (2.1)

Ad(X) = hex (x ∈ X) | ex∗y = e−1 y ex ey i

which is called the adjoint group, and a conjugation closed subset R of a group G is regarded as a quandle by the binary operation ∗ : R × R → R,

g ∗ h = h−1 gh.

Then in particular, from a given conjugation closed subset X of a group one gets a new group Ad(X). and moreover, it is easy to verify that the map φ : Ad(X) → G,

ex 7→ x

is a well-defined surjection. An element of the form w −1 sw for w ∈ W, s ∈ S is called a reflection of W , and we denote by XW the set of all reflections of W . Then XW is a quandle by the above construction and we call it the Coxeter quandle associated with a Coxeter system (W, S). Our object to study in this paper is the adjoint group of the Coxeter quandle Ad(XW ). In [A], the following basic properties of Ad(XW ) are proved. Proposition 2.2 (Akita [A]). (2) The map

(1) Ad(XW ) is generated by es for s ∈ S. Φ : AW → Ad(XW ),

as 7→ es

is a well-defined homomorphism. We here note that the precise statement of Theorem 1.1 is that the projection π : AW → W is decomposed as the composite of the surjections Φ

φ

AW − → Ad(XW ) − → W. 2.3. Akita’s results. Let Gab be the abelianization of a group G and ab : G → Gab be the abelianization map. It is known that the abelianization of a Coxeter group W is a product of Z/2, and we define the integer c(W ) to be its rank. Andruskiewitsch, Fantino, Garc´ıa, and Vendramin [AFGV] determined the kernel of the map φ : Ad(XΣn ) → Σn , and Eisermann [E] also determined the abelianization Ad(XΣn )ab and the commutator subgroup of Ad(XΣn ). All proofs of these results are direct calculations which cannot be applied to a general Coxeter system (W, S). Recently, Akita [A] generalized these results to an arbitrary Coxeter system (W, S) by using group cohomology. Proposition 2.3 (Akita [A]).

(1) There is an isomorphism Ad(XW )ab ∼ = Zc(W ) .

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DAISUKE KISHIMOTO

(2) The map φ : Ad(XW ) → W induces an isomorphism ∼ =

→ [W, W ]. [Ad(XW ), Ad(XW )] − Lemma 2.4. Suppose there is a commutative square of groups G1

f1

// H1

g

G2

h f2

// H2

such that f1 , f2 are surjective. Then the square is a pullback if and only if the natural map Ker f1 → Ker f2 is an isomorphism. f2

h

Proof. Let G be the pullback of the triad G2 − → H2 ← − H1 , that is, G = {(x, y) ∈ G2 × H1 | f2 (x) = h(y)}. We aim to prove that the map e : G1 → G given by e(z) = (g(z), f1 (z)) for z ∈ G1 is an isomorhism if and only if the natural map Ker f1 → Ker f2 is an isomorphism. Since f1 , f2 are surjective, there is a commutative diagram of groups with exact rows 1

// Ker π

// G

1

// Ker f1

// G1

1

// Ker f2

// G2

π

// H1

// 1

// H1

// 1

e f1

g

h f2

// H2

// 1

where π is the projection. By definition, we have Ker π = {(x, 1) ∈ G2 × H1 | f2 (x) = 1}, so the composite of the natural maps Ker π → Ker f1 → Ker f2 is an isomorphism. Then Ker π → Ker f1 is an isomorphism if and only if so is Ker f1 → Ker f2 . On the other hand, e is an isomorphism if and only if so is Ker π → Ker f1 . Thus the proof is completed. We combine the two properties of Proposition 2.3 to obtain the following. Theorem 2.5. The commutative square Ad(XW )

ab

// Zc(W )

ab

// (Z/2)c(W )

proj

φ

W is a pullback.


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Proof. Consider the commutative diagram with exact rows 1

// [Ad(XW ), Ad(XW )]

// Ad(XW )

// W

ab

// Ad(XW )ab

// 1

ab

// Wab

// 1.

φ

1

// [W, W ]

Since the map φ is surjective, the right vertical map is surjective too. Then by Proposition 2.3 (1), we get a commutative square in the statement, and by Proposition 2.3 (2) and Lemma 2.4, this square turns out to be a pullback, completing the proof. We will give a homotopical translation of Theorem 2.5. For a homomorphism of groups f : G → H, we denote the induced map between the classifying spaces BG → BH by the same symbol f . Lemma 2.6. Suppose there is a commutative square of groups G1 G2

f1

f2

// H1 // H2

such that f1 , f2 are surjective. Then the square is a pullback if and only if the induced homotopy commutative square BG1 BG2

f1

f2

// BH1 // BH2

is a homotopy commutative square. Proof. Since f1 , f2 are surjective, there is a homotopy commutative diagram BKer f1

// BG1

BKer f2

// BG2

f1

f2

// BH1 // BH2

such that the rows are homotopy fibrations. Then the right square is a homotopy pullback if and only if the induced map BKer f1 → BKer f2 is a homotopy equivalence. Obviously, the induced map BKer f1 → BKer f2 is a homotopy equivalence if and only if the induced homomorphism Ker f1 → Ker f2 is an isomorphism. Thus by Lemma 2.4, the proof is completed.

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DAISUKE KISHIMOTO

Theorem 2.7. There is a homotopy pullback // (S 1 )c(W )

BAd(XW ) φ

incl

BW

// (RP ∞ )c(W ) .

Proof. Combine Theorem 2.5 and Lemma 2.6.

3. Right-angled Coxeter groups and polyhedral products 3.1. Right-angled Coxeter groups. A Coxeter system (W, S) is called right-angled if m(s, t) = 1, 2, ∞ for any s, t ∈ S. We characterize the right-angularity of a Coxeter system (W, S) in terms of graph products of groups. Let G be a group and Γ be a simple graph with the vertex set V . The graph product of GΓ is the quotient of the free product of Gv = G for v ∈ V by the commutator relations [Gu , Gv ] = 1 whenever the vertices u, v are adjacent in Γ. Graph products of groups are natural in the sense that for a homomorphism G → H of groups and a subgraph inclusion Θ → Γ, there are homomorphisms GΓ → H Γ

and GΘ → GΓ .

We regard a set S to be a discrete graph so that the graph product ZS is identified with the free group generated by S. We then set θ : ZS → W, θ˜ : ZS → AW , θ¯: ZS → Ad(XW ) to be the projections induced from the inclusions of generators. Define a graph ΓW for a Coxeter system (W, S) such that its vertex set is S and s, t ∈ S are adjacent in ΓW whenever m(s, t) = 2. Then if (W, S) is right-angled, we have W = hs ∈ S | s2 = 1, [s, t] = 1 for m(s, t) = 2i ∼ = (Z/2)Γ . W

Moreover, taking the generating set S into account, we get: Proposition 3.1. A Coxeter system (W, S) is right-angled if and only if there is an isomorphism W ∼ = (Z/2)ΓW such that the map θ : ZS → W is identified with the projection ZS → (Z/2)ΓW . If a Coxeter system (W, S) is right-angled, we also have AW hs ∈ S | [s, t] = 1 for m(s, t) = 2i ∼ = ZΓW . Then we similarly get: Proposition 3.2. If a Coxeter system (W, S) is right-angled, there is an isomorphism AW ∼ = ZΓ W

such that the map θ˜: ZS → AW is identified with the projection ZS → ZΓW .


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3.2. Polyhedral products. We translate Proposition 3.1 into homotopy theory by using spaces called polyhedral products. Let (X, A) be a pair of spaces and K be an abstract simplicial complex on the vertex set [m]. We associate to σ ∈ K, possibly empty, a subspace D(σ) of X m such that ( X i∈σ (3.1) D(X,A) (σ) = Y1 × · · · × Ym , Yi = A i 6∈ σ. The polyhedral product of (X, A) associated with K is now defined by [ Z(K; (X, A)) = D(X,A) (σ). σ∈K

Although polyhedral products are defined more generally for a sequence of pairs of spaces, a single pair of spaces is enough for our purpose here. We refer to [BBCG, IK] for the basic homotopy theory of polyhedral products. Note that D is regarded as a functor from the face poset of K to the category of topological spaces. In particular, we may write (3.2)

Z(K; (X, A)) = colim D(X,A) .

By definition, polyhedral products are natural in the sense that for a map (X, A) → (Y, B) and a subcomplex inclusion L → K, there are induced maps Z(K; (X, A)) → Z(K; (Y, B)) and Z(L; (X, A)) → Z(K; (X, A)). We now express the classifying space of a graph product of a group. The following lemmas are useful in identifying the homotopy types of polyhedral products. Lemma 3.3 (cf. Farjoun [F, Proposition, pp.180]). Let {Fi → Ei → B}i∈I be a diagram of homotopy fibrations with fixed base. Then hocolim Fi → hocolim Ei → B i∈I

i∈I

is a homotopy fibration. Lemma 3.4. Let (F, F ′ ) → (E, E ′ ) → B be a pair of homotopy fibrations where (F, F ′), (E, E ′ ) are NDR pairs. Then the sequence Z(K; (F, F ′)) → Z(K; (E, E ′ )) → B m is a homotopy fibration. Proof. There is a diagram of homotopy fibrations {D(F,F ′ ) (σ) → D(E,E ′) (σ) → B m }σ∈K over a fixed base, where D(X,A) is as in (3.1). Then by Lemma 3.3 we get a homotopy fibration hocolim D(F,F ′ ) → hocolim D(E,E ′ ) → B m . If (X, A) is an NDR pair, the map D(X,A) (σ) → D(X,A) (τ ) is a cofibration for any σ ⊂ τ ⊂ [m]. Then the natural projection hocolim D(X,A) → colim D(X,A) is a homotopy equivalence, so by (3.2), the proof is completed.

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DAISUKE KISHIMOTO

A simplicial complex is called flag if every collection of pairwise adjacent vertices is its simplex. For a simple graph Γ, we denote by ∆(Γ) the flag complex whose 1-skeleton is Γ. Proposition 3.5. For a non-trivial K(π, 1)-space X, Z(K; (X, ∗)) is a K(π, 1)-space if and only if K is a flag complex. Proof. The if part was proved by J.H.C. Whitehead [W] and we prove the only of part. Suppose K is not a flag complex. Then K includes the boundary of an n-simplex ∂∆n for n ≥ 2 as a full subcomplex, so Z(∂∆n ; (X, ∗)) is a retract of Z(K; (X, ∗)). By Lemma 3.4, the homotopy fiber of the inclusion Z(∂∆n ; (X, ∗)) → X n+1 is homotopy equivalent to Z(∂∆n ; (CΩX, ΩX)), and as in [IK], we have Z(∂∆n ; (CΩX, ΩX)) ≃ Σn ΩX · · ∧ ΩX} | ∧ ·{z n+1

where the right hand side is a wedge of n-dimensional spheres since X is a non-trivial K(π, 1) − space. Then for n ≥ 2, Z(∂∆n ; (CΩX, ΩX)) is not a K(π, 1)-space, so Z(K; (X, ∗)) is not a K(π, 1)-space neither, completing the proof. Then by the van Kampen theorem, we get: Corollary 3.6. For a group G and a simple graph Γ, there is a homotopy equivalence BGΓ ≃ Z(∆(Γ); (BG, ∗)) which is natural with respect to G and Γ. Combining Proposition 3.1 and Corollary 3.6, we obtain the following. Proposition 3.7. A Coxeter system (W, S) is right-angled if and only if there is a homotopy equivalence BW ≃ Z(∆(ΓW ); (RP ∞, ∗)) such that the map θ : BZS → BW is identified with the inclusion Z(S; (S 1, ∗)) → Z(∆(ΓW ); (RP ∞ , ∗)). By Proposition 3.2, we can prove the following quite similarly to Proposition 3.7, so the homotopical form of the right-angularity of (W, S) in Proposition 3.7 may be used to consider its inheritance to groups related to W . Proposition 3.8. If a Coxeter system (W, S) is right-angled, there is a homotopy equivalence BAW ≃ Z(∆(ΓW ); (S 1 , ∗)) such that the map θ˜: ZS → AW is identified with the inclusion Z(S; (S 1, ∗)) → Z(∆(ΓW ); (S 1 , ∗)). For a later use, we calculate the cohomology of W and AW when (W, S) is right-angled. Let K be a simplicial complex on the vertex set S. In a ring generated by elements vs (s ∈ S), the Stanley-Reisner ideal of K is defined by IK = (vs1 · · · vsk | {s1, . . . , sk } 6∈ K).


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Then the Stanley-Reisner ring of K over a commutative ring R is defined by R[K] = R[vs | s ∈ S]/IK ,

|vs | = 2

and its exterior analogue is defined by ΛR (K) = ΛR (vs | s ∈ S)/IK ,

|vs | = 1.

Similarly to the calculation of the cohomology of Z(K; (CP ∞ , ∗)) in [DJ], we can prove: Proposition 3.9. There are natural isomorphisms H ∗ (Z(K; (RP ∞, ∗)); Z/2) ∼ = Z/2[K],

H ∗ (Z(K; (S 1, ∗)); R) ∼ = ΛR (K).

Combining Theorem 3.8 and 3.8 and Proposition 3.9, we obtain: Corollary 3.10. If a Coxeter system (W, S) is right-angled, there are natural isomorphisms H ∗ (W ; Z/2) ∼ = Z/2[∆(ΓW )],

H ∗ (AW ; R) ∼ = ΛR (∆(ΓW )).

4. Main theorem and its applications 4.1. Main theorem. Proposition 4.1. Let F → E → B be a homotopy fibration such that the fiber inclusion is a cofibration. Then the commutative square (4.1)

Z(K; (E, F ))

incl

// E m

Z(K; (B, ∗))

incl

// B m

is a homotopy pullback, where m is the number of vertices of K. Proof. Since we are assuming that spaces have the homotopy types of CW-complexes, it is sufficient to show that the natural map between the homotopy fibers of the horizontal arrows of the square diagram is a homotopy equivalence. Since (E, F ) is an NDR pair and the basepoint of B is non-degenerate, it follows from Lemma 3.4 that the homotopy fibers of the both horizontal arrows of (4.1) are homotopy equivalent to Z(K(CΩB, ΩB)). Moreover, by the construction, the induced maps between these homotopy fibers is identified with the identity map of Z(K(CΩB, ΩB)). Therefore the proof is completed. We now prove the main theorem which indicates a homotopical inheritance of the rightangularity of a Coxeter system (W, S) by the adjoint group Ad(XW ) as in Proposition 3.7 and 3.8. Let I2 be the mapping cylinder of the degree 2 map S 1 → S 1 , and we consider the pair (I2 , S 1 ) such that the inclusion S 1 → I2 ≃ S 1 is of degree 2. Note that we also have an inclusion S 1 → I2 which is a homotopy equivalence.

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DAISUKE KISHIMOTO

Theorem 4.2. If a Coxeter system (W, S) is right-angled, then there is a homotopy equivalence BAd(XW ) ≃ Z(∆(ΓW ); (I2 , S 1 )) such that the map θ¯ : BZS → BAd(XW ) is identified with the inclusion Z(S; (S 1 , ∗)) → ≃ Z(∆(ΓW ); (I2 , S 1 )) induced from a homotopy equivalence S 1 − → I2 . Proof. Suppose a Coxeter system (W, S) is right-angled. Then we have c(W ) = |S|. By Proposition 2.7 and 3.7, there is a homotopy pullback BAd(XW )

// I2|S|

// (RP ∞ )|S| .

∞

Z(∆(ΓW ); (RP , ∗))

Then by applying Proposition 4.1 to the homotopy fibration S 1 → I2 → RP ∞ , we get a homotopy equivalence BAd(XW ) ≃ Z(∆(ΓW ); (I2 , S 1 )). By definition, the map θ¯ : ZS → Ad(XW ) coincides with the canonical map obtained by applying Proposition 2.5 to the commutative diagram ZS θ

W

proj

// Z|S| proj

ab // (Z/2)|S|.

Considering this observation at the space level, we obtain that the map θ¯ : ZS → Ad(XW ) is identified with the inclusion Z(S; (S 1, ∗)) → Z(∆(ΓW ); (I2 , S 1 )) induced from a homotopy ≃ equivalence S 1 − → I2 . Therefore the proof is completed. 4.2. Applications of Theorem 4.2. We first consider a torsion in the adjoint group Ad(XW ). By definition, W always has a torsion whereas one of the biggest problems on Artin groups to show whether they are torsion free or not [P]. Since Ad(XW ) is an intermediate object between W and AW by Theorem 1.1, one may ask: Problem 4.3. Is Ad(XW ) torsion free? Suppose (W, S) is right-angled. Since BAW ≃ Z(∆(ΓW ); (S 1 , ∗)) is finite dimensional by Proposition 3.2, AW is torsion free. So one may expect that Ad(XW ) is also torsion free. Indeed, by Theorem 4.2 BAd(XW ) ≃ Z(∆(ΓW ); (I2 , S 1 )) is finite dimensional, so we get: Corollary 4.4. If a Coxeter system (W, S) is right-angled, then Ad(XW ) is torsion free. We next consider a property of the map Φ : AW → Ad(XW ) when (W, S) is right-angled. We assume (W, S) is right-angled and |S| = m so that we may identify S with [m]. Note that Zm is identified with the graph product Z∆m−1 . Then by Theorem 3.2, the map ab : BAW → BZm is


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identified with the inclusion Z(∆(ΓW ); (S 1 , ∗)) → Z(∆m−1 ; (S 1 , ∗)) = (S 1 )m , so by Proposition 3.9, the map ab : BAW → BZm is the projection ΛR (v1 , . . . , vm ) → ΛR (∆(ΓW )) in cohomology. In particular, since ΛR (∆(ΓW )) is a free R-module, the map ab∗ : H ∗ (Zm ; R) → H ∗ (Ad(XW ); R) has a section as R-modules. On the other hand, there is a commutative diagram ab // c(W ) AW Z Φ

Ad(XW )

ab

// Zc(W )

even when (W, S) is not right-angled. Thus by considering the induced commutative diagram in cohomology, we obtain: Proposition 4.5. If a Coxeter system (W, S) is right-angled, the map Φ∗ : H ∗ (Ad(XW ); R) → H ∗ (AW ; R) has a section as R-modules. We will show that this property of Φ can be recovered by a homotopical property of the induced map Φ : BAW → BAd(XW ). To this end, we recall the natural homotopy decomposition of a suspension of a polyhedral product. Let (X, A) be a pair of spaces and K be a simplicial complex on the vertex set [m]. For any σ ∈ K, we put ( b (X,A) (σ) = Y1 ∧ · · · ∧ Ym , Yi = X i ∈ σ D A i 6∈ σ

and define the polyhedral smash product of (X, A) with respect to K by [ b b (X,A) (σ) (⊂ X ∧n ) Z(K; (X, A)) = D σ∈K

where X ∧n is the smash product of n-copies of X. For a subset I ⊂ [m], we put KI = {σ ∈ K | σ ⊂ I}. Theorem 4.6 (Bahri, Bendersky, Cohen, and Gitler [BBCG]). There is a homotopy equivalence _ b I ; (X, A)) ΣZ(K; (X, A)) ≃ Σ Z(K ∅6=I⊂[m]

which is natural with respect to (X, A).

Proposition 4.5 is recovered by the following homotopical property of the map Φ : AW → Ad(XW ). Theorem 4.7. If a Coxeter system (W, S) is right-angled, the map ΣΦ : ΣBAW → ΣBAd(XW ) has a left homotopy inverse.

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DAISUKE KISHIMOTO

Proof. Put K = ∆(ΓW ). For a non-empty subset I ⊂ S, we have ( |I| I∈K b I ; (S 1 , ∗)) = S Z(K ∗ I 6∈ K

b I ; (I2 , S 1)) = (I2 )∧|I| whenever I ∈ K. Let j : (S 1 , ∗) → (I2 , S 1 ) be the incluand Z(K sion such that j : S 1 → I2 is of degree 1. Then the map j induces a homotopy equivW ≃ 1 b I ; (S 1 , ∗)) − b I ; (I2 , S 1 )) for I ∈ K, so the map b alence Z(K → Z(K ∅6=I⊂S Z(KI ; (S , ∗)) → W 1 b ∅6=I⊂S Z(KI ; (I2 , S )) induced from the map j has a left homotopy inverse. Thus by the naturality of the homotopy decomposition of Theorem 4.6, the map ΣZ(KI ; (S 1 , ∗)) → ΣZ(KI ; (I2 , S 1 )) induced from the map j has a left homotopy inverse. By Theorem 4.2 and Proposition 3.2, this map is identified with ΣΦ : ΣBAW → ΣBAd(XW ), completing the proof. 5. Mod 2 cohomology We determine the mod 2 cohomology of Ad(XW ) when a Coxeter system (W, S) is rightangled. We first observe a property of graph products of groups. For a non-empty subset T ⊂ S, let WT be a subgroup of W generated by T . Suppose (W, S) is right-angled. Then (WT , T ) is a Coxeter system and it is right-angled. Moreover, by sending s ∈ S − T to 1, we get a leftinverse of the inclusion WT → W . Then by Proposition 2.5, we get: Proposition 5.1. If a Coxeter system (W, S) is right-angled and T ⊂ S is non-empty, then φ : Ad(XWT ) → WT is a retract of φ : Ad(XW ) → W . By Theorem 2.7, there is a homotopy fibration (5.1)

(S 1 )c(W ) → BAd(XW ) → BW.

Corollary 5.2. If a Coxeter system (W, S) is right-angled and T is a non-empty subset of S, then the homotopy fibration (S 1 )|T | → BAd(XWT ) → BWT is a retract of (5.1), where c(W ) = |S|. By construction, the map BW → BWT corresponds to the projection Z(∆(ΓW ); (RP ∞ , ∗)) → Z(∆(ΓW )T ; (RP ∞ , ∗)) since ΓWT = (ΓW )T , the induced subgraph on T , and ∆((ΓW )T ) = ∆(ΓW )T . Then by Proposition 3.9, its induced map in mod 2 cohomology is the natural projection Z/2[∆(ΓW )] → Z/2[∆(ΓW )T ] which sends vs to 0 for s ∈ S − T . Suppose that a Coxeter system (W, S) is right-angled. Let us now calculate the mod 2 cohomology of Ad(XW ) by the Serre spectral sequence of the homotopy fibration (5.1). Let ΓW be the complement graph of ΓW . Then by Proposition 3.9 we have H ∗ (BW ; Z/2) = Z/2[vs | s ∈ S]/(vs vt | {s, t} is an edge of ΓW ),

|vs | = 1.

By Theorem 2.7, the homotopy fibration (5.1) is a homotopy pullback of a homotopy fibration 2 (S 1 )c(W ) − → (S 1 )c(W ) → (RP ∞ )c(W ) . Then the local coefficient system of the homotopy fibration


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(5.1) is trivial, so the E2 -term is isomorphic to Z/2[vs | s ∈ S]/(vs vt | {s, t} is an edge of ΓW ) ⊗ Λ(us | s ∈ S) where |us | = 1. Moreover, we have d2 vs = 0 and d2 us = vs2 . Then the kernel of d2 is generated by as,σ = vs uσ−s (s ∈ σ ∈ ∆(ΓW )) where uσ−s = us1 · · · usk for σ − s = {s1 , . . . , sk }. Obviously these generators are permanent cycles and satisfy (5.2)

as,σ at,τ = 0 for {s, t} ∈ ∆(ΓW ) or (σ − s) ∩ (τ − t) 6= ∅

and (5.3)

as,σ1 at,τ1 + as,σ2 at,τ2

for σ1 ∪ τ1 = σ2 ∪ τ2 and σ1 ∩ τ1 = σ2 ∩ τ2 = ∅.

Moreover, the only differential hitting to these generators is X (5.4) d2 u σ = as,s as,σ for σ ∈ ∆(ΓW ). s∈σ

Let IW be an ideal of Z/2[as,σ | s ∈ σ ∈ ∆(ΓW )] generated by as,σ at,τ for {s, t} ∈ ∆(ΓW ) or (σ − s) ∩ (τ − t) 6= ∅, as,σ1 at,τ1 + as,σ2 at,τ2 for σ1 ∪ τ1 = σ2 ∪ τ2 and σ1 ∩ τ1 = σ2 ∩ τ2 = ∅, and P s∈σ as,s as,σ . Then summarizing, we obtain that the E∞ -term is Z/2[as,σ | s ∈ σ ∈ ∆(ΓW )]/IW .

We next consider the extension of the E∞ -term to H ∗ (BAd(XW ); Z/2). For a degree reason, the relation (5.2) holds in H ∗ (BAd(XWσ∪τ ); Z/2), so it also holds in H ∗ (BAd(XW ); Z/2) since H ∗ (BAd(XWσ∪τ ); Z/2) is a retract of H ∗ (BAd(XW ); Z/2). Quite similarly, we see that the relations (5.3) and (5.4) hold in H ∗ (BAd(XW ); Z/2). Thus we finally obtain: Theorem 5.3. Suppose that a Coxeter system (W, S) is right-angled. Then we have H ∗ (BAd(XW ); Z/2) = Z/2[as,σ | s ∈ σ ∈ ∆(ΓW )]/IW where the ideal IW is given as above. References [A] T. Akita, The adjoint group of a Coxeter quandle, arXiv:1702.07104v2. [AFGV] N. Andruskiewitsch, F. Fantino, G. A. Garc´ıa, and L. Vendramin, On Nichols algebras associated to simple racks, Groups, algebras and applications, Contemp. Math., vol. 537, Amer. Math. Soc., Providence, RI, 2011, pp. 31-56. [BBCG] A. Bahri, M. Bendersky, F.R. Cohen, and S. Gitler, The polyhedral product functor: a method of decomposition for moment-angle complexes, arrangements and related spaces, Advances in Math. 225 (2010), 1634-1668. [D] M.W. Davis, The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series 32, Princeton University Press, Princeton, NJ, 2007. [DJ] M.W. Davis and T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke Math. J. 62 (1991), 417-451. [E] M. Eisermann, Quandle coverings and their Galois correspondence, Fund. Math. 225 (2014), no. 1, 103-168.

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DAISUKE KISHIMOTO

[F] E.D. Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics 1622, Springer-Verlag, Berlin, 1996. [IK] K. Iriye and D. Kishimoto, Fat wedge filtrations and decomposition of polyhedral products, to appear in Kyoto J. Math. [N] T. Nosaka, Central extensions of groups and adjoint groups of quandles, arxiv:1505.03077. [P] L. Paris, Lectures on Artin groups and the K(π, 1) conjecture, Groups of exceptional type, Coxeter groups and related geometries, Springer Proc. Math. Stat., vol. 82, Springer, New Delhi, 2014, pp. 239-257. [W] J.H.C. Whitehead, On the asphericity of regions in a 3-sphere, Fund. Math. 32 (1939), no. 1, 149-166. Department of Mathematics, Kyoto University, Kyoto, 606-8502, Japan E-mail address: [email protected]