aa r X i v : . [ m a t h . A T ] F e b Second Mod Homology of Artin Groups T OSHIYUKI A KITA Y E L IU In this paper, we compute the second mod 2 homology of an arbitrary Artin group,without assuming the K ( π,
1) conjecture. The key ingredients are (A) Hopf’sformula for the second integral homology of a group and (B) Howlett’s result onthe second integral homology of Coxeter groups.20F36,20J06; 20F55
An Artin group (or an Artin Tits group) is a finitely presented group with at most onesimple relation between a pair of generators. Examples includes finitely generatedfree abelian groups, free groups of finite rank, Artin’s braid groups with finitely manystrands and right angled Artin groups, etc. Artin groups appear in diverse branchesof mathematics such as singularity theory, low dimensional topology, geometric grouptheory and the theory of hyperplane arrangements, etc.Artin groups are closely related to Coxeter groups. For a Coxeter graph Γ and thecorresponding Coxeter system ( W ( Γ ) , S ), we associate an Artin group A ( Γ ) obtainedby, informally speaking, dropping the relations that each generator has order 2 fromthe standard presentation of W ( Γ ). The symmetric group S n is the Coxeter groupassociated to the Coxeter graph of type A n − and the braid group Br ( n ) is the corre sponding Artin group. The Coxeter group W ( Γ ) can be realized as a reflection groupacting on a convex cone U (called Tits cone) in R n with n = S the rank of W . Let A be the collection of reflection hyperplanes. The complement M ( Γ ) = (int( U ) + √− R ) \ [ H ∈A H ⊗ C admits the free W ( Γ ) action, and the resulting orbit space N ( Γ ) = M ( Γ ) / W ( Γ ) has thefundamental group isomorphic to A ( Γ ) ([vdL83]). The celebrated K ( π,
1) conjecturestates that N ( Γ ) is a K ( A ( Γ ) ,
1) space. See Subsection 2.3 for a list of Γ for which the K ( π,
1) conjecture is proved.
Toshiyuki Akita and Ye Liu
Existing results about (co)homology of Artin groups all focus on particular typesArtin groups, for which the K ( π,
1) conjecture has been proved. There are very fewproperties that can be said for (co)homology of all
Artin groups (except for their firstintegral homology, which are simply abelianizations). In this paper, we compute thesecond mod 2 homology of all
Artin groups, without assuming an affirmative solutionof the K ( π,
1) conjecture. Our main tools are Hopf’s formula on the second homology(or the Schur multiplier) of groups, together with Howlett’s theorem (Theorem 3.2) onthe second integral homology of Coxeter groups. We are inspired by [KS03], wherethe authors computed the second integral homology of the mapping class groups oforiented surfaces using Hopf’s formula.Our main result is the following.
Theorem 1.1
Let A ( Γ ) be theArtin group associated to aCoxeter graph Γ . Then H ( A ( Γ ); Z ) ∼ = Z p ( Γ ) + q ( Γ )2 , where p ( Γ ) and q ( Γ ) are non negative integers associated to Γ ; see Theorem 2.6 fordefinitions.As a corollary, we obtain a sufficient condition that the classifying map c : N ( Γ ) → K ( A ( Γ ) ,
1) induces an isomorphism c ∗ : H ( N ( Γ ); Z ) → H ( A ( Γ ); Z ) . Furthermore, we conclude that the induced homomorphism c ∗ ⊗ id Z : H ( N ( Γ ); Z ) ⊗ Z → H ( A ( Γ ); Z ) ⊗ Z is always an isomorphism. This provides affirmative evidence for the K ( π,
1) conjec ture.A part of contents of this paper is based on the second author’s Ph.D. thesis.
We collect relevant definitions and properties of Coxeter groups and Artin groups. Werefer to [Bou68, Hum90] for Coxeter groups and [Par09, Par14a, Par14b] for Artingroups. econd Mod Homology of Artin Groups Let S be a finite set. A Coxeter matrix over S is a symmetric matrix M = ( m ( s , t )) s , t ∈ S such that m ( s , s ) = s ∈ S and m ( s , t ) = m ( t , s ) ∈ { , , · · · } ∪ {∞} fordistinct s , t ∈ S . It is convenient to represent M by a labeled graph Γ , called the Coxeter graph of M defined as follows: • The vertex set V ( Γ ) = S ; • The edge set E ( Γ ) = {{ s , t } ⊂ S | m ( s , t ) ≥ } ; • The edge { s , t } is labeled by m ( s , t ) if m ( s , t ) ≥ Γ odd be the subgraph of Γ with V ( Γ odd ) = V ( Γ ) and E ( Γ odd ) = {{ s , t } ∈ E ( Γ ) | m ( s , t ) odd } inheriting labels from Γ . By abuse of notations, we frequently regard Γ (hence also Γ odd ) as its underlying 1 dimensional CW complex. Definition 2.1
Let Γ be a Coxeter graph and S its vertex set. The Coxeter system associated to Γ is the pair ( W ( Γ ) , S ), where the Coxeter group W ( Γ ) is defined by thefollowing standard presentation W ( Γ ) = h S | ( st ) m ( s , t ) = , ∀ s , t ∈ S with m ( s , t ) = ∞i . Each generator s ∈ S of W has order 2. For distinct s , t ∈ S , the order of st is precisely m ( s , t ) if m ( s , t ) = ∞ . In case m ( s , t ) = ∞ , the element st has infinite order.However, in this paper, we adopt an equivalent definition. To do so, we first introduce anotation. For two letters s , t and an integer m ≥
2, we shall use the following notationof the word of length m consisting of s and t in an alternating order.( st ) m : = m z }| { sts · · · . For example, ( st ) = st , ( st ) = sts , ( st ) = stst . Definition 2.2
Let Γ be a Coxeter graph and S its vertex set. The Coxeter group associated to Γ is the group defined by the following presentation W ( Γ ) = h S | R W ∪ Q W i . The sets of relations are R W = { R ( s , t ) | m ( s , t ) < ∞} and Q W = { Q ( s ) | s ∈ S } ,where R ( s , t ) : = ( st ) m ( s , t ) ( ts ) − m ( s , t ) and Q ( s ) : = s .Note that since R ( s , t ) = R ( t , s ) − , we may reduce the relation set R W by introducing atotal order on S and put R W : = { R ( s , t ) | m ( s , t ) < ∞ , s < t } . We have the followingpresentation with fewer relations W ( Γ ) = h S | R W ∪ Q W i . Toshiyuki Akita and Ye Liu
We shall omit the reference to Γ if there is no ambiguity. The rank of W is defined tobe S .Let ( W , S ) be a Coxeter system. For a subset T ⊂ S , let W T denote the subgroup of W generated by T , called a parabolic subgroup of W . In particular, W S = W and W ∅ = { } . It is known that ( W T , T ) is the Coxeter system associated to the Coxetergraph Γ T (the full subgraph of Γ spanned by T inheriting labels)(cf. Th´eor`eme 2 inChapter IV of [Bou68]). The Artin group A ( Γ ) associated to a Coxeter graph Γ is obtained from the presentationof W ( Γ ) by dropping the relation set Q W . Definition 2.3
Given a Coxeter graph Γ (hence a Coxeter system ( W , S )), we intro duce a set Σ = { a s | s ∈ S } in one to one correspondence with S . Then the Artinsystem associated to Γ is the pair ( A ( Γ ) , Σ ), where A ( Γ ) is the Artin group of type Γ defined by the following presentation: A ( Γ ) = h Σ | R A i , where R A = { R ( a s , a t ) | m ( s , t ) < ∞} and R ( a s , a t ) = ( a s a t ) m ( s , t ) ( a t a s ) − m ( s , t ) .As in the Coxeter group case, we introduce a total order on S and put R A : = { R ( a s , a t ) | m ( s , t ) < ∞ , s < t } . We have the following presentation with fewer relations A ( Γ ) = h Σ | R A i . There is a canonical projection p : A ( Γ ) → W ( Γ ), a s s ( s ∈ S ), whose kernel iscalled the pure Artin group of type Γ .We say that an Artin group A ( Γ ) is of finite type (or spherical type) if the associatedCoxeter group W ( Γ ) is finite, otherwise A ( Γ ) is of infinite type (or non spherical type). K ( π, conjecture Consider a Coxeter graph Γ and the associated Coxeter system ( W , S ) with rank S = n . Recall that W can be realized as a reflection group acting on a Tits cone U ⊂ R n (see [Par14a]). Let A be the collection of the reflection hyperplanes. Put M ( Γ ) : = (cid:16) int( U ) + √− R n (cid:17) \ [ H ∈A H ⊗ C . econd Mod Homology of Artin Groups Then W acts on M ( Γ ) freely and properly discontinuously. Denote the orbit space by(2–1) N ( Γ ) : = M ( Γ ) / W . It is known that
Theorem 2.4 ([vdL83]) The fundamental group of N ( Γ ) is isomorphic to the Artingroup A ( Γ ).In general, N ( Γ ) is only conjectured to be a classifying space of A ( Γ ). Conjecture 2.5
Let Γ be an arbitrary Coxeter graph, then the orbit space N ( Γ ) is a K ( π,
1) space, hence is aclassifying space of theArtin group A ( Γ ).This conjecture is proved to hold for a few classes of Artin groups. Here is a list ofsuch classes known so far. • Artin groups of finite type ([Del72]). • Artin groups of large type ([Hen85]). • • Artin groups of FC type ([CD95]). • Artin groups of affine types f A n , f C n ([Oko79]). • Artin groups of affine type f B n ([CMS10]). • Artin group A ( Γ ) such that the K ( π,
1) conjecture holds for all A ( Γ T ) where T ⊂ S and Γ T does not contain ∞ labeled edges ([ES10]). N ( Γ ) Clancy and Ellis [CE10] computed the second integral homology of N ( Γ ) using theSalvetti complex for an Artin group. We recall their result and follow their notations.Let us first fix some notations. Let Γ be a Coxeter graph with vertex set S . Define Q ( Γ ) = {{ s , t } ⊂ S | m ( s , t ) is even } and P ( Γ ) = {{ s , t } ⊂ S | m ( s , t ) = } . Write { s , t } ≡ { s ′ , t ′ } if two such pairs in P ( Γ ) satisfy s = s ′ and m ( t , t ′ ) is odd. Thisgenerates an equivalence relation on P ( Γ ), denoted by ∼ . Let P ( Γ ) / ∼ be the set ofequivalence classes. An equivalence class is called a torsion class if it is representedby a pair { s , t } ∈ P ( Γ ) such that there exists a vertex v ∈ S with m ( s , v ) = m ( t , v ) = Toshiyuki Akita and Ye Liu
Theorem 2.6 ([CE10]) Let Γ be aCoxeter graph and N ( Γ ) as in(2–1),then H ( N ( Γ ); Z ) ∼ = Z p ( Γ )2 ⊕ Z q ( Γ ) , where p ( Γ ) : = number of torsion classes in P ( Γ ) / ∼ , q ( Γ ) : = number of non torsion classes in P ( Γ ) / ∼ , q ( Γ ) : = Q ( Γ ) − P ( Γ )) = {{ s , t } ⊂ S | m ( s , t ) ≥ } , q ( Γ ) : = rank H ( Γ odd ; Z ) , q ( Γ ) : = q ( Γ ) + q ( Γ ) + q ( Γ ) . Remark
Note that H ( N ( Γ ); Z ) ∼ = H ( A ( Γ ); Z ) is isomorphic to the abelianization of A ( Γ ), which is a free abelian group with rank equals to rank H ( Γ odd ; Z ), the numberof connected components of Γ odd . The (co)homology of the orbit space N ( Γ ) coincides with that of the Artin group A ( Γ ), provided the K ( π,
1) conjecture for A ( Γ ) holds. There are many results about(co)homology of N ( Γ ) in the literature, for example [DCPSS99, DCPS01, CMS08,CMS10]. The K ( π,
1) conjecture is known to hold in these cases.In this section, nevertheless, we shall work on the second homology of arbitrary
Artingroups, without assuming that the K ( π,
1) conjecture holds. Our main result is thefollowing theorem.
Theorem 3.1
Let Γ be an arbitrary Coxeter graph and A ( Γ ) the associated Artingroup. Then the second mod 2 homology of A ( Γ ) is H ( A ( Γ ); Z ) ∼ = Z p ( Γ ) + q ( Γ )2 , where p ( Γ ) and q ( Γ ) are asin Theorem 2.6.The outline of our proof is as follows. In Subsection 3.1, we state Howlett’s theoremon the second integral homology group H ( W ( Γ ); Z ) of the Coxeter group W ( Γ ). Nextin Subsection 3.2, we recall Hopf’s formula of the second homology of a group. Thekey of the proof is that, by virtue of Hopf’s formula, we are able to find explicitlya set Ω ( W ) of generators of H ( W ( Γ ); Z ) (Subsection 3.3), as well as a set Ω ( A ) ofgenerators of H ( A ( Γ ); Z ) (Subsection 3.4). On the other hand, Howlett’s theorem econd Mod Homology of Artin Groups implies that Ω ( W ) forms a basis of H ( W ( Γ ); Z ), which is an elementary abelian2 group of rank p ( Γ ) + q ( Γ ). Furthermore, we will show that the homomorphism p ∗ : H ( A ( Γ ); Z ) → H ( W ( Γ ); Z ) induced by the projection p : A ( Γ ) → W ( Γ ) maps Ω ( A ) onto Ω ( W ). Hence p ∗ is actually an epimorphism and becomes an isomorphismwhen tensored with Z . As mentioned in the previous paragraph, we shall study the homomorphim p ∗ : H ( A ( Γ ); Z ) → H ( W ( Γ ); Z ) induced by the projection p : A ( Γ ) → W ( Γ ). A rea son for doing so is that we have the following Howlett’s theorem. Theorem 3.2 ([How88]) Thesecond integral homology oftheCoxeter group W ( Γ )associated to aCoxeter graph Γ is H ( W ( Γ ); Z ) ∼ = Z p ( Γ ) + q ( Γ )2 , where p ( Γ ) and q ( Γ ) are asin Theorem 2.6. Remark
The original statement in [How88] was H ( W ( Γ ); Z ) ∼ = Z − n ( Γ ) + n ( Γ ) + n ( Γ ) + n ( Γ )2 , where n ( Γ ) : = S , n ( Γ ) : = {{ s , t } ∈ E ( Γ ) | m ( s , t ) < ∞} , n ( Γ ) : = P ( Γ ) / ∼ , n ( Γ ) : = rank H ( Γ odd ; Z ) . For a Coxeter graph Γ , the above numbers are related to those used by Clancy Ellis asfollows − n ( Γ ) + n ( Γ ) + n ( Γ ) + n ( Γ ) = p ( Γ ) + q ( Γ ) . In fact, n ( Γ ) = V ( Γ odd ), n ( Γ ) = q ( Γ ) + E ( Γ odd ) and n ( Γ ) = p ( Γ ) + q ( Γ ). Theabove equation follows from the Euler Poincar´e theorem applied to Γ odd , V ( Γ odd ) − E ( Γ odd ) = rank H ( Γ odd ; Z ) − rank H ( Γ odd ; Z ) . Example 3.3
We shall make use of the following example later.
Toshiyuki Akita and Ye Liu
Let
Γ = I ( m ). Thus W ( Γ ) = D m is the dihedral group of order 2 m . Theorem 3.2shows that H ( W ( Γ ); Z ) ∼ = ( Z , m is even;0 , m odd . See Corollary 10.1.27 of [Kar93] for a complete list of integral homology of dihedralgroups.
Hopf’s formula gives a description of the second integral homology of a group. Wefirst recall some notations. For a group G , the commutator of x , y ∈ G is the element[ x , y ] = xyx − y − . The commutator subgroup [ G , G ] of G is the subgroup of G generated by all commutators. In general, we define [ H , K ] as the subgroup of G generated by [ h , k ] , h ∈ H , k ∈ K for any subgroups H and K of G . Theorem 3.4 (Hopf’s formula) Ifagroup G has apresentation h S | R i ,then H ( G ; Z ) ∼ = N ∩ [ F , F ][ F , N ] , where F = F ( S ) isthefreegroup generated by S and N = N ( R ) isthenormalclosureof R (subgroup of F normally generated by the relation set R ).See Section II.5 of [Bro82] for a topological proof. Moreover, Hopf’s formula admitsthe following naturality (see Section II.6, Exercise 3(b) of [Bro82]). Proposition 3.5
Let G = F / N = h S | R i and G ′ = F ′ / N ′ = h S ′ | R ′ i as in Theorem3.4. Suppose a homomorphism α : G → G ′ lifts to e α : F → F ′ . Then the followingdiagram commutes, H ( G ; Z ) N ∩ [ F , F / [ F , N ] H ( G ′ ; Z ) N ′ ∩ [ F ′ , F ′ ] / [ F ′ , N ′ ] H ( α ) ∼ = ∼ = α ∗ where α ∗ is induced by e α . econd Mod Homology of Artin Groups For simplicity we denote by h x i G = x [ F , N ] ∈ F / [ F , N ] the coset of [ F , N ] representedby x ∈ F and h x , y i G = [ x , y ][ F , N ] ∈ [ F , F ] / [ N , F ] for x , y ∈ F . Thanks to Hopf’sformula, second homology classes of G can be regarded as h x i G for x ∈ N ∩ [ F , F ].To see how the representatives look like, we make the following simple observations,which we learned from [KS03]. Lemma 3.6
Thegroup N / [ F , N ] is abelian. Proof
Note that N / [ F , N ] is a quotient group of N / [ N , N ] and the latter is theabelianization of N .Thus we write the group N / [ F , N ] additively. It is clear h n i G = −h n − i G for n ∈ N . Lemma 3.7
In the abelian group N / [ F , N ], wehave h n i G = h fnf − i G for n ∈ N and f ∈ F . Proof
Since [ f , n ] ∈ [ F , N ], h f , n i G = h fnf − n − i G = h fnf − i G − h n i G = N / [ F , N ] is represented by an element of the form Q r ∈ R r n ( r ) ( n ( r ) ∈ Z ). Hopf’s formula implies that a second homology class of G can be represented byan element Q r ∈ R r n ( r ) ∈ [ F , F ].The next lemma is useful. Lemma 3.8
Let G = F / N beasinTheorem3.4. If x , y , z ∈ F suchthat [ x , y ] , [ x , z ] ∈ N ∩ [ F , F ],then h x , yz i G = h x , y i G + h x , z i G , h x , y − i G = −h x , y i G , Proof
Note that [ x , yz ] = [ x , y ] y [ x , z ] y − . Then in the abelian group N / [ F , N ], h x , yz i G = h x , y i G + h y [ x , z ] y − i G . The term h y [ x , z ] y − i G = h x , z i G since[ x , z ] − y [ x , z ] y − = [[ x , z ] − , y ] ∈ [ N , F ] . Hence the first equality holds. The second follows immediately from the first. Toshiyuki Akita and Ye Liu
The aim of this subsection is to construct an explicit set Ω ( W ) of generators of H ( W ( Γ ); Z ). Combined with Howlett’s theorem (Theorem 3.2), we show that Ω ( W )is a basis of H ( W ( Γ ); Z ).Let us describe the construction of Ω ( W ). Let Γ be a Coxeter graph and ( W , S ) theassociated Coxeter system with S totally ordered. Then W = h S | R W ∪ Q W i is asin Definition 2.2. Let F W = F ( S ) be the free group on S and N W = N ( R W ∪ Q W )be the normal closure of R W ∪ Q W . Therefore W = F W / N W . Using Hopf’s formulawe identify H ( W ; Z ) ∼ = ( N W ∩ [ F W , F W ]) / [ F W , N W ]. We shall construct three sets Ω i ( W ) ⊂ ( N W ∩ [ F W , F W ]) / [ F W , N W ] ( i = , , W is of the form h x i W with x expressed bya word Q R ( s , t ) ∈ R W R ( s , t ) n ( s , t ) Q Q ( s ) ∈ Q W Q ( s ) n ( s ) ∈ [ F W , F W ]. We decompose h x i W = h x i W + h x i W + h x i W as in the proof of Theorem 3.15, such that h x i i W is generatedby Ω i ( W ). Then Ω ( W ) = Ω ( W ) ∪ Ω ( W ) ∪ Ω ( W ) generates H ( W ; Z ). Now weexhibit respectively the constructions of Ω i ( W ) ( i = , , Ω ( W )Let Ω ( W ) = {h s , t i W | s , t ∈ S , s < t , m ( s , t ) = } . Recall that h s , t i W = [ s , t ][ F W , N W ] ∈ ( N W ∩ [ F W , F W ]) / [ F W , N W ] and R ( s , t ) = [ s , t ]when m ( s , t ) =
2. Note that the above expression may have repetitions. In fact, wehave the following.
Proposition 3.9 Ω ( W ) ≤ p ( Γ ) + q ( Γ ). Proof
We shall show that h s , t i W = h s , t ′ i W in Ω ( W ) if { s , t } ≡ { s , t ′ } in P ( Γ ).Suppose s < t and s < t ′ with { s , t } ≡ { s , t ′ } in P ( Γ ), that is m ( s , t ) = m ( s , t ′ ) = m ( t , t ′ ) is odd. Then in N W / [ F W , N W ], h s , t i W − h s , t ′ i W = h s , R ( t , t ′ ) i W = h sR ( t , t ′ ) s − R ( t , t ′ ) − i W = h sR ( t , t ′ ) s − i W + h R ( t , t ′ ) − i W = h R ( t , t ′ ) i W − h R ( t , t ′ ) i W = , where the first and the third equalities follow from Lemma 3.8, the fourth from Lemma3.7. Similarly, h s , t i W = h s ′ , t i W in Ω ( W ) if { s , t } ≡ { s ′ , t } in P ( Γ ). Hence Ω ( W ) ≤ (cid:0) P ( Γ ) / ∼ (cid:1) = p ( Γ ) + q ( Γ ). econd Mod Homology of Artin Groups Ω ( W )Let Ω ( W ) = {h R ( s , t ) i W | s , t ∈ S , s < t , m ( s , t ) ≥ } . Recall that R ( s , t ) = ( st ) m ( s , t ) ( ts ) − m ( s , t ) . Note that when m ( s , t ) is even, R ( s , t ) is inthe kernel of the abelianization map Ab : F W → F W / [ F W , F W ] and hence R ( s , t ) ∈ [ F W , F W ]. The following is an obvious observation. Proposition 3.10 Ω ( W ) ≤ q ( Γ ). Ω ( W )The construction of Ω ( W ) requires more preparations. Recall that Γ odd is the subgraphof Γ considered as a 1 dimensional CW complex with 0 cells S and 1 cells {h s , t i | s , t ∈ S , s < t , m ( s , t ) odd } oriented by ∂ h s , t i = t − s . We define a group C W = { ( α, β ) ∈ C ( Γ odd ) ⊕ C ( Γ odd ) | ∂α = β } where 2 C ( Γ odd ) = { γ | γ ∈ C ( Γ odd ) } is the group of 0 chains with all coefficientseven, and a subgroup D W of C W generated by (2 h s , t i , − s + t ) for all 1 cells h s , t i .Consider the following homomorphism Φ W : C W → N W ∩ [ F W , F W ][ F W , N W ]defined by Φ W X s < t , m ( s , t ) odd n ( s , t ) h s , t i , X s ∈ S n ( s ) s = * Y s < t , m ( s , t ) odd R ( s , t ) n ( s , t ) Y s ∈ S Q ( s ) n ( s ) + W . The definition is indeed valid by the following easy lemma.
Lemma 3.11
Thefollowing are equivalent.(A) (cid:16)P s < t , m ( s , t ) odd n ( s , t ) h s , t i , P s ∈ S n ( s ) s (cid:17) ∈ C W .(B) Q s < t , m ( s , t ) odd R ( s , t ) n ( s , t ) Q s ∈ S Q ( s ) n ( s ) ∈ [ F W , F W ]. Proof
We suppress the ranges since they should be clear.(A) ⇔ ∂ (cid:16)X n ( s , t ) h s , t i (cid:17) = X n ( s ) s ⇔ X n ( s ) s + X n ( s , t )( s − t ) = ⇔ Ab (cid:16)Y R ( s , t ) n ( s , t ) Y Q ( s ) n ( s ) (cid:17) = ⇔ (B) , Toshiyuki Akita and Ye Liu where Ab : F W → F W / [ F W , F W ] is the abelianization map and we write F W / [ F W , F W ]additively. Note that Ab( R ( s , t )) = s − t if m ( s , t ) is odd.The following is a consequence of Example 3.3. Proposition 3.12 D W lies in thekernel of Φ W . Proof
It suffices to show that any generator (2 h s , t i , − s + t ) of D W is mapped tothe identity by Φ W , or equivalently, the word (cid:0) ( st ) m ( ts ) − m (cid:1) ( s ) − t lies in [ F W , N W ]when m is odd. Let s , t ∈ S with m : = m ( s , t ) odd, consider the parabolic subgroup W ′ : = W { s , t } of W , which is isomorphic to the dihedral group D m of order 2 m . FromExample 3.3, we know H ( W ′ ; Z ) =
0. On the other hand, Hopf’s formula appliedto W ′ shows that H ( W ′ ; Z ) ∼ = ( N W ′ ∩ [ F W ′ , F W ′ ]) / [ F W ′ , N W ′ ]. Therefore the word (cid:0) ( st ) m ( ts ) − m (cid:1) ( s ) − t ∈ N W ′ ∩ [ F W ′ , F W ′ ] represents the trivial homology class. Thatis to say (cid:0) ( st ) m ( ts ) − m (cid:1) ( s ) − t ∈ [ F W ′ , N W ′ ] ⊂ [ F W , N W ] . This proves the proposition.As a consequence, the homomorphism Φ W factors through C W ։ C W / D W → ( N W ∩ [ F W , F W ]) / [ F W , N W ] . Let Z ( Γ odd ; Z ) denote the group of 1 cycles of Γ odd with integral coefficients and Z ( Γ odd ; Z ) the group of 1 cycles of Γ odd with coefficients in Z . Define a homo morphism Ξ W : C W → Z ( Γ odd ; Z ) by ( α, ∂α ) α , where α ∈ C ( Γ odd ) such that ∂α ∈ C ( Γ odd ) and α ∈ C ( Γ odd ; Z ) is the mod 2 reduction of α . The condition ∂α ∈ C ( Γ odd ) asserts that α is indeed a 1 cycle of Γ odd with coefficients in Z . Proposition 3.13
The homomorphism Ξ W : C W → Z ( Γ odd ; Z ) factors through anisomorphism C W C W / D W Z ( Γ odd ; Z ) Ξ W ∼ = econd Mod Homology of Artin Groups Proof
The homomorphism Ξ W is obviously an epimorphism and( α, ∂α ) ∈ Ker Ξ W ⇔ α = ∈ Z ( Γ odd ; Z ) ⇔ α ∈ C ( Γ odd ) ⇔ ( α, ∂α ) ∈ D W . Hence Ker Ξ W = D W .Via the isomorphism in Proposition 3.13, we obtain a homomorphism Ψ W : Z ( Γ odd ; Z ) → N W ∩ [ F W , F W ][ F W , N W ] , which fits into the following commutative diagram(3–1) C W N W ∩ [ F W , F W ] / [ F W , N W ] C W / D W Z ( Γ odd ; Z ) Φ W Ψ W ∼ = We fix a basis Ω ( Γ odd ; Z ) of Z ( Γ odd ; Z ) ∼ = Z q ( Γ ) once and for all and denote by Ω ( Γ odd ; Z ) the basis of Z ( Γ odd ; Z ) ∼ = Z q ( Γ )2 obtained from Ω ( Γ odd ; Z ) by mod 2reduction Z ( Γ odd ; Z ) ։ Z ( Γ odd ; Z ).Define Ω ( W ) to be the image of Ω ( Γ odd ; Z ) under Ψ W , Ω ( W ) = Ψ W ( Ω ( Γ odd ; Z )) . To be precise, Ω ( W ) = * Y s < t , m ( s , t ) odd R ( s , t ) n ( s , t ) + W (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X s < t , m ( s , t ) odd n ( s , t ) h s , t i ∈ Ω ( Γ odd ; Z ) Proposition 3.14 Ω ( W ) ≤ q ( Γ ).Let Ω ( W ) = Ω ( W ) ∪ Ω ( W ) ∪ Ω ( W ), we conclude that Theorem 3.15 Ω ( W ) is abasis of H ( W ; Z ). Toshiyuki Akita and Ye Liu
Proof
Since Ω ( W ) ≤ p ( Γ ) + q ( Γ ) and H ( W ; Z ) ∼ = Z p ( Γ ) + q ( Γ )2 (Theorem 3.2). Itsuffices to show that Ω ( W ) generates H ( W ; Z ). An arbitrary homology class in H ( W ; Z ) is represented by the coset h x x x i W with x = Y s < t , m ( s , t ) = [ s , t ] n ( s , t ) , x = Y s < t , m ( s , t ) ≥ R ( s , t ) n ( s , t ) , x = Y s < t , m ( s , t ) odd R ( s , t ) n ( s , t ) Y Q ( s ) ∈ Q W Q ( s ) n ( s ) . with x , x , x ∈ [ F W , F W ]. Thus h x x x i W = h x i W + h x i W + h x i W . We claim that h x i i W is generated by Ω i ( W ). In fact, the claim for i = , i =
3, let α = X s < t , m ( s , t ) odd n ( s , t ) h s , t i , β = X s ∈ S n ( s ) s . Thus ( α, β ) ∈ C W by Lemma 3.11 with Φ W ( α, β ) = h x i W . By the commutativediagram (3–1), the mod 2 reduction α ∈ Z ( Γ odd ; Z ) of α is mapped to h x i W by Ψ W . This proves the claim. Remark
It is worth noting that in the previous proof, we have managed to get rid ofthe relations Q ( s ) without altering the homology class h x i W . This will be crucial inthe proof of Theorem 3.20. Now we turn to the Artin group case. The arguments here are parallel to those in theCoxeter group case.Let Γ be a Coxeter graph with the vertex set S totally ordered, A = A ( Γ ) be the Artingroup of type Γ with the presentation A = h Σ | R A i given in Definition 2.3. Let F A = F ( Σ ) be the free group on Σ and N A be the normal closure of R A . Hopf’sformula yields H ( A ; Z ) ∼ = ( N A ∩ [ F A , F A ]) / [ F A , N A ]. For the same reason as before,a second homology class of A is represented by a coset h x i A with x of the form Q R ( a s , a t ) ∈ R A R ( a s , a t ) n ( s , t ) ∈ [ F A , F A ].We construct a set Ω ( A ) of generators of H ( A ; Z ) using the same method as in theprevious subsection. econd Mod Homology of Artin Groups Ω ( A ) and Ω ( A )The constructions of Ω ( A ) and Ω ( A ) are exactly parallel to those in the Coxeter case.Let Ω ( A ) = {h a s , a t i A | s , t ∈ S , s < t , m ( s , t ) = } , Ω ( A ) = {h R ( a s , a t ) i A | s , t ∈ S , s < t , m ( s , t ) ≥ } . The same reasoning shows
Proposition 3.16 Ω ( A ) ≤ p ( Γ ) + q ( Γ ) , Ω ( A ) ≤ q ( Γ ). Ω ( A )Consider the following homomorhpism Ψ A : Z ( Γ odd ; Z ) → N A ∩ [ F A , F A ][ F A , N A ] , defined by Ψ A X s < t , m ( s , t ) odd n ( s , t ) h s , t i = * Y s < t , m ( s , t ) odd R ( a s , a t ) n ( s , t ) + A . The definition is valid by the following lemma.
Lemma 3.17
Thefollowing are equivalent.(A) P s < t , m ( s , t ) odd n ( s , t ) h s , t i ∈ Z ( Γ odd ; Z ).(B) Q s < t , m ( s , t ) odd R ( a s , a t ) n ( s , t ) ∈ [ F A , F A ]. Proof
We suppress again the ranges.(A) ⇔ ∂ (cid:16)X n ( s , t ) h s , t i (cid:17) = ⇔ X n ( s , t )( t − s ) = ⇔ Ab (cid:16)Y R ( a s , a t ) n ( s , t ) (cid:17) = ⇔ (B) , where Ab : F A → F A / [ F A , F A ] is the abelianization map. Note that Ab( R ( a s , a t )) = a s − a t if m ( s , t ) is odd. Toshiyuki Akita and Ye Liu
Recall that we have chosen a basis Ω ( Γ odd ; Z ) for Z ( Γ odd ; Z ). Let Ω ( A ) be the imageof Ω ( Γ odd ; Z ) under Ψ A , Ω ( A ) = Ψ A ( Ω ( Γ odd ; Z )) . To be precise, Ω ( A ) = * Y s < t , m ( s , t ) odd R ( a s , a t ) n ( s , t ) + A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X s < t , m ( s , t ) odd n ( s , t ) h s , t i ∈ Ω ( Γ odd ; Z ) Proposition 3.18 Ω ( A ) ≤ q ( Γ ).Let Ω ( A ) = Ω ( A ) ∪ Ω ( A ) ∪ Ω ( A ), hence Ω ( A ) ≤ p ( Γ ) + q ( Γ ). We have thefollowing Theorem 3.19 Ω ( A ) is aset ofgenerators of H ( A ; Z ). Proof
The proof is similar to that of Theorem 3.15 so we omit it.
Theorem 3.1 will follow from the next more precise theorem.
Theorem 3.20
The projection p : A ( Γ ) → W ( Γ ) induces an epimorphism betweenthe second integral homology p ∗ : H ( A ( Γ ); Z ) ։ H ( W ( Γ ); Z ) . Proof
The epimorphism p : A → W defined by p ( a s ) = s lifts to e p : F A → F W .Then by Proposition 3.5, we obtain the explicit formulation of p ∗ , p ∗ : H ( A ; Z ) ∼ = N A ∩ [ F A , F A ][ F A , N A ] → N W ∩ [ F W , F W ][ F W , N W ] ∼ = H ( W ; Z ) * Y R ( a s , a t ) ∈ R A R ( a s , a t ) n ( s , t ) + A * Y R ( s , t ) ∈ R W R ( s , t ) n ( s , t ) + W We claim that p ∗ maps Ω i ( A ) onto Ω i ( W ). The claim is obvious for i = ,
2. As forthe case i =
3, consider the following diagram Ω ( Γ odd ; Z ) Ω ( A ) Ω ( Γ odd ; Z ) Ω ( W ) Ψ A mod 2 p ∗ Ψ W econd Mod Homology of Artin Groups Take α = P s < t , m ( s , t ) odd n ( s , t ) h s , t i ∈ Ω ( Γ odd ; Z ), then p ∗ ◦ Ψ A ( α ) = * Y s < t , m ( s , t ) odd R ( s , t ) n ( s , t ) + W ∈ N W ∩ [ F W , F W ][ F W , N W ] . Recall the construction of Φ W , we have Φ W ( α, ∂α ) = p ∗ ◦ Ψ A ( α ). Thus we obtain Ψ W ( α ) = p ∗ ◦ Ψ A ( α ) , by the commutative diagram (3–1). This proves that p ∗ maps Ω ( A ) into Ω ( W )and the above diagram commutes. Since the mod 2 reduction restricts to a bi jection Ω ( Γ odd ; Z ) → Ω ( Γ odd ; Z ) and by definition the horizontal maps are onto, p ∗ : Ω ( A ) → Ω ( W ) is onto. The proof is complete. Proof of Theorem 3.1
Consider the following composition of epimorphisms(3–2) Z Ω ( A ) φ −−→ H ( A ; Z ) p ∗ −−→ H ( W ; Z ) , where Z Ω ( A ) is the free abelian group generated by Ω ( A ) and φ ( ω ) = ω for ω ∈ Ω ( A ).Taking tensor product with Z for terms in the above sequence (3–2), Z Ω ( A )2 φ ⊗ id Z −−−−→ H ( A ; Z ) ⊗ Z p ∗ ⊗ id Z −−−−−→ H ( W ; Z ) ⊗ Z , where Z Ω ( A )2 is the elementary abelian 2 group generated by Ω ( A ) with rank Ω ( A ) ≤ p ( Γ ) + q ( Γ ) and H ( W ; Z ) ⊗ Z ∼ = Z p ( Γ ) + q ( Γ )2 (Theorem 3.2). Since tensoring with Z preserves surjectivity, this forces Ω ( A ) = p ( Γ ) + q ( Γ ) and both maps are in factisomorphisms. Thus H ( A ( Γ ); Z ) ⊗ Z ∼ = Z p ( Γ ) + q ( Γ )2 . On the other hand, we have thefollowing exact sequence by universal coefficient theorem,0 → H ( A ; Z ) ⊗ Z → H ( A ; Z ) → Tor( H ( A ; Z ) , Z ) → , where Tor( H ( A ( Γ ); Z ) , Z ) = H ( A ( Γ ); Z ) is torsion free (Theorem 2.6 andthe Remark following). Now we conclude H ( A ( Γ ); Z ) ∼ = Z p ( Γ ) + q ( Γ )2 and finish theproof of Theorem 3.1.As a byproduct of the proof, we have the following corollaries. Recall that M ( Γ ) isthe complement of the complexified arrangement of reflection hyperplanes associatedto the Coxeter group W ( Γ ). The orbit space N ( Γ ) = M ( Γ ) / W ( Γ ) has fundamentalgroup π ( N ( Γ )) ∼ = A ( Γ ). Let c : N ( Γ ) → K ( A ( Γ ) ,
1) be the classifying map. Then c always induces an isomorphism c ∗ : H ( N ( Γ ); Z ) → H ( A ( Γ ); Z ) and an epimorphism c ∗ : H ( N ( Γ ); Z ) → H ( A ( Γ ); Z ). We give a sufficient condition on Γ such that c induces an isomorphism c ∗ : H ( N ( Γ ); Z ) → H ( A ( Γ ); Z ). Toshiyuki Akita and Ye Liu
Corollary 3.21 If Γ satisfies thefollowing conditions • P ( Γ ) / ∼ consists of torsion classes. • Γ = Γ odd . • Γ is atree.Then H ( A ( Γ ); Z ) ∼ = Z p ( Γ )2 . Hence c induces anisomorphism c ∗ : H ( N ( Γ ); Z ) → H ( A ( Γ ); Z ). Proof
Since N ( Γ ) is path connected and has fundamental group π ( N ( Γ )) ∼ = A ( Γ ),there is an exact sequence (see for example Section II.5 Theorem 5.2 of [Bro82]),(3–3) π ( N ( Γ )) h −−→ H ( N ( Γ ); Z ) c ∗ −−→ H ( A ( Γ ); Z ) → , where h is the Hurewicz homomorphism. Suppose that Γ satisfies the three conditions,then q ( Γ ) = q ( Γ ) = q ( Γ ) =
0. Theorem 2.6 implies that H ( N ( Γ ); Z ) ∼ = Z p ( Γ )2 .Then by Theorem 3.20, H ( A ( Γ ); Z ) sits in the following sequence Z p ( Γ )2 ։ H ( A ( Γ ); Z ) ։ Z p ( Γ )2 , the composition must be an isomorphism, hence H ( A ( Γ ); Z ) ∼ = Z p ( Γ )2 . As a result, c ∗ must be an isomorphism. Corollary 3.22
IfthethreeconditionsinCorollary3.21aresatisfied,then p : A → W induces an isomorphism p ∗ : H ( A ; Z ) → H ( W ; Z ) . Proof
The corollary follows from Howlett’s Theorem 3.2, Theorem 3.20 and Corol lary 3.21.
Corollary 3.23
For any Coxeter graph Γ , the induced map c ∗ : H ( N ( Γ ); Z ) → H ( A ( Γ ); Z ) becomes an isomorphism after tensoring with Z . Proof
By right exactness of tensor functor, taking tensor product with Z preservesthe exactness of (3–3), π ( N ( Γ )) ⊗ Z h ⊗ id Z −−−−−→ H ( N ( Γ ); Z ) ⊗ Z c ∗ ⊗ id Z −−−−−→ H ( A ( Γ ); Z ) ⊗ Z → . Note that c ∗ ⊗ id Z is an isomorphism as a consequence of Theorem 3.1 and Clancy Ellis’ Theorem 2.6. econd Mod Homology of Artin Groups Example 3.24
The Coxeter graphs of affine type f D n ( n ≥
4) and e E i ( i = , ,
8) allsatisfy the conditions in Corollary 3.21. Therefore we compute the second integralhomology of the associated Artin groups as follows. H ( A ( f D n ); Z ) ∼ = ( Z , n = Z , n ≥ . H ( A ( e E i ); Z ) ∼ = Z , i = , , . Besides the above cases, the Coxeter graphs of certain hyperbolic Coxeter groups alsoprovide plenty of examples satisfying the conditions in Corollary 3.21. We point outthat to the best of the authors’ knowledge, the K ( π,
1) conjecture has not been provedin the above mentioned cases.
We mention a corollary concerning homological stability in the end of this paper.Consider a family of Coxeter graphs { Γ i } i ≥ , starting from Γ with a base vertex s and each Γ i ( i ≥
2) is obtained by adding a vertex s i connected to s i − by an unlabelededge. The embedding Γ i ֒ → Γ i + of Coxeter graphs induces inclusion of Coxetergroups W ( Γ i ) ֒ → W ( Γ i + ), as well as inclusion of Artin groups A ( Γ i ) ֒ → A ( Γ i + )(cf. [vdL83, Par97]). It is known that the families of Artin groups { A ( A n ) } , { A ( B n ) } and { A ( D n ) } possess integral cohomological stability ([Arn69, DCS99]). Hepworthproved a more general result for Coxeter groups. Theorem 3.25 ([Hep16]) The map H k ( W ( Γ n − )) → H k ( W ( Γ n )) is an isomorphismfor 2 k ≤ n witharbitrary constant coefficient.As for the sequence of Artin groups { A ( Γ i ) } , it is not difficult to see that the firstintegral homology admits stability. We prove a stability result for the second mod 2homology of the sequence { A ( Γ i ) } . Theorem 3.26
The map H ( A ( Γ n − ); Z ) → H ( A ( Γ n ); Z ) is an isomorphism for n ≥ Toshiyuki Akita and Ye Liu
Proof
Consider the commutative diagram(3–4) H ( A ( Γ n − ); Z ) H ( A ( Γ n ); Z ) H ( A ( Γ n − ); Z ) ⊗ Z H ( A ( Γ n ); Z ) ⊗ Z H ( W ( Γ n − ); Z ) ⊗ Z H ( W ( Γ n ); Z ) ⊗ Z p ∗ ⊗ id Z p ∗ ⊗ id Z where the commutativity of the upper square follows from the naturality of the universalcoefficient theorem and the lower from tensoring with Z to the following commutativediagram(3–5) H ( A ( Γ n − ); Z ) H ( A ( Γ n ); Z ) H ( W ( Γ n − ); Z ) H ( W ( Γ n ); Z ) p ∗ p ∗ Since all vertical maps in (3–4) are isomorphisms and the bottom horizontal map is anisomorphism when n ≥ n ≥ Corollary 3.27 It Γ satisfies the three conditions in Corollary 3.21, then the map H ( A ( Γ n − ); Z ) → H ( A ( Γ n ); Z ) is anisomorphism for n ≥ Proof
The corollary follows from Corollay 3.22, Theorem 3.25 and the commutativediagram (3–5).
Acknowledgement
The first author was partially supported by JSPS KAKENHI Grant Number 26400077.The second author was supported by JSPS KAKENHI Grant Number 16J00125.
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