The Betti Numbers of Real Toric Varieties Associated to Weyl Chambers of Type B∗

Suyoung CHOIBoram PARKHanchul PARK

1 Introduction

A toric variety of complex dimension n is a normal algebraic variety over C with an effective algebraic action of(C{O})nhaving an open dense orbit.A compact smooth toric variety is called a toric manifold.One of the most important facts on toric geometry is that there is a 1-1 correspondence between the class of toric varieties of complex dimension n and the class of fans in Rn.This fact is called the fundamental theorem of toric geometry.In particular,a toric manifold X of complex dimension n corresponds to a complete regular fan ΣXin Rn.

Among toric manifolds,the class of toric manifolds associated to Weyl chambers has been considered since it is introduced by Procesi[12].A classical construction associates each root system to a toric manifold whose fan corresponds to the re fl ecting hyperplanes of the root system and its weight lattice.It is natural to ask about the topology of the corresponding toric manifold.Note that the integral cohomology of a toric manifold is well-established by Jurkiwicz[11]for the projective cases and by Danilov[7]for general cases.For a coefficient field k,the ith k-Betti number of a topological space X is the rank of Hi(X;k)over k,and it is denoted by βi(X;k).One remarkable fact is that the Betti numbers of a toric manifold X depend only on the face numbers of its associated fan ΣX.Especially,the structures of the cohomology of toric manifolds associated to Weyl chambers have been studied by[1,9,12,15].

On the other hand,the subset consisting of points with real coordinates of a toric manifold is called a real toric manifold.Unlike toric manifolds,little is known about the topology of real toric manifolds.Let X be a toric manifold and XRits real toric manifold.By Davis and Januszkiewicz[8],the ith Z2-Betti number of XRis equal to the 2ith Z-Betti number of X,and,hence,it depends only on the face numbers.However,the Betti numbers with rational coefficients are not only determined by the face numbers.For instance,both the torus and the Klein bottle are real toric manifolds and their corresponding fans have the face structure combinatorially equivalent to the 4-gon.Hence,their Z2-Betti numbers are the same while their Q-Betti numbers are different.From this sense,the computation of the rational Betti numbers of real toric manifolds is difficult,and only a few examples have been computed so far.One known example is the real toric manifolds associated to Weyl chambers of type Andue to Henderson[10].Interestingly,their rational Betti numbers are the Euler zigzag numbers.Arnol′d[2]has defined the notion of snake numbers as a generalization of the Euler zigzag numbers as follows:A snake of type An(respectively Bn),or an An-snake(respectively,Bn-snake),is a sequence of integers xisatisfying the conditions:

(1)for An:x0＜ x1＞ x2＜ ···xn,xixjfor ij;

(2)for Bn:0＜ x1＞ x2＜ ···xn,xi±xjfor ij,

where 0≤xi≤n for all i for An,and 1≤|xi|≤n for all i for Bn.Denote by an(respectively,bn)the number of An-snakes(respectively,Bn-snakes).The number anis also known as the Euler zigzag number(see[13,A000111]),and the number bnis also known as the generalized Euler number or the Springer number(see[13,A001586]).

Table 1 The list of anand bnfor small n

The formula of the rational Betti numbers by Henderson was recovered by Suciu[16]later using the general formula for rational Betti numbers of real toric manifolds established by Suciu and Trevisan[17].

Theorem 1.1(see[10,16])Denote bythe real toric manifold associated to the Weylchambers of typeAn.ThekthQ-Betti number ofXRAnis

We note that Choi and Park[5]showed that the formula used in[17]works for not only Q coefficient but also arbitrary field k coefficient whose characteristic is not equal to 2.Combining it with[16],we obtain the following corollary.

Corollary 1.1The integral cohomology ofisp-torsion free for all odd primesp.

In this paper,we compute the rational Betti numbers of real toric manifolds associated to the Weyl chambers of type Bn,and show that their integral cohomologies are p-torsion free for all odd primes p.We prove the following theorem.

Theorem 1.2Denote bythe real toric manifold associated to the Weyl chambers of typeBn.Then,we have

Furthermore,their integral cohomologies arep-torsion free for all odd primesp.

It is worthwhile to note that the same techniques to prove the above theorem do not directly apply to the case of type C and D,the other regular types.This is because the analogues for the shellability results like Lemma 3.3 fail for type C or D,making it hard to compute homology of the corresponding posets.

This paper is organized as follows.In Section 2,we introduce preliminary facts including the formula of Suciu-Trevisan to compute the Betti numbers of real toric manifolds and the way to define projective toric manifolds associated to Weyl chambers.In Section 3,we prove the main theorem,that is,we compute the Betti numbers of real toric manifolds of type Bn.

2 Preliminaries

2.1 The Betti numbers of real toric manifolds

In this subsection,we shall introduce a formula of the Betti numbers of real toric manifolds.From now on,we restrict our interests in the projective toric manifolds and its real toric manifolds.Let X be a projective toric manifold of complex dimension n and XRits real toric manifold.We assume that the associated fan ΣXof X has m rays r1,···,rm.Then,ΣXcan be regarded as a pair of an(n−1)-dimensional polytopal simplicial sphere K with the vertex set[m]={1,···,m}and a map λ:[m]→ Znsuch that

(1) σ ={i1,···,iℓ} ∈ K if and only if{ri1,···,riℓ}forms a cone in ΣX,and

(2)λ(i)is the primitive vector in the direction of ri.

We call K the underlying simplical complex of X and λ the characteristic map of X.Furthermore,since X is projective,there is a convex simple polytope P with m facets F1,···,Fm,whose face structure is isomorphic to K and the outward normal vector of Fiis λ(i)for i=1,···,m.

Similarly to the fundamental theorem for toric geometry,it is known that as a Z2-space,a real toric manifold XRis determined by the pair(K,λR),where λRthe composition map of λ and the canonical quotient map Z→Z/2Z,i.e.,λR:[m]λ→Z→Z/2Z.We call λRthe Z2-characteristic map,and we note that λRcan be represented as a Z2-matrix of size n × m,called the Z2-characteristic matrix.For each subset S of{1,···,n},writewhere λiis the ith row of λR.Let[m]S:={j ∈ [m]|the jth entry of λSis nonzero} ⊂ [m].For such S we define KS:={σ∈K|σ⊂ [m]S},and,as dual,We note that the topological realization of KSis homotopy equivalent to PS.Throughout this paper,we denote by K the topological realization of a simplicial complex K if there is no danger of confusion.

Theorem 2.1(see[5,17])LetXbe a toric manifold andXRits real toric manifold.Letkbe a ring where2is invertible ink.Then theith Betti numberβi(XR;k)ofXRwith coefficientkis given by

Suciu and Trevisan in their unpublished paper[17]established the formula for the rational Betti numbers of real toric manifolds.Later,Choi and Park[5]also derived a cohomology formula of real toric manifolds with the coefficient ring G,where 2 is invertible in G.It should be noted that if the reduced cohomology of PSis p-torsion free for all S⊆[n]and all odd primes p,then so is the cohomology of XR.This formula also determines a stable homotopy decomposition of a wider class of spaces called real toric spaces(see[4]).

2.2 Projective toric manifold associated to Weyl chambers

As mentioned in Section 1,we mainly deal with the class of(real)toric manifolds associated to the decomposition given by Weyl chambers.Let V be a finite dimensional real Euclidean space,Φ ⊂ V a root system,and W its Weyl group.In V we have the lattice Λ ={v∈ V|(v,α)∈ Z for any α ∈ Φ}which defines an integral structure,where(−,−)is the natural inner product.For each set Δ of simple roots in Φ we consider the cone CΔ={v ∈ V|(v,α) ＞0 for any α ∈ Δ}.These cones provide the rational polyhedral decomposition of V,i.e.,the set of cones is a fan in V.Hence,it defines a projective toric variety,which is in fact smooth.

From now on,let us consider the Weyl groups of regular types.Throughout this paper,for the Weyl group of type An,the corresponding toric variety,its fan,the underlying simplicial complex,the characteristic map,the corresponding real toric variety,and the Z2-characteristic map are denoted by XAn,ΣAn,KAn,λAn,and,respectively.For the Weyl group of type Bn,the corresponding notions are similarly denoted by XBn,ΣBn,KBn,λBn,andrespectively.

2.3 Type An

In this subsection,we shall review a sketch of proof of Theorem 1.1 and Corollary 1.1.The proof presented here is essentially the same with that by[16]or[6]for the special case when the corresponding graph is a complete graph.However,we enclose this subsection for the sake of self-contained readability.

It is well-known that the vertices of KAncan be identified by the nonempty proper subsets I of[n+1]and each(ℓ− 1)-dimensional simplex of KAnis related to a nested ℓ nonempty proper subsets of[n+1],that is,{Ii1,···,Iiℓ} ∈ KAnif and only if there is a permutation σ on[ℓ]such that Iiσ(1)⊂ ···⊂ Iiσ(ℓ).In addition,the characteristic map λAnis

where εiis the ith standard vector of Zn.As a consequence,

where eiis the ith standard vector of

From now on,let us compute the Q-Betti number ofBy Theorem 2.1,we have to consider(KAn)Sfor all subsets S⊂[n].Here are three important nontrivial steps.

For an odd number r,defineas

(1)is homotopy equivalent to the wedge of spheres of dimension

(2)The reduced Euler characteristic

(3)For S⊂[n]with|S|=r or|S|=r+1 for some odd number r,(KAn)Sis homotopy equivalent to

By(1)–(3)together with Theorem 2.1,both Theorem 1.1 and Corollary 1.1 are immediately proved.

3 Type Bn

Let Φ be a root system of type Bn.It consists of 2n2roots

where εiis the ith standard vector of Rn=V.One can see that the lattice Λ consists of all integral vectors in Rn.We note that a line containing a ray of ΣBnis the intersection of n − 1 hyperplanes normal to Δ {α},where Δ is a set of simple roots of type Bnand α ∈ Δ,and the direction of the ray is determined by α.A set of simple roots of type Bnforms

where µj= ±1 and σ:[n]→ [n]is a permutation.For α ∈ Δ,there exists a unique primitive integral vector β =(b1,···,bn)such that(β,α′)=0 for all α′∈ Δ {α}and(β,α) ＞ 0.We note that each component bjof β is either ±1 or 0.Then,we label the ray of ΣBncorresponding to α ∈ Δ by the set I={jbj|j=1,···,n} ⊂ [±n]={±1,±2,···,±n}.More precisely,by putting xi= µiεσ(i)− µi+1εσ(i+1)for i=1,···,n − 1 and xn= µnεσ(n),if α =xi,thenand,hence,the corresponding label is{µ1σ(1),···,µiσ(i)}.Therefore,the vertices of KBncan be labelled by the nonempty subsets I of[±n]satisfying

and the characteristic map λBnis

Consequently,

where eiis the ith standard vector of Zn2.

Furthermore,one can see that each n-dimensional cone CΔin ΣBncorresponds to n subsets I1,···,Insatisfying(∗)such that I1··· Inand vice versa.This implies that each(ℓ−1)-dimensional simplex of KBnis labelled by a nested ℓ subsets of[±n]satisfying(∗),that is,{Ii1,···,Iiℓ} ∈ KBnif and only if there is a permutation σ on[ℓ]such that Iiσ(1)⊂ ···⊂ Iiσ(ℓ).

Example 3.1 Let us consider ΣB2.The corresponding toric variety XB2is CP2♯5,and the corresponding real toric varietyis the connected sum of six RP2s.Let us compute the Betti number ofusing Theorem 2.1.We expressby a matrix and draw the geometric realization of KB2as below,respectively:

where the numbers over the horizontal lines are indicators for vertices of KB2.Then,(KB2){1}≃S1{{2},is homotopy equivalent to S0,and similarly,we have(KB2){2}≃S0and(KB2){1,2}≃3S0.Therefore,the ith Betti number ofis

From now on,let us compute the Q-Betti number of XRBn.By Theorem 2.1,we have to consider(KBn)Sfor all subsets S⊂[n].Given a subset S⊂[n],then(KBn)Sis the restriction of KBnby{I∈V(KBn)||S∩I±|is odd},where V(K)is the vertex set of a simplicial complex K.

Now let us consider the case where S=[n].defineas

=(KBn)[n]={σ ∈ KBn|σ consists of I such that|I|is odd}.

We define the posetwhose vertices are the vertices ofand the partial order is given by inclusion,and define another poset:=∪{Ø,[±n]}with inclusion.Note that the order complex ofisand hence,=µ(Ø,[±n])whereµis the M¨obius function of(see[14,Section 3]for details),that is,

When n=0,we define by convention µ(Ø,[±n])= µ(Ø,{0})= −1.

Lemma 3.1The absolute value ofµ(Ø,[±n])isbn.More precisely,

Proof In this proof,we use i to denote the imaginary unit such that i2=−1.

For a vertex I insuch that I is neither Ø nor[±n],put|I|=2k+1.Note that the M¨obius function µ(Ø,I)depends only on|I|and µ(Ø,I)=(−1)k+1a2k+1(see the proof of Theorem 2.9 of[6]).Hence,we have

Recall that the exponential generating functions of anand bnare

and

respectively.Since ex=cos(−ix)+isin(−ix),we have

Therefore,M(ix)=(cosx−isinx)sec2x.Since the exponential generating function of secx has only even degree terms,cosxsec2x contributes the even degree terms of M(ix)and sinxsec2x contributes the odd degree terms of−iM(ix).Therefore,the lemma immediately follows by comparing the coefficients of B(x)and M(x).

We shall use the following well-known lemma in[3].This can be regarded as an alternative definition of shellability.Recall that a simplicial complex is called shellable if it admits a shelling.

Lemma 3.2(see[3,Lemma 2.3])An orderF1,F2,···,Ftof the facets of a simplicial complex is a shelling if and only if for everyiandkwith1≤i＜k≤t,there is ajwith1≤j＜ksuch thatFi∩Fk⊆Fj∩Fkand|Fj∩Fk|=|Fk|−1.

Lemma 3.3For any integern,is shellable.

Proof Note that since KBnbounds a convex polytope,it is shellable.Choose a shelling σ:F1,···,Ftof KBn.For each m∈[t],letbe the face obtained from Fmby deleting all vertices of Fmcorresponding to even subsets of[±n].Note that for any m∈[t],is a facet ofThen consider an ordering σ′:···,of the facets of,and then we deletewhenever=for some ℓ such that ℓ＜m.Let σ∗:···,be the resulting ordering,that is,the ordering obtained from σ′by dropping all facets ofnot firstly appeared in σ′.Clearly, σ∗is an ordering of the facets ofWe shall show that σ∗is a shelling of.By Lemma 3.2,it is enough to show that,for every i and k with 1≤i＜k≤s,there is j with 1≤j＜k such that

(1)∩⊆∩,and

(2)|∩|=||−1.

For each m∈[s],let dm∈[t]be the smallest integer such that⊂Fdm,i.e.,Fdmis the first facet in σ containing.Note that for all ℓ,m∈[s],

Take i and k with 1≤i＜k≤s.Then⊂Fdiand⊂Fdk.Since di＜dkby(3.1),by considering two facets Fdiand Fdkof KBntogether with Lemma 3.2,there is J with 1≤J＜dksuch that Fdi∩Fdk⊆FJ∩Fdkand|FJ∩Fdk|=|Fdk|−1.Then we consider a facetofLet j be the smallest integer such that=Then dj≤J by definition,and so we have dj≤J＜dk.Thus j＜k by(3.1),and it indeed satisfies the conditions(1)and(2)as follows.

Let V be the set of vertices of.Note that∩=Fdi∩Fdk∩V and∩=FJ∩Fdk∩V.Therefore(1)follows from the fact that Fdi∩Fdk⊆FJ∩Fdk.Moreover,since∩=FJ∩Fdk∩V and|FJ∩Fdk|=|Fdk|−1,we have|∩|≥||−1.Since jk implies that(2)is proved.

Note thatis homotopy equivalent to a wedge of uniform spheres Sdas it is shellable.One can easily see that the dimension of the sphere is d=by observing the dimension of the facets.Since the absolute value of the reduced Euler characteristic ofis bnby Lemma 3.1,we conclude that

Remark 3.1 Here we give an explicit shelling ofWe define an ordering ≺ on[±n],

(just fix an ordering so that the positive integers proceed to the negative integers)and we define an order lexicographically induced by≺on the set of all maximal chains of(comparing the smaller element).We also denote by the same symbol≺the order on the set of all maximal chains.More precisely,for two maximal chains σ and σ′such that

we say σ′≺ σ if there exists 1≤i≤r such that＜lexiIi(comparing lexicographically under the ordering ≺ on[±n])and I′j=Ijfor any j ＜ i.Then it can be shown that this ordering on maximal chains gives a shelling of

Now,let us return to the case where S[n].If S is an empty set,so is(KBn)S.

Lemma 3.4(see[6,Lemma 5.2])LetIbe a vertex of a simplicial complexKand suppose that the link ofI,LkI,is contractible.ThenKis homotopy equivalent to the complexKStI,whereStIis the star ofI.

Lemma 3.5For a positive integern≥3,forS⊂[n],(KBn)Sis homotopy equivalent towhereis obtained from(KBn)Sby deleting verticesIin(KBn)Ssuch that

Proof For simplicity,we let K=(KBn)SandWe shall show that we can eliminate stars of vertices in KK′,one by one,from K to K′,without changing the homotopy type.First,for any vertex I of K,I∩SØ.In addition,two vertices I and J meet in K if and only if I⊂J or J⊂I.

Let I be a vertex of KK′such that|I±∩S|=1,say I±∩S={x}.Let J be a vertex in K such that J±={x}and JI.Take any L∈LkI.If I⊂L,then J⊂L,and so L meets J.Suppose that L⊂I.Then L±∩S is a subset of I±∩S.Since L is a vertex of K,L±∩SØ.Therefore L±∩S=I±∩S={x}=J±and so J±⊂ L±.Since J±⊂ L±⊂ I±,J⊂ I and L⊂I,it follows that J⊂L,and so L meets J.

Hence,LkI is contractible,and so K is homotopy equivalent to KStI by Lemma 3.4.By rede fi ning K:=KStI and repeating the argument,we can conclude that the star of any vertex I with|I±∩S|=1 can be eliminated.

Inductively,assume that we could eliminate all vertices I∈KK′such that|I±∩S|＜j,and let K∗be the simplicial complex obtained by deleting stars of all those vertices,where j≥ 2.Take a smallest vertex I∈K∗K′such that|I±∩S|=j.Let J be a vertex in K∗such that I±∩S=J±and J⊂I.(Note that J∈K′and so J is in K∗and IJ.)Take any L∈LkI in K∗.If I⊂L,then J⊂L,and so L meets J.Suppose that L⊂I.Then L±∩S is a subset of I±∩S=J±.If|L±∩S|＜j,then such L should have already been deleted by our induction hypothesis.Thus|L±∩S|=j and so L±∩S=J±.Therefore J±⊂L±.Since J⊂I and L⊂I,we have J⊂L,and so L meets J.

Hence,LkI is contractible,and so K∗is homotopy equivalent to K∗StI by Lemma 3.4.By rede fi ning K∗:=K∗StI and repeating the argument,we can conclude that the star of any vertex I with|I±∩S|=j can be eliminated in an increasing order of the size|I±∩S|.

Lemma 3.6Letr=|S|.Then(KBn)Sis homotopy equivalent to

Proof By Lemma 3.5,it clearly follows.

Theorem 3.1TheithQ-Betti number

Furthermore,their integral cohomologies ofarep-torsion free for all odd primesp.

Proof Let S⊂[n]and assume that|S|=r.By Lemma 3.6,(KBn)SWe recall thatHence,the homotopy type of(KBn)Sonly depends on the cardinality of S.For a fixed i and a field k whose characteristic is not equal to 2,by Theorem 2.1,

where δi,j=1 if i=j and 0 otherwise.It proves the theorem.

AcknowledgementsThe authors thank to Prof.Soojin Cho for helpful discussions,and Prof.Jang Soo Kim for suggesting nice proof of Lemma 3.1.They are also thankful to the anonymous referee for the thorough reading and kind comments.

[1]Abe,H.,Young diagrams and intersection numbers for toric manifolds associated with Weyl chambers,Electron.J.Combin.,22(2),2015,24.

[2]Arnol’d,V.I.,Snake calculus and the combinatorics of the Bernoulli,Euler and Springer numbers of Coxeter groups,Uspekhi Mat.Nauk,47(1),1992,3–45.

[3]Bj¨orner,A.and Wachs,M.L.,Shellable nonpure complexes and posets.I,Trans.Amer.Math.Soc.,348(4),1996,1299–1327.

[4]Choi,S.,Kaji,S.and Theriault,S.,Homotopy decomposition of a suspended real toric manifold,Bol.Soc.Mat.Mex.,3,23(1),2017,153–161.

[5]Choi,S.and Park,H.,On the cohomology and their torsion of real toric objects,Forum Math.,29(3),2017,543–553.

[6]Choi,S.and Park,H.,A new graph invariant arises in toric topology,J.Math.Soc.Japan,67(2),2015,699–720.

[7]Danilov,V.I.,The geometry of toric varieties,Uspekhi Mat.Nauk,33(2),1978,85–134,247.

[8]Davis,M.W.and Januszkiewicz,T.,Convex polytopes,Coxeter orbifolds and torus actions,Duke Math.J.,62(2),1991,417–451.

[9]Dolgachev,I.and Lunts,V.,A character formula for the representation of a Weyl group in the cohomology of the associated toric variety,J.Algebra,168(3),1994,741–772.

[10]Henderson,A.,Rational cohomology of the real Coxeter toric variety of type A,Con figuration spaces,CRM Series,Vol.14,Ed.Norm.,Pisa,2012,313–326.

[11]Jurkiewicz,J.,Chow ring of projective nonsingular torus embedding,Colloq.Math.,43(2),1980,261–270.

[12]Procesi,C.,The toric variety associated to Weyl chambers,Mots,Lang.Raison.Calc.,Herm`es,Paris,1990,153–161.

[13]Sloane,N.J.A.,The on-line encyclopedia of integer sequences,http://oeis.org.

[14]Stanley,R.P.,Enumerative combinatorics,Vol.1,Cambridge Studies in Advanced Mathematics,Vol.49,Cambridge University Press,Cambridge,1997,With a foreword by Gian-Carlo Rota,Corrected reprint of the 1986 original.

[15]Stembridge,J.R.,Some permutation representations of Weyl groups associated with the cohomology of toric varieties,Adv.Math.,106(2),1994,244–301.

[16]Suciu,A.I.,The rational homology of real toric manifolds,Oberwolfach Reports,4,2012,2972–2976.

[17]Suciu,A.I.and Trevisan,A.,Real toric varieties and abelian covers of generalized davis-januszkiewicz spaces,unpublished,2012.