Zhi L()Song ZHANG()

School of Mathematical Sciences,Fudan University,Shanghai 200433,China

E-mail:zlu@fudan.edu.cn;18110180040@fudan.edu.cn

Abstract The purpose of this paper is to represent the cohomology ring of a moment-angle manifold over an m-gon explicitly in terms of the quotient of an exterior algebra,and to count the Betti numbers of the cohomology groups of a special class of quotients of moment-angle manifolds.

Key words moment-angle manifold;spectral sequence,;cohomology ring;Betti number

Dedicated to Professor Banghe LI on the occasion of his 80th birthday

1 Introduction

Moment-angle complexes,which were first formulated by Buchstaber and Panov [6]in 1998,belong to the special class of spaces known as polyhedral products.Some special cases of moment-angle complexes have been studied since the 1960’s [19].In the early 1990’s,the seminar work of M.Davisand J.Januszkiewic [11]introduced quasi-toric manifolds,which were a topological generalization of projective toric varieties.Every quasi-toric manifold could be considered as the quotient of a moment-angle complex by the free action of a real torus.Generally,there is a family of manifolds which are called the quotients of moment-angle manifolds,as was shown in [9].Each of these quotient manifolds is equivariantly homeomorphic to the orbit space of a moment-angle manifold with a canonical torus action.These manifolds have recently been further studied for the reason that they support complex-analytic structures which are usually non-K?hler,and that have some interesting geometrical structures [18].

In recent years,toric topology and geometry has developed rapidly.In 2004,The cohomology rings of moment-angle complexes were first computed by Baskakov,Buchstaber and Panov [5].Later,the homotopy types of polyhedral products were studied by Grbi′c and The riault [14,15].Bahri,Bender sky,Cohen and Gitler [1,2]gave a natural decomposition of the suspension of polyhedral products,and expressed this decomposition as the wedge of smash moment-angle complexes,which provided a precise identification of the stable homotopy type of the“polyhedral product functor”.Following these works,fruitful results regarding polyhedral products showed up in the 2010’s,for example [3,4,16].Lately,more general situations were studied by Zheng [24]and Yu [22,23],who both gave nice descriptions of the cohomology rings.

In terms of a quotient of moment-angle manifold,however,there is only one main result for its cohomology ring [12,18],and it is represented in the form of a Tor-algebra.As was shown in [12],Franz discussed the different multiplications of this Tor-algebra between the case of when the ring contains 2 as a unit and the case of when the ring is a principle ideal domain.Later,Franz and Fu made a further move to obtain the naturality of these isomorphisms with respect to the morphisms of toric varieties [13].This Tor-algebra was not easy to compute in practice,for the reason that one would face huge difficulties when trying to obtain the generators.In addition,it is not clear whether there exists a Hochster’s type formula for the cohomology groups in general.In this paper,we decompose the quotient of moment-angle manifold with several disc-product-torus pieces,which will lead to a filtration of the whole space,with a spectral sequence for(co)homology induced.Then we simplify the situation by restricting the dimension of the polytope by two,and we use this spectral sequence to calculate the cohomology rings of moment-angle manifolds over polygons and the Betti numbers of a special class of quotient manifolds.

2 Backgrounds,Notations,and Main Results

In this paper,we always denotePnas ann-simple polytope withmfacets,written as F(P)={F1,···,Fm}.In particular,whenn=2,P2is anm-gon.

2.1 Moment-angle Complexes and Manifolds

In the special case whenXi=XandAi=Afor alli,it is convenient to denote the polyhedral product by Z(K;(X,A)).A moment-angle complex ZK=Z(K;(D2,S1))is a special case of the polyhedral product.In the work of Baskakov,Buchstaber and Panov [5],the Tor-algebra of the Stanley-Reisner algebra(the face ring ofK)

was given to represent the equivariant cohomology of the moment-angle complex ZK,and then the ordinary cohomology

Furthermore,in Buchstaber and Panov’s book [9],we see that the cohomology ringH?(ZK)is described by the Hochster formula

with a multiplicative structure

whereKIis the full subcomplex ofKonI.Obviously the calculation will be more tractable under this Hochster formula.However,it seems to be difficult to find out the generators with their relations specifically.

wherekis a field with arbitrary characteristics.For the general case of polyhedral products,an important description of the cohomology rings was shown in the work of Bahri,Bendersky,Cohen and Gitler [3]:

This may also be considered a Hochster type formula.

Another method of generalizing ZKis a functor from the face category CAT(S)to the category of topological spaces TOP,which just replaces the simplicial complexKby a simplicial poset S.A poset S is a partially ordered set given by the relation,and it is called simplicial when it has an initial elementsuch that

for eachσ∈S,the lower segmentis a boolean lattice(the face poset of a simplex).A rank function|·|endowed on the simplicial poset S is defined by|σ|=kforσ∈S ifis the face poset of a(k?1)-simplex.We assume that S is on the vertices set [m],the vertices of which are rank-1 elements.As defined,simplicial posets are a class of more general combinatorial objects than simplicial complexes.The face category CAT(S)consists of the objectsσ∈S and morphisms fromσtoτwheneverστ.Eachσ∈S we assign to a topological space

wherej∈[m],and we have the inclusionsBστforστ,from which it follows that the assignmentσBσgives a diagram(D2,S1)Sfrom CAT(S)to TOP.Then the momentangle complex over the simplicial poset S is defined by ZS=colim(D2,S1)S,which can be considered as a functor from CAT(S)to TOP.Lü and Panov [17]proved that the cohomology ring of ZSis isomorphic to the Tor-algebra of the face ring of S,just as in the case of the original moment-angle complex ZK,written as

where the face ring given by Stanley [21]is written as

They also gave a direct generalization of the Hochster formula [17],expressed as

to represent the algebraic Betti numbers in terms of the reduced homology of full subposets Saof S,wherea∈{0,1}m.

Throughout this paper,we consider a special case,called moment-angle manifolds overPn,which are formally defined as ZP=Z(KP;(D2,S1))?(D2)mwith a canonical action(S1)m×ZP→ZP,by

whereKPis the boundary complex of the dual ofPn;i.e.,it is an abstract simplicial complex on the vertices set [m]defined with the rule

As in Buchstaber and Panov’s book [9,Section 6.2],there is an equivalent way to define the moment-angle manifold overPn,which is the space(S1)m×Pn/～constructed by the trivial colouring onPn,namely a characteristic functionλ:F(P)→Zm,withFiei(thei-th standard unit vector of Zm),where the equivalence relation is defined as

2.2 Quotients of Moment-angle Manifolds Associated with Non-degenerate Colourings

This concept,introduced in Panov’s work [18],could be considered as a generalization of the quasi-toric manifolds defined by Davis and Januszkiewicz [11].If there is a subtorusTH?(S1)mwith a dimensionm?r(n≤r≤m)that acts freely on ZP,whereHis an(m?r)-subspace in Zmgenerated by vectorsh1,···,hm?rand

then its orbit space ZP/THis an(n+r)-manifold with an(S1)m/TH-action.

On the other hand,there exists a non-degenerate surjective Zr-characteristic functionλ:F(P)→Zrassociated with an(n+r)-manifoldM(Pn,λ),which is homeomorphic toP/TH.The property of the non-degeneracy of the functionλmeans that,for any vertexv=∩···∩are unimodular in Zr;i.e.,they span ann-dimensional direct summand of Zr.In addition,in this paper,wealwaysregard the epimorphism Λ :Zm→Zr, Λ (ei)=λ(Fi)as the characteristic functionλ:F(P)→Zr.The manifoldM(Pn,λ)=(S1)r×Pn/～λwith an(S1)r-action is constructed by the equivalence relation～λ,which is given by

BothM(Pn,λ)andhave the same orbit spacePn,and the homeomorphism between them is(S1)r(S1)m/TH-equivariant.This family of manifolds are known as quotients of moment-angle manifolds over simple polytopes.Quasi-toric manifolds overPnarea special type of manifolds belonging to this family,one of which is constructed by a non-degenerate surjective Zn-characteristic function,or equivalently,the orbit space ofPby a freeTm?n-action.

Remark 2.1Such a subtorusTHdetermines a surjective charateristic functionλ.On the other hand,given a non-degenerate epimorphism Λ :Zm→ZronPn,it is true thatTker Λfreely acts on the moment-angle manifold ZP.Furthermore,there is still an equivariant homeomorphism betweenM(Pn,λ)andP/Tker Λ.

2.3 Cubical Decompositon

GivenPn,Cone(KP),thecone of the boundary complexKPof the dual ofPn,will lead to a cubical decomposition ofPn[11].On the other hand,as was seen in Buchstaber and Panov’s book [9,Section 2.9],there is a canonical method to imbedPninto the boundary?Imof anmcube Im=[0,1]m,which is a more explicit geometrical realization of the cubical decomposition ofPn.

2.4 Main Results

For a moment-angle manifoldoverP2,Buchstaber and Panov gave an explicit description of the Betti numbers [7]and [8,Proposition 7.23]by considering theE2term of the Leray-Serre spectral sequence for the fibre bundleTm?2→→M4,whereM4is a quasi-toric manifold overP2,andE3=E∞for the spectral sequence.

Lemma 2.2([7])The(co)homology groups can be shown as

However,Buchstaber and Panov just gave a vague description of“up to a sign”,instead of computing the sign precisely.Motivated by this,we give an explicit presentation ofH?(;Z).

In addition,given a non-degenerate colouringλ:F(P2)→Zr,where 2≤r≤M,we say that the dependent colouring condition means that all the 3 vectors {λ(Fm),λ(F1),λ(F2)},···,{λ(Fm?1),λ(Fm),λ(F1)}coming from adjacent edges are dependent.Then the Betti numbers of the manifoldM(P2,λ)constructed by this colouring can be shown as follows:

Theorem 2.4When a non-degenerate colouringλ:F(P2)→Zrsatisfies the dependent colouring condition,we have that

3 A Sp ectral Sequence for M(P n,λ)

In this section,we are going to induce a spectral sequence from the filtration ofM(Pn,λ)given as follows.This is a preparation for the calculation of the(co)homology groups.

3.1 A filtration of Quotients of Moment-angle Manifolds

3.2 The Induced Sp ectral Sequence

The first step is to figure out the algebraic structure ofHp(Xr+l,Xr+l?1)andHp(Xr+l,Xr+l?1).We have

Lemma 3.1For allp,l∈Z,the homology and cohomology groups of the pairs(Xr+l,Xr+l?1)are free.

ProofFor dimf=n?l,we have the commutative diagram

Note that induced homomorphisms of the inclusionsiboundare isomorphic because of deformation retracts,and the excision theorem gives rise to the isomorphisms between those of the inclusionsiinter.Hence,we have the description

Apparently,the relative CW-complex（(D2)l×(S1)r?l,?(D2)l×(S1)r?l）has a cellular chain complex with trivial boundary operations(or equivalently consider the CW-complex(D2)l×(S1)r?l/?(D2)l×(S1)r?l),and,it also has 1(r+l)-cell,r?l(r+l?1)-cells,···,1 2l-cell,and 1 0-cell.Thus,it is clear that

LetQ?{l+1,···,r}with|Q|=j,(S1)Qbe the productYl+1×···×Yq×···×Yr,whereYq=S1ifq∈QandYq={1}otherwise.The differential can be determined by the map

Thus,we can use a large partitioned matrix

4 The Cohomology Ring of the Moment-angle Manifold over an mgon

In this section we concentrate on“highest-dimension”objects:moment-angle manifolds over polygons,which can be constructed by the colouringλ:F(P2)→Zm,Fiei.

4.1 Betti Numbers of

Using the filtration given before,we set

and

Step III:E2toEmp ageNow we can write all the next pages of this spectral sequence for the cohomology as

4.2 Generators of the Cohomology Groups

Notation:the“gap s”Letting Q={q0,···,qj+1}?{1,···,m},the“gaps of Q”is a subset of footnotes {0,···,j+1}defined as

Remark 4.2It is not necessary to make clear all the coefficients above,for the reason that each of those is 1 or?1,which will not affect the results,nor the calculation of the multiplication in the following subsection.

4.3 The Cup Prod uct on H?(;Z)

A filtration ofM(P2,λ)×M(P2,λ)After all the specific forms of the generators are represented,it is possible to figure out the structure of the ringH?（M(P2,λ);Z）.We endow a filtration to the manifoldM(P2,λ)×M(P2,λ)as follows:

Using the Künneth theorem and the Mayer-Vietoris sequence,it is easy to get the cohomology groups

Remark 4.3Note that all the group generators of the formare still the generators of the cohomology ring,however the highest-dimension generatore∈can be represented by the multiplication of lower-dimension elements,which is listed in the relations in Theorem 2.3.

Remark 4.4In the work [5],the cohomology of the moment-angle complex ZKwas already described as a Tor-algebra with the form ofIn addition,to be more tractable,from [9]we can divide the cohomology groups with the Hochster formula

and represent the ring structure by the multiplications

5 (Co)homology Groups of the Quotients of Moment-angle Manifolds under Special Colourings over an m-gon

In this section we consider a special class of characteristic functions on polygons,instead of the trivial ones for moment-anglemanifolds.Wetakea non-degeneratecolouringλ:F(P)→Zrsuch that all 3 vectors {λ(Fm),λ(F1),λ(F2)},···,{λ(Fm?1),λ(Fm),λ(F1)}from adjacent edges are dependent.

Remark 5.1It would be interesting if the Betti numbers of all the quotients of momentangle manifolds over polygons were given,however for general situations,the differentialsdwould be too complicated to deal with,as it would be hard to represent their kernels and images.Nevertheless,there would be some hope to achive this goal with this spectral sequence if we could find some way to simplify the calculation.