Homology Groups of Simplicial Complements∗

Jun MAFeifei FANXiangjun WANG

1 Introduction

Throughout this paper,k is a field or an integer ring Z.k[m]=k[v1,···,vm]is the graded polynomial algebra on m variables,and deg(vi)=2.The face ring(also known as the Stanley-Reisner ring)of a simplicial complex K on[m]is the quotient ring where IKis the ideal generated by those square free monomials vi1···visfor which{i1,···,is}is not a simplex in K.

For any simple polytope Pn,Davis and Januszkiewicz introduced a Tm-manifold ZPwith an orbit space Pnin[5].After that,Buchstaber and Panov generalized this definition to any simplicial complex K with vertices[m]={1,···,m},and named it the moment-angle complex（i.e.,the moment-angle complexi∈ σ and Yi=S1if i/σ）.

The following theorem is proved by Buchstaber and Panov[3]for the case over a field by using Eilenberg-Moore spectral sequence,and by[1]for the general case.

Theorem 1.1(see[7,Theorem 4.7]) Let K be a simplicial complex.Then the following isomorphism of algebras holds:

In their proof,they proved that H∗(ZK;k)=?H∗[Λ[u1,u2,···,um]⊗ k[x],d] first.Then they used the Koszul resolution on k to get

Since(k,k(K))has a natural Z⊕Zm-bigrade,the bigraded cohomology ring can be decomposed as follows:

Hochster gave a combinatorial description of the Tor-groups(k,k(K)).

Theorem 1.2(see[6])

where KI={ω ⊆ I|ω ∈ K},and?H−1(∅;k)=k.

Then in[4]they developed a more precise description.

Theorem 1.3 (see[4,Theorem 3.2.9])

where(∅;k)=k.

Recently,Zheng and Wang has proposed another way to compute(k(K),k)by using Taylor resolution on Stanley-Reisner ring k(K)in[8].This method was presented firstly by Yuzvinsky in[9].

They defined the simplicial complement P of a simplicial complex K as below.

Definition 1.1(Missing Face and Simplicial Complement)Let K be a simplicial complex on the set[m]as above.A missing face of K is the subset τ⊆[m],where τ/∈K and every proper subset of τ is a simplex of K.

A simplicial complement P is a subset of all non-faces of K containing all missing faces.

The Stanley-Reisner ideal IPis the homogeneous ideal generated by all square-free monomials xτ=xi1xi2···xis,where τ={i1,···,is} ∈ P.Obviously,for any two simplicial complements P and P′of the complex K,IP=IP′=IK.

Then one can define exterior algebra Λ∗[P]generated by all faces of the simplicial complement P.For any monomial u= τi1τi2···τis,define the total set Su= τi1∪ τi2∪ ···∪ τis.So Λ∗[P]has a natural Z⊕Zm-bigrade,which means

where Λi,I[P]is generated by the monomial u satisfying Su=I and the degree of monomial u is i.

Theorem 1.4(see[8])Let K be a simplicial complex on the set[m],and let P be one of the simplicial compliments of K.Give a differential d:Λr[P]−→ Λr−1[P],generated by

whereotherwise δ∂su=0.The differential d keeps the second grade.Then

Remark 1.1 Let K be a simplicial complex and P be one of its simplicial complements.By Theorem 1.4,we know that the homology group Hi(Λi,I[P],d)of simplicial complement P is not related to the choice of P.It just depends on the simplicial complex K.So if we fix the second degree by the set of all vertices[m],then we can get a homology group which just depends on the simplicial complex K.We call it homology group of simplicial complements.

In this paper,we will first give the geometric description of the new differential d on Λ∗,[m][P].And the following theorem is proved by using the simplicial Alexander duality.

Theorem 1.5 For any simplicial complex K on the set[m],let P be one of the simplicial compliments of K.Then we have the following group isomorphism:

where we assume H−1(Λ[∅],d)=k.

It is easy to check that PI={τ⊆ I|τ∈P}is a simplicial complement of KI,where KI={ω⊆I|ω∈K}.So we have following corollary.

Corollary 1.1

Remark 1.2 Consider the following commutative diagram:

The isomorphisms φ,ψ and η come from Theorem 1.4,Corollary 1.1 and a classical result in homological algebra theory respectively,and ζ= ηφψ−1is also an isomorphism.Thus we give a new proof of the Hochster theorem.

2 Geometric Description of the Differential d

If K is a simplex,the theorem is trivial.So in this paper,we assume that K is a simplicial complex on the set[m],but not a simplex.Denote P0is obviously one of the simplicial complements of the complex K.For any τi∈ P0,we have simplicial complex star∂Δm−1τi={τ∈ ∂Δm−1|τ∪ τi∈ ∂Δm−1}.Clearly,the star∂Δm−1τiis a triangulation of Dm−2.We denote by Ui=Int|star∂Δm−1τi|the interior of the geometric realization of the complex star∂Δm−1τi.

Proposition 2.1 U={Ui}i=1,2,···,sis an open cover of the topological space U(K),where U(K)=|∂Δm−1||K|.

Proof If x ∈ |∂Δm−1||K|,x must be an interior point of some simplex of∂Δm−1.Since xK,there is a simplex τ∈P0satisfying x∈Int|τ|.In other words,x∈Int|star∂Δm−1τ|,where τ∈P0.

Definition 2.1(The Nerve and theech Homology of an Open Cover)For any topological space X,let U={Ui}i=1,2,···,sbe an open cover of the space X.To every open set Ui,we assign a vertex i.If Ui1∩ Ui2∩ ···∩ Uirwe get a simplex(i1,i2,···,ir).Then we get a complex called the nerve of U,denoted by N(U),where

Define(X;U;k)=(N(U);k),called the reducedˇCech homology groups of an open cover U.

Theorem 2.1 Let K be a simplicial complex on[m],P0={τ1,τ2,···,τs}be defined as above.By Proposition 2.1,U={Ui}i=1,2,···,sforms an open cover of the topological space U(K).Then the homology groups H∗(Λ∗,[m][P0],d)is exactly the reducedech homology groups of the open cover U.Precisely,we have the following isomorphisms:

Before proving Theorem 2.3,we are going to work on the following lemma first.

Lemma 2.1 All notations are as above,i,j=1,2,···,s.Then

(1)if τi∪ τj[m],then

(2)Ui∩Uj∅⇔τi∪τj[m].

Proof(1)Obviously holds,by definition.

(2)If τ= τi∪ τj[m],then star∂Δm−1τ⊂ star∂Δm−1τi,since τ⊂ τi.So

Similarly,Int|star∂Δm−1τ|⊂ Int|star∂Δm−1τj|.Since τ[m],Int|star∂Δm−1τ|∅,and then Ui∩Uj∅.

On the other hand,if Ui∩Uj∅,then from(1)

Thus τi∪ τj[m].

Proof of Theorem 2.1 By Proposition 2.1 and Definition 2.2,we know U={Ui}i=1,2,···,sforms an open cover of the topological space U(K).And the nerve of the cover U is the complex

Lemma 2.4 shows that Ui1∩ Ui2∩ ···∩ Uir∅ if and only if τi1∪ τi2···∪ τir[m].

So the nerve complex can be written as

Let Λ∗[P0]be the exterior algebra generated by{τ1,τ2,···,τs}.We define another differential∂ :Λr[P0]→ Λr−1[P0]by

where∂su=τi1·····τirfor any monomial u=τi1τi2···τis.

We define a map

generated by Φ((i1,i2···,ir)):= τi1τi2···τir∈ Λr[P],where(i1,i2,···,ir)is an(r−1)-simplex of N(U).Obviously,Φ is a monomorphism.

Then we get a short exact sequence of the chain complexes,

where Λ∗+1[P0]/(N(U),k)is generated by all monomials u ∈ Λ∗,[m][P0](i.e.,Su=[m]).The differential d′is induced by ∂.

It is easy to see that there is a chain isomorphism

where(Λ∗,[m][P0],d)is as in Theorem 1.4.

Since(Λ∗[P0],∂)is isomorphic to the chain complex of the simplex with s+1 vertices,clearly(Λ∗[P0],∂)=0,and from the long exact sequence induced by the short exact sequence above,we get that

3 Barycentric Subdivision and Inflation Complex

Let K be a simplicial complex on the set[m]as above.Here come two new complexes constructed from K.

Definition 3.1(Barycentric Subdivision and Inflation Complex)The barycentric subdivision of the simplicial complex K is a simplicial complex K′on the set{σ ∈ K},where

The in fl ation complex of the complex K is also a simplicial complex F(K)on the set{σ∈K},where

Remark 3.1 The barycentric subdivision K′and in fl ation complex F(K)of the same complex K are both the complexes on the set{σ ∈ K}.For a simple x(σ0,σ1,···,σn)of K′,it is clear that σ0σ1··· σn,which means σ0∩σ1∩···∩σn∅.So(σ0,σ1,···,σn)is a simplex in F(K).Thus the barycentric subdivision K′is a subcomplex of the in fl ation complex F(K).

Definition 3.2 Let K be a simplicial complex.For any sunbclomlex L⊂K,define the(closed)combinatorial neighborhood UK(L)of L in K by

Lemma 3.1 Let K be a simplicial complex on[m].Then the geometric realization of the barycentric subdivision K′is a deformation retract of the geometric realization of the in fl ation complex F(K).

Before proving Lemma 3.1,we need the following statement coming from homotopy theory.

Statement A Given that a pair(X,A)satisfies the homotopy extension property,if the inclusion A→X is a homotopy equivalence,then A is a deformation retraction of X.

Proof of Lemma 3.1 In this proof,we do not distinguish simplicial complexes and their geometric realizations.

We prove this by induction on the number l of simplices of K.If l=1,the lemma is clearly true.For the induction step,choose a maximal simplex τ of K.Then K0=Kτ is a simplicial complex.Let L=∂τ={σ|σ τ}.Clearly L′is a subcomplex of.There is a deformation retraction r′:UK′0(L′)→L′corresponding to the vertex set map σ→σ∩τ(easy to verify that this map is simplicial).

Meanwhile,define a subcomplex L of F(K0)by

Similarly,there is a deformation retraction r′′:L → F(L)corresponding to the vertex set map:σ→σ∩τ.Since F(L)⋍L′by induction,the two deformation retractions give L⋍UK′0(L′),and then by statement A,UK′0(L′)is a deformation retraction of L.It is easy to see that UK′0(L′)=∩L.So there is a deformation retraction

which satisfiesSince⋍F(K0)by induction andL is a subcomplex of F(K0),applying statement A again,we get a deformation retraction:

The composition r1◦r2is a deformation retraction from F(K0)to K′0which satisfies r1◦r2(L)=U(L′).

From the definition of K0and L,we haveconeL.So r1◦r2can be naturally extended to a deformation retraction

Note thatis a subcomplex of coneUK′0(L′)and they are both contractible spacesextends to a deformation retraction fromThen by applying statement A again,there is a deformation retraction from coneto

which can be extended to a deformation retraction

Thus the composition r◦r0is the desired deformation retraction and the induction step is finished.

Remark 3.2 By Lemma 3.1,we have the following isomorphisms of homology groups(reduced or unreduced):

4 Proof of Theorem 1.5

Following the definitions in[4,7],we have Alexander dual simplicial complex of a complex and simplicial Alexander duality theorem.

Definition 4.1(Alexander Dual Simplicial Complex)Let K be a simplicial complex on[m],but not the simplex Δm−1.The Alexander dual simplicial complex is defined as

Theorem 4.1(Simplicial Alexander Duality,see[4])Let K be a simplicial complex on[m],but not Δm−1.Then the following duality holds:

where−1≤ j≤ m−2 and we use the agreement

Remark 4.1 As before,we use the notation P0=2[m]−K−[m]to denote a simplicial complement of K.

The in fl ation complex of the dual complexis the complex on the set{σ |[m]σ ∈ P0},i.e.,

There is a one-to-one map from{σ[m]|[m]σ/∈K}to P0(σ→[m]σ).Moreover,it is easy to check that σ1∩ σ2∩ ···∩ σi∅,[m]σj∈ P0for j=1,···,i,if and only if τ1∪ τ2∪ ···∪ τi[m],where τj=[m]σj∈ P0,j=1,···,i.

So the in fl ation complex of the dual complexis isomorphic to the complex on P0,which is also denoted by F():

We recall the proof of Theorem 2.3.If we assume P0={τ1,τ2,···,τs},U={Ui}i=1,2,···,swould form an open cover of the topological space U(K).The nerve complex is the complex below:

It is obvious that the in fl ation complex F()is isomorphic to the nerve complex N(U).

Now we can finish the proof of Theorem 1.5.

Proof of Theorem 1.5 By Theorem 2.3,we have

Remark 4.3 tells us F()∼=N(U).Then by Lemma 3.1,we have

Combining with the simplicial Alexander duality

we get the final result

Remark 4.2 Proposition 2.1 told us that U={Ui}i=1,2,···,sis an open cover of the topological space U(K),where U(K)=|∂Δm−1||K|.In[2,Corollary 13.3],there is a theory that if the cover U of the topological space X is good enough,then theˇCech homology of this cover is exactly the homology of the space X.Here “good” means that for each simplex σ =(i1,i2,···,in)∈ N(U),(Uσ)=0,where Uσ=Ui1∩Ui2∩···∩Uin.Luckily,it is easy to prove that the open cover U given by any simplicial complement of the complex P is “good”.It will give us another proof of Theorem 1.5,combining with the geometric Alexander duality theorem.

AcknowledgementThe authors are grateful to Q.Zheng for his helpful suggestions,without his help,the proof of Lemma 3.1 would be much more complicated.

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