Ivan LIMONCHENKO

1 Introduction

The main aim of this work is to show that one of the key objects of study in toric topology–the moment-angle manifold ZPof a simple convex n-dimensional polytope P–gives us an example of a smooth closed 2-connected manifold with a compact torus action such that its rational cohomology ring may contain a nontrivial higher Massey product of order n.These polytopes P are 2-truncated cubes and,moreover,flagnestohedra(see[22–23]).The class of 2-truncated polytopes was studied in toric topology by Buchstaber and Volodin,who proved that flagnestohedra can be realized as 2-truncated cubes and that Gal conjecture on γ-vectors of simple polytopes holds for 2-truncated cubes and,therefore,for all flagnestohedra(see[7]).We generalize in the polytopal sphere case the result of Baskakov[2]who constructed a family of triangulated spheres K whose moment-angle complexes ZKhave nontrivial triple Massey products of 3-dimensional classes in H∗(ZK).In the lowest dimension Baskakov’s construction gives a 2-sphere with 8 vertices K–the only K with a nontrivial triple Massey product in H∗(ZK)among all the fourteen 2-spheres on 8 vertices.Denham and Suciu[9]generalized the result of Baskakov by proving a combinatorial criterion for K to give a ZKwith a nontrivial triple Massey product of 3-dimensional classes in H∗(ZK).

Denote by K a simplicial complex of dimension n − 1 on the vertex set[m]={1,···,m}and by k the base field or the ring of integers.Let k[v1,···,vm]be a graded polynomial algebra on m variables,deg(vi)=2.The Stanley-Reisner ring(or the face ring)of K over k is the quotient ring

where IKis the ideal generated by square free monomials vi1···viksuch that{i1,···,ik}is not a simplex in K.The monomial ideal IKis called the Stanley-Reisner ideal of K.Then k[K]has a structure of a k-algebra and a module over k[v1,···,vm]via the quotient projection.

In what follows we denote by P a simple n-dimensional convex polytope with m facets(i.e.,faces of codimension 1)F1,···,Fm.Such a polytope P can be defined as a bounded intersection of m halfspaces:

where ai∈ Rn,bi∈ R.We assume that the hyperplanes defined by the equations〈ai,x〉+bi=0 are in general position,that is,at most n of them meet at a single point.We also assume that there are no redundant inequalities in(1.1),that is,no inequality can be removed from(1.1)without changing P.Then the facets of P are given by

Let APbe the m×n matrix of row vectors ai,and denote by bPthe column vector of scalars bi∈R.Then we can rewrite(1.1)as

Consider the affine map

which embeds P into

definition 1.1We define the spaceZPas a pullback in the following commutative diagram(see[5,Lemma3.1.6,Construction3.1.8]):

whereµ(z1,···,zm)=(|z1|2,···,|zm|2).The latter map may be thought of as the quotient map for the coordinatewise action of the standard torus

onCm.Therefore,Tmacts onZPwith a quotient spaceP,iZis aTm-equivariant embedding with a trivial normal bundle,andZPis embedded intoCmas a nondegenerate intersection of Hermitian quadrics.One can easily see thatZPhas a structure of a smooth closed manifold of dimensionm+n,called the moment-angle manifold ofP.

Suppose(X,A)=is a set of topological pairs.The following construction appeared firstly in the work of Buchstaber and Panov[5]and then was studied intensively and generalized in the works of Bahri,Bendersky,Cohen,Gitler[1],Grbi´c and Theriault[13],Iriye and Kishimoto[16],and others.

definition 1.2A polyhedral product is a topological space:

whereParticular casesof a polyhedral product(X,A)Kinclude moment-angle-complexesZK=(D2,S1)Kand real moment-angle complexesRK=(D1,S0)K.

Denote by KPthe nerve complex of P,i.e.,the boundary∂P∗of the dual simplicial polytope.It can be viewed as an(n−1)-dimensional simplicial complex on the set[m],whose simplices are subsets{i1,···,ik}such that Fi1∩ ···∩ FikØ in P.By[6,Theorem 6.2.4],ZPis Tm-equivariantly homeomorphic to the moment-angle-complex ZKP.

The Tor-groups of K acquire a topological interpretation by means of the following result due to Buchstaber and Panov.

Theorem 1.1(see[6,Theorem 4.5.4]or[21,Theorem 4.7])The cohomology algebra of the moment-angle-complexZKis given by the isomorphisms

where bigrading and differential in the cohomology of the differential bigraded algebra are defined by

In the third row,∗(KI)denotes the reduced simplicial cohomology of the induced subcomplexKIofK(the restriction ofKtoI⊂[m]).The last isomorphism is the sum of isomorphisms

and the ring structure is given by the maps

which are induced by the canonical simplicial mapsKI∪J→ KI∗ KJ(join of simplicial complexes)forI∩J=Øand zero otherwise.

Additively the following theorem of Hochster holds.

Theorem 1.2(see[15])For any simplicial complexKonmvertices,we have

The ranks of the bigraded components of the Tor-algebra

are called the bigraded Betti numbers of k[K]or K,when k is fixed.In what follows we need a particular case of the Hochster result for j=i+1.One has

whereand cc(PJ)equals the number of connected components of PJ.

Due to[6,Construction 3.2.8,Theorem 3.2.9],the Tor-algebra of K acqures a multigrading and the multigraded components can be calculated in terms of induced subcomplexes.

Theorem 1.3For any simplicial complexKonmvertices,we have

whereifais not a(0,1)-vector.

Moreover,if we denote by R(K)= Λ[u1,···,um]⊗ k[K]/(vi2=uivi=0,1 ≤ i≤ m)a graded algebra with the differential d as in Theorem 1.1,then R(K)also acquires multigrading and the following isomorphism holds:

for any simplicial complex K.

2 Nestohedra and Graph-Associahedra

We begin with a definition of a family of simple polytopes called nestohedra and state the result of Buchstaber and Volodin on geometric realization of flagnestohedra.

definition 2.1Let[n+1]={1,2,···,n+1},n ≥ 2.A building set on[n+1]is a family of nonempty subsetsB={S⊆[n+1]},such that

(1){i}∈Bfor all1≤i≤n+1,

(2)ifS1∩S2Ø,thenS1∪S2∈B.

A building set is called connected if[n+1]∈B.

Then a nestohedron is a simple convexn-dimensional polytopewhere in theMinkowski sum,one has

Note that facets ofPBare in1−1correspondence with proper elementsSinB(see[10]and[6,Proposition1.5.11]).

Example 2.1 If P is a combinatorial n-simplex,then the subset of 2[n+1]consisting of all the singletons{i},1≤i≤n+1 and the whole set[n+1]gives a connected building set B,such that P=PBfor any n≥2.

If P is a combinatorial n-cube,then the following set B consisting of

will be a connected building set for P for any n≥2.

Any n-dimensional nestohedron PBon a connected building set B can be obtained from an n-simplex by a sequence of its face truncations.In order to give the precise statement,suppose B0⊂B1being building sets on[n+1],and S∈B1.Then define a decomposition of S into elements of B0as S=S1⊔···⊔Sk,where Sjare pairwise nonintersecting elements of B0and k is minimal among such disjoint representations of S.One can see easily that this decomposition exists and is unique.

Theorem 2.1(see[6,Lemma 1.5.17,Theorem 1.5.18])Every nestohedronPBcorresponding to a connected building setBcan be obtained from a simplex by a sequence of face truncations.

More precisely,letB0⊂B1be connected building sets on[n+1].ThenPB1is combinatorially equivalent to the polytope obtained fromPB0by a sequence of truncations at the facesGi=corresponding to the decompositionsSi=⊔ ···⊔of elementsSi∈ B1B0,numbered in any order that is inverse to inclusion(i.e.,Si⊃ Si′⇒ i≤ i′).

Buchstaber suggested to call a simple convex n-dimensional polytope P a 2-truncated cube if it can be obtained from an n-cube by a sequence of cut offsome faces of codimension 2 only.It is allowed to cut offany codimension 2 face that we have on a previous step of the sequence of face truncations.

Example 2.2 Here is an example of a 3-dimensional 2-truncated cube P which we shall use later.

Figure 1 A 2-truncated cube P.

Then any flag nestohedron can be realized as a 2-truncated cube.The following statement holds.

Theorem 2.2(see[7,Proposition 6.1,Theorem 6.5])A nestohedronPBis a flag polytope if and only if it is a2-truncated cube.

More precisely,ifPBis a flag polytope,then there exists a sequence of building setsB0⊂B1⊂ ···⊂ BN=B,wherePB0is a combinatorial cube,Bi=Bi−1∪ {Si},andPBiis obtained fromPBi−1by a2-truncation at the faceFSj1∩ FSj2⊂ PBi−1of codimension2,whereSi=Sj1⊔ Sj2,andSj1,Sj2∈ Bi−1.

The next family of polytopes introduced by Carr and Devadoss[8]are flagnestohedra and,therefore,by Theorem 2.2 can be realized as 2-truncated cubes.

definition 2.2A graphical building setB(Γ)for a(simple)graphΓon the vertex set[n+1]consists of suchSthat the induced subgraphΓSon the vertex setS ⊂ [n+1]is a connected graph.

ThenPΓ=PB(Γ)is called a graph-associahedron.

Example 2.3 The following families of graph-associahedra are of particular interest in convex geometry,combinatorics and representation theory.

(1)Γ is a complete graph on[n+1].

Then PΓ=Penis a permutohedron,see Figure 2.

Figure 2 3-dimensional permutohedron and the corresponding graph.

(2)Γ is a stellar graph on[n+1].

Then PΓ=Stnis a stellahedron,see Figure 3.

Figure 3 3-dimensional stellahedron and the corresponding graph.

(3)Γ is a cycle graph on[n+1].

Then PΓ=Cynis a cyclohedron(or Bott-Taubes polytope,see[4]),see Figure 4.

Figure 4 3-dimensional cyclohedron and the corresponding graph.

(4)Γ is a chain graph on[n+1].

Then PΓ=Asnis an associahedron(or Stasheff polytope,see[24]),see Figure 5.

Figure 5 3-dimensional associahedron and the corresponding graph.

In order to determine the nerve complex KPof a graph-associahedron P=PΓ,we should describe the combinatorial structure of its face poset.The following is a reformulation of the general property stated in[6,Theorem 1.5.13].

Proposition 2.1Facets ofPΓare in1-1correspondence with non-maximal connected subgraphs ofΓ.

Moreover,a set of facets corresponding to such subgraphsΓi1,···,Γishas a nonempty intersection if and only if

(1)For any two subgraphsΓik,Γil,either they do not have a common vertex or one is a subgraph of another;

(2)If any two of the subgraphsΓik1,···,Γikl,l≥ 2do not have common vertices,then their union graph is disconnected.

Note that if P is a permutohedron,then its facets F1and F2have a nonempty intersection if and only if the corresponding subgraphs Γ1and Γ2are subgraphs of one another.

3 Bigraded Betti Numbers of Graph-Associahedra

In this section we describe certain bigraded Betti numbers of associahedra P in terms of combinatorics of their graphs Γ.This approach can be viewed as another argument to prove our previous result(see[18,Theorem 2.9])and can be used to compute bigraded Betti numbers β−i,2(i+1)(P)of all graph-associahedra P=PΓ.We begin with a following generalization of a result of Fenn(see[11,Theorem 4.6.4]).

Proposition 3.1SupposeP=PB1andQ=PB2aren-dimensional nestohedra on connected building setsBi,i=1,2andJ⊂B1⊂B2.Consider the following set

Thenis homeomorphic to

Proof By Theorem 2.1,any nestohedron PBon a connected building set B⊂2[n+1]can be obtained as a result of a sequence of face truncations starting with a simplex Δn.Thus the nerve complex of our nestohedron KP= ∂P∗can be obtained from a boundary of a simplex as a result of a number of barycentric subdivisions in some of its simplices.Moreover,Theorem 2.1 states that the new vertices(barycenters of those simplices)correspond to the decompositions of the elements in B2B1in a disjoint unions of elements of B1.Applying the descriprion of the face poset of Q in[6,Theorem 1.5.13] finishes the proof as any triangulation of a topological space is homeomorphic to the space itself.

Another way to prove this statement is similar to that of the proof in[11,Theorem 4.6.4].Indeed,the centers of the geometric realizations of P and Q in Rn+1are Minkowski sums of the centers of their simplices from the definition of a nestohedron.Then we can translate P and Q so that their centers coincide and project the boundary of P onto the boundary of Q outwards from their common center.Obviously,the image of PJis inand every facet incontains a point in the image of PJ.Finally,we make a continuous bijective transformation of the image(on each of the facet in)onto the whole

In particular,when B2=2[n+1]and Q is a permutohedron,we get the result of Fenn[11,Theorem 4.6.4].In order to describe the bigraded Betti numbers of associahedra combinatorially,we introduce the following notion of a special subgraph γ in Γ.

definition 3.1SupposeΓis a graph.For any of its connected subgraphsγ,one can compute the numberi(γ)of such connected subraphsinΓthat eitherγ∩Ø,γ,(in this case we say they have a nontrivial intersection)orγ ∩=Ø,γ ⊔is a connected subgraph inΓ.From now on we describe a subgraph inΓas a vertex set meaning that the subgraph consists of its vertices and all edges inΓconnecting these vertices(induced subgraph).We denote byimax=imax(Γ)the maximal value ofi(γ)over all connected subgraphsγinΓ.A connected subgraphγ,on whichimaxis achieved,will be called a special subgraph.

Example 3.1 On Figure 5,we have 3 special subgraphs:{1,2},{1,4}and{2,3}.The number imaxis equal to 4 and is achieved,for example,on γ={1,2}with the graphs eγ being{3},{4},{1,4},{2,3}(the latter two intersect γ nontrivially).

The following statement for the bigraded Betti numbers of the type β−i,2(i+1)(P)for associahedra P holds.

Theorem 3.1LetP=PΓbe an associahedron of dimensionn ≥ 3.Then fori＞ imax(Γ),one has

Denote the number of special subgraphs inΓbys.Letω = −imax,2(imax+1).Then

Proof By Theorem 1.2 and Proposition 2.1 it is sufficient to prove the following three cases.

(a)We have cc(PJ)≤ 2 if|J|＞ imax.In the latter case,if PJ=PJ1⊔PJ2with|J1,2|≥ 2,then there exists another J′⊂ B(Γ)such that PJ′=⊔PJ2′with||=1 and|J′|＞|J|.

(b)Suppose cc(PJ)=2,PJ=PJ1⊔PJ2,|J|＞imax.Then either|J1|=1 or|J2|=1.Moreover,if|J|=imax+1,|J1|=1 then J1consists of a special subgraph of Γ and J2consists of all the imaxconnected subgraphs in Γ determined in the definition of a special graph above.

(c)Suppose|J|＞imax+1.Then cc(PJ)=1.

For an associahedron Asnthe statement(a)follows from[18,Lemmas 2.13–2.14],the statement(b)follows from[18,Lemmas 2.15–2.16]and the statement(c)follows from[18,Lemma 2.17].

Remark 3.1 Using Propostion 2.1 one can see easily that Theorem 3.1 states that the last nonzero bigraded Betti number βω(P)in the sequence of β−i,2(i+1)(P),1 ≤ i≤ m − n is achieved precisely on PJwhich is a union of a facet of P corresponding to a special subgraph in Γ and all the facets of P that do not intersect this facet.All the PJwith a greater cardinality|J|of J are connected spaces in Rn.An argument similar to that in the proof of Theorem 3.1 shows the same holds for a permutohedron Pen,n≥3 and applying Proposition 3.1 one can get the same result for any graph-associahedron on a connected graph Γ.

As an application of Theorem 3.1,the values of imax(Γ)and s can be computed explicitly in terms of the combinatorics of the graph Γ.Using induction on the polytope dimension n for combinatorial enumerations in Γ it can be seen that a special graph γ is a path graph in Γ on eithervertices.This follows also from the proof of[18,Theorem 2.9],where the special graphs correspond to the longest diagonals in a regular(n+3)-gon G and the numbers of the vertices in such a graph are the numbers of vertices of G lying in one of the open halves of G divided by the diagonal.Thus,we get the following result(see[18,Theorem 2.9]).

Corollary 3.1For an associahedronPΓof dimensionn≥3,one has the following values ofimax=q(n)and s:

whereq=q(n)is

As graph-associahedra are flag polytopes,we can apply the previous result to studying the loop homology algebra H∗(ΩZP)for associahedra P.Namely,due to[12,Theorem 4.3]the minimal number of multiplicative generators of H∗(ΩZP)is equal toThen Theorem 3.1 gives us lower bounds for the number of multiplicative generators in the Pontryagin algebra of ZP.

4 Massey Products

In this section we prove the main result of this article concerning Massey higher products in H∗(ZP)(see Theorem 4.2)and a criterion when a nontrivial triple Massey product of 3-dimensional classes exists in H∗(ZPΓ)(see Proposition 4.1).We first prove the statement on triple Massey products in the graph-associahedron PΓcase,where Γ is an arbitrary(possibly disconnected)graph.

Let us state the following theorem due to Denham and Suciu which gives a combinatorial criterion for a simplicial complex K to produce a nontrivial triple Massey product of 3-dimensional classes in H∗(ZK).

Theorem 4.1(see[9,Theorem 6.1.1])The following are equivalent:

(1)There exist cohomology classesαi∈ H3(ZK),i=1,2,3for which〈α1,α2,α3〉is defined and nontrivial.

(2)The underlying graph(1-skeleton)ofKcontains an induced subgraph isomorphic to one of the five graphs in Figure6.

Moreover,all Massey products arising in this fashion are decomposable.

Figure 6 The five obstruction graphs.

Applying Theorem 4.1 to the graph-associahedra case gives us the following result.

Proposition 4.1There is a nontrivial triple Massey product〈α1,α2,α3〉of3-dimensional cohomology classesαi∈ H3(ZPΓ)fori=1,2,3if and only if there is a connected component ofΓonm ≥ 4vertices which is different from a complete graphK4.

Proof We start with a connected graph Γ case.Suppose that the number of vertices in Γ is less than 4.Then PΓis either a point,a segment,a pentagon or a hexagon.The corresponding moment-angle manifold ZPis either a disk D2,a sphere S3,or a connected sum of products of spheres respectively(see[3,20]).These manifolds are formal spaces,therefore,there are no nontrivial higher Massey products in H∗(ZP).

Suppose that there are 4 vertices in Γ.There are 6 combinatorially different connected graphs Γ on 4 vertices,thus,giving 6 combinatorially different 3-dimensional graph-associahedra PΓ.If Γ is a complete graph K4,then P=PΓis a permutohedron and the boundary of its dual simplicial polytope K=KPis combinatorially equivalent to a barycentric subdivision of a boundary of a 3-simplex.As there are no induced 5-cycles in K and the first two graphs in Figure 6 can not also be induced graphs in K,by Theorem 4.1 there are no nontrivial triple Massey products in H∗(ZP).On the other hand,using Figures 2–5 and Theorem 2.1 one can check easily that the third(middle)of the five graphs in the Figure 6 is an induced subgraph in the underlying graph(1-skeleton)of KPfor P being any of the other five 3-dimensional graph-associahedra on a connected graph with 4 vertices.The case of a connected graph on 4 vertices now holds from Theorem 4.1.

Suppose now,that Γ is a connected graph on more than 4 vertices.Using induction on the number of edges in Γ,we get an induced subgraph γ in Γ on 4 vertices.Using Proposition 2.1 the induced subcomplex in KP,P=PΓon the vertex set corresponding to all connected subgraphs in γK4will give us a nontrivial triple Massey product in H∗(ZP)by the argument above.On the other hand,if any connected subgraph on 4 vertices in Γ is a complete graph K4,then Γ is a complete graph Kn+1.Indeed,consider two different vertices α and β in Γ.Then there is a connected subgraph containing them in Γ.Such a graph γ with a minimal number of edges will obviously be a path between α and β.If it has more than 2 edges then it has more than 3 vertices and thus contains K4as an induced graph on some 4 of its vertices,thus γ being not minimal(any pair of vertices in K4is connected by one edge).Similarly,if γ has 2 edges then one of its 3 vertices is conected to another vertex of Γ (as Γ has more than 4 vertices and is connected)and we get K4as an induced subgraph.So,γ is not minimal again.Thus,γ has one edge,i.e., α and β are connected by an edge in Γ and Γ is a complete graph.

It remains to consider the case when Γ is a complete graph Kn+1,n ≥ 4 and P=PΓis a permutohedron.Note that KQis an induced subcomplex in KPfor any such P when Q is a 4-dimensional permutohedron.Consider the graph Γ=K5for Q and an induced subgraph in KQon the following vertices:

One can see easily that this induced subgraph is the first(left)graph in Figure 6.By Theorem 4.1 and Theorem 1.1(see(1.2)),any permutohedron P of dimension 4 and greater gives us a nontrivial triple Massey product in H∗(ZP).

Finally,the case of a disconnected graph Γ follows from Proposition 2.1 and Theorem 1.1 and the connected graph case as if two graphs Γ1and Γ2are disjoint,then for their union graph Γ,one gets PΓ=PΓ1× PΓ2and the moment-angle functor Z preserves products of polytopes(see[6,Chapter 4]).This finishes the proof.

Remark 4.1 Note that each of the six 3-dimensional graph-associahedraP=PΓmentioned above is a 2-truncated cube and,moreover,Pe3can be obtained from Cy3by cut offits 2 nonadjacent edges,if realized as a simplex truncation(see Theorem 2.1 and Figures 2 and 4).As ZPfor P=Inis a product of spheres and,therefore,is a formal manifold,it follows that a nontrivial higher Massey product in H∗(ZP)can either appear or vanish after a(codimension 2)face truncation(or after a stellar subdivision in the dual simplicial sphere KP).

Example 4.1 Consider P=Pe3(see Figure 2).It has n=3 and m=14.Letting us label its facets by the numbers 1,···,14 such that the bottom and upper 6-gon facets are 1 and 14 respectively,the bottom facets are labeled by 2,···,7 and the upper facets are labeled by 8,···,13,both clock wisely.

Consider the following 3-dimensional cocycles:

They correspond to 4 pairs of parallel facets of P if realized as a result of face truncations from Δ3.Suppose that they are representatives of the cohomology classes αi∈ H3(ZP),that is,αi=[ai]for i=1,···,4.

Then we get the following defining system A(see[17])for the Massey 4-product〈α1,α2,α3,α4〉(up to signs):

so 0 ∈ 〈α1,α2,α3,α4〉.Thus the two 3-products 〈α1,α2,α3〉and 〈α2,α3,α4〉are defined and vanish simultaneously and the 4-product 〈α1,α2,α3,α4〉is defined and trivial.

Remark 4.2 Note that the same calculation works in full generality,namely,if P=Pen,n ≥ 2 and the classes αi∈ H3(ZP),1≤ i≤ n+1 are represented by(n+1)pairs of the parallel permutohedra facets(see Figure 2),then 〈α1,···,αn+1〉is defined and trivial.Similarly,if P=Stn,n ≥ 2 and the classes αi∈H3(ZP),1≤ i≤ n are represented by n pairs of the parallel stellahedra facets(see Figure 3),then 〈α1,···,αn〉is defined and trivial.

We next consider a particular family of 2-truncated n-cubes P,one for each dimension n,for which ZPhas a nontrivial Massey product of order n.

definition 4.1Suppose thatIn=[0,1]n,n≥2is ann-dimensional cube with facetsF1,···,F2n,such thatFi,1 ≤ i ≤ ncontain the origin0,a unit inner normal vector toFi,1 ≤ i ≤ nis(0,···,1,···,0)with1in theith position,FiandFn+i,1 ≤ i ≤ nbeing parallel.Then we definePas a result of a consecutive cut of faces of codimension2fromIn,having the following Stanley-Reisner ideal:

wherevicorrespond toFi,1≤i≤2nand in the dots are the monomials corresponding to the new facets(i.e.,facets obtained after performing truncations).This determines uniquely the combinatorial type ofP.

Example 4.2 For n=2 we get a 2-dimensional cube(the square)P and its Stanley-Reisner ideal is the following one:

For n=3 we get a simple polytope P from Figure 1,for which K=KPis a simplicial complex with a nontrivial triple Massey product in H∗(ZK)due to the result of Baskakov(see[2]).Moreover,using the computer software Plantri it can be seen that K is the only one of the 14 combinatorially different 2-spheres with 8 vertices giving nontrivial higher Massey products in H∗(ZK)(see[9]).

The Stanley-Reisner ideal of P can be written as follows(see Figure 1):

Remark 4.3 The 2-truncated cube P is not a graph-associahedron as its number of facets(see[7,Theorem 9.2]).However,we can easily construct the building set B for P on the vertex set[n+1]by identifying Fiwith{1,···,i}for 1≤i≤n and identifying Fiwith{i−n+1}for n+1≤i≤2n.Then,by Theorem 2.2,we consecutively cut the following faces:

Thus,P=PBfor the building set B consisting of the building set B0of an n-cube from Example 2.1,the above subsets of[n+1]and all the subsets of[n+1]which are the unions of nontrivially intersecting elements in B.

Theorem 4.2Letαi∈ H3(ZP)be represented by a3-cocycleviun+ifor1 ≤ i≤ nandn ≥ 2.Then all Massey products of consecutive elements fromα1,···,αnare defined and the wholen-product〈α1,···,αn〉is nontrivial.

ProofLet us prove the theorem by induction on n.The base case n=2 is trivial:α1and α2are the classes of 3-dimensional spheres in ZPS3× S3and their cup-product(i.e.,Massey 2-product)is the dual to the fundamental class of ZP.

We first note that all Massey products of orders less than n vanish simultaneously in H∗,∗(ZP)HΛ[u1,···,um]⊗ k[P],d,i.e.,contain coboundaries.Starting with the representing cocycles viui+nof αi,it can be seen by induction on the dimension n of P that if a defining system C for the n-product 〈α1,···,αn〉can be extended from ith diagonal of the matrix C to its(i+1)th diagonal for all 2≤i≤n,then clm,m−l=i≥2 have either a form vkuj1···uj2i−1or a form vkuj1···uj2i−1+d(ukuj1···uj2i−1)(up to the signs).The latter can be checked as the differential in the cohomology algebra preserves multigrading and by using the codimension 2 face cuts from the definition of P(see also the example below).

Then the Massey n-product 〈α1,···,αn〉is defined and any cohomology class belonging to it lies in the multigraded component H−(2n−2),(2,···,2,0,···,0)(ZP)of the moment-angle manifold ZPwith one of its representatives being the class of the cocycle v1v2nu2···u2n−1.Up to sign we have the following equality for any representative c of an element in 〈α1,···,αn〉for any defining system C(see[17]):

where(n+1)×(n+1)-matrix C is upper triangular with zeros on the diagonal and ci,i+1=−viun+ifor 1≤i≤n,such that the following condition holds:

and=(−1)|cij|cijdepends on the degree|cij|of a matrix element cij.

By definition of higher Massey operations(see[17]),one has:d(c2,n+1)is a representative in〈α2,···,αn〉and d(c1,n)is a representative in 〈α1,···,αn−1〉.To prove that v1v2nu2···u2n−1is the only representing cocycle for the n-product,we use induction on n,the represnting monomials for the indeterminacies and the multigrading in H∗(ZP),see Theorem 1.3.For instance,the indeterminacy for the first of the(n−1)-products above is lying in the multigraded component of v2u3···unun+2···u2nand the only cocycle in that component is the coboundary d(u2···unun+2···u2n).The indeterminacy for the second of the(n − 1)-products above is lying in the multigraded component of v1u2···un−1un+1···u2n−1and the only cocycle in that component is the coboundary d(u1···un−1un+1···u2n−1).

Thus,the Massey n-product 〈α1,···,αn〉is defined and nontrivial,consisting only of the cohomology class of v1v2nu2···u2n−1.

Remark 4.4 Note that the nontrivial n-product constructed above is decomposable.Namely,one has[v1v2nu2···u2n−1]= ±[v1un+1···u2n−1]·[v2nu2···un].

Example 4.3 Consider the case n=4.Then the Stanley-Reisner ideal of P is

and the cohomology classes αi,1 ≤ i≤ 4 are represented by the cocycles ai=viu4+i,1≤ i≤ 4.One has(up to sign)

Then one has the following cocycle representing a class in 〈α1,α2,α3〉(here the Massey 2-product of a and b is equal to a ·b,a=(−1)|a|a):

and the following cocycle representing a class in 〈α2,α3,α4〉:

Alternatively,one has(up to sign)

The representing cocycle for 〈α1,α2,α3〉will be d(v3u1u2u5u6u7)=d(c1,4)and for 〈α2,α3,α4〉,one gets d(v4u2u3u6u7u8)=d(c2,5).

Thus,the Massey products 〈α1,α2,α3〉and 〈α2,α3,α4〉vanish simultaneously and the 4-product〈α1,α2,α3,α4〉is defined.More precisely,the representing cocycle c for〈α1,α2,α3,α4〉is equal to

Considering the multigrading in H∗(ZP)it is easy to see that the latter 4-fold product consists of the only class with a representative(up to sign)v1v8u2···u7in H−6,(2,···,2,0,···,0)(ZP) ⊂H−6,16(ZP)⊂H10(ZP),where ZPis a closed smooth 17-dimensional manifold.

Finally,one has[v1v8u2···u7]= −[v1u5u6u7]·[v8u2u3u4].

Using Theorem 4.2 we can construct a smooth closed 2-connected manifold M with a compact torus action,such that there are nontrivial higher Massey products of any prescribed orders n1,···,nr,r ≥ 2 in H∗(M).Namely,consider the building sets Bi,1 ≤ i≤ r for Pni,1 ≤ i ≤ r.Let M=ZP,where P=PB′,B′=B(B1,···,Br)(see[6,Construction 1.5.19])and B be a connected building set of a(r−1)-dimensional cube.Then P is a flag polytope combinatorially equivalent to Ir−1× Pn1× ···× Pnr(see[6,Lemma 1.5.20])and H∗(ZP)contains nontrivial Massey products of orders ni,1≤i≤r as the functor Z preserves products for simple polytopes.Note that P=PB′is still a flag nestohedron and,therefore,can be realized as a 2-truncated cube.

AcknowledgementsThe author is grateful to Victor Buchstaber and Taras Panov for many helpful discussions and advice,and also thanks James Stasheffand the referee of this article for their valuable comments and suggestions on improving the text.

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