Li YU Mikiya MASUDA

Abstract The authors give several new criteria to judge whether a simple convex polytope in a Euclidean space is combinatorially equivalent to a product of simplices. These criteria are mixtures of combinatorial, geometrical and topological conditions that are inspired by the ideas from toric topology. In addition, they give a shorter proof of a well known criterion on this subject.

Keywords Convex polytope, Product of simplices, Moment-angle complex

1 Background

A convex polytopePis the convex hull of a finite set of points in a Euclidean space Rd.The dimension ofPis the dimension of the affine hull of these points. Any codimension-one face ofPis called a facet ofP. We call ann-dimensional convex polytopePsimple if each vertex ofPis the intersection of exactlyndifferent facets ofP. Two convex polytopes are combinatorially equivalent if their face lattices are isomorphic. Topologically, combinatorial equivalence corresponds to the existence of a (piecewise linear) homeomorphism between the two polytopes that restricts to homeomorphisms between their facets, and hence all their faces(see [20, Chapter 2.2]).

IfP1⊂Rn1andP2⊂Rn2are two convex polytopes, then their productP1×P2is a convex polytope in Rn1+n2= Rn1×Rn2. Products of simplices are special type of simple polytopes with very delicate combinatorial structures. They play an important role in Coxeter’s famous work[10]on the discrete reflection groups in Euclidean spaces and also appear in many different researches in combinatorics(see [1, 12–13]). In this paper, we give several new criteria to judge whether a convex polytope is combinatorially equivalent to a product of simplices (Theorems 2.2–2.3) and at the same time, list some known ones (Proposition 2.1). Some of these criteria are purely combinatorial, while others are phrased in geometrical or topological terms. Since some of our new criteria are inspired from the ideas in toric topology, we first explain some basic constructions and facts in toric topology that are relevant to our discussion.

An abstract simplicial complex on a set [m] = {v1,···,vm} is a collectionKof subsetsσ⊆[m] such that ifσ∈K, then any subset ofσalso belongs toK. We always assume that the empty set belongs toKand refer toσ∈Kas a simplex ofK. In particular, one-element simplices are called vertices ofK. IfKcontains all one-element subsets of [m], then we say thatKis a simplicial complex on the vertex set [m]. To avoid ambiguity in our argument, we also useV(K) andV(σ) to refer to the vertex sets ofKand any simplexσinK, respectively.For any subsetω⊆[m], we callK|ω={σ∈K|σ⊆ω} the full subcomplex ofKby restricting toω.

Any abstract simplicial complexKadmits a geometric realization in some Euclidean space.Also sometimes we useKto denote its geometric realization when the meaning is clear in the context.

Given a finite abstract simplicial complexKon a set [m] and a pair of spaces (X,A) withA⊂X, we can construct a topological space (X,A)Kby:

Here ∏means Cartesian product. So (X,A)Kis a subspace of the Cartesian product ofmcopies ofX. It is called the polyhedral product or the generalized moment-angle complex ofKand (X,A). In particular, ZK= (D2,S1)Kand RZK= (D1,S0)Kare called the momentangle complex and the real moment-angle complex ofK,respectively(see[4, Section 4.1]). The natural actions of (Z2)mon (D1)mand (S1)mon (D2)minduce canonical actions of (Z2)mon RZKand (S1)mon ZK, respectively.

WhenKis the boundary of the dual of a simple convex polytopeP, the ZKand RZKare closed manifolds, also denoted by ZPand RZPrespectively. In this case, ZPand RZPare called the moment-angle manifold and the real moment-angle manifold ofP, respectively (see[3, Section 6.1]). These manifolds can be constructed in another way as described below (see[11, Construction 4.1]).

LetPnbe ann-dimensional simple convex polytope. Let F(Pn)={F1,···,Fm}be the set of facets ofPn. Let {e1,···,em} be a basis of (Z2)mand define a mapλ:F(Pn)→(Z2)mbyλ(Fi)=ei. Then we can construct a space

where (p,g)～(p′,g′) if and only ifp=p′andg−1g′∈Gλp,whereGλpis the subgroup of (Z2)mgenerated by the set {λ(Fi)|p∈Fi}. Letπλ:M(Pn,λ)→Pnbe the quotient map. One can show that RZPnis homeomorphic toM(Pn,λ) and the canonical action of (Z2)mon RZPncan be written onM(Pn,λ) as:

The moment-angle manifold ZPncan be similarly constructed fromPnand a map Λ :F(Pn) →Zm,where {Λ(F1),···,Λ(Fm)} is a unimodular basis of Zm. Indeed, if we identify the torus (S1)m=Rm/Zm, then we have

where (p,g)～(p′,g′) if and only ifp=p′andg−1g′∈Tλp,whereTλpis the subtorus of (S1)mdetermined by the linear subspace of Rmspanned by the set {Λ(Fi)|p∈Fi}.

In addition, RZPnand ZPnare smooth manifolds. In fact, there exists an equivariant smooth structure on RZPn(or ZPn) with respect to the canonical (Z2)m-action (or (S1)maction). The reader is referred to [3, Chapter 6] or [4, Chapter 6] for the discussion of smooth structures on (real)moment-angle manifolds. Moreover,for any proper facefofPn,is an embedded closed smooth submanifold of RZPnwhich is the fixed point set of the subgroup of (Z2)mgenerated by {λ(Fi)|f∈Fi} under the canonical (Z2)m-action.

2 Descriptions of Products of Simplices

For anyk∈N, let Δkdenote the standardk-dimensional simplex, which is

For anyn1,···,nq∈N,consider Δn1×···×Δnqas a product of Δn1,···,Δnqin the Cartesian product Rn1+1×···×Rnq+1.

We first list some descriptions of products of simplices appearing in Wiemeler’s paper [19].

Theorem 2.1(see[19])Let Pnbe an n-dimensional simple convex polytope with m facets,n≥3. Then the following statements are equivalent:

(a)Pnis combinatorially equivalent to a product of simplices.

(b)Any2-dimensional face of Pnis either a3-gon or a4-gon.

(c)There exists a quasitoric manifold M2nover Pnwhich admits a nonnegatively curved Riemannian metric that is invariant under the canonical(S1)n-action on M2n.

A quasitoric manifoldM2noverPnis the quotient space of ZPnunder a free action of a rankm−ntoral subgroup of (S1)m(see [11]). There is a canonical (S1)n-action onM2ninduced from the canonical action of (S1)mon ZPn, which makesMna torus manifold (see[14]).

The equivalence of Theorem 2.1 (a) and (b) is a corollary of [19, Proposition 4.5]. The equivalence of Theorem 2.1 (a) and (c) is a corollary of [19, Lemma 4.2]. Note that Theorem 2.1(b) is a particularly useful description of products of simplices. Indeed, the proofs of many other descriptions of products of simplices in this paper boil down to this one first. But the proof of [19, Proposition 4.5] is a little long and not particularly easy to follow. We will give a shorter proof of the equivalence of Theorem 2.1 (a) and (b) in the appendix to make our paper more self-contained.

Remark 2.1The equivalence of Theorem 2.1(a)and(b)also implies that a simple convex polytope is combinatorially equivalent to a product of simplices if and only if every facet of the polytope is combinatorially equivalent to a product of simplices. In fact this statement appeared in [10, Lemma 2.7] where a product of simplices is called a “simplicial prism”. But the proof of [10, Lemma 2.7] in [10] is a bit vague in the final step.

Next, we give more descriptions of products of simplices from combinatorial and topological viewpoints. For convenience, let us introduce some notations first.

• For any topological spaceXand any field k, let

whereHi(X;k) is the singular cohomology ofXwith coefficient in k.

• For any vertexvin a simplicial complexK, we denote by linkKvthe link ofvinK. We denote a simplex spanned by verticesv0,v1,···,vpinKby [v0,v1,···,vp] and its boundary complex by∂[v0,v1,···,vp].

In addition, for a simplicial complexKon the vertex set [m]={v1,···,vm}, we can define a new simplicial complexL(K) fromK, called the double ofK, whereL(K) is a simplicial complex on the vertex set [2m] = {v1,v′1,···,vm,v′m} determined by the following condition:ω⊂[2m] is a minimal (by inclusion) missing simplex ofL(K) if and only ifωis of the form {vi1,v′i1,···,vik,v′ik},where {vi1,···,vik} is a minimal missing simplex ofK. Note that any minimal missing simplex inL(K) must have even number of vertices. The double ofKis a special case of iterated simplicial wedge construction (also called simplicial wedgeJconstruction). Indeed, by the notation introduced in [2],L(K)=K(2,···,2).

The following are some basic facts aboutL(K) (see [18–19]).

• dim(L(K))=m+dim(K) (see [18, Lemma 1.2]).

•L(K1∗K2)=L(K1)∗L(K2) (here ∗is the join of two simplicial complexes).

•IfK=∂P∗,whereP∗is the simplicial polytope dual to a simple convex polytopeP,thenL(K) =∂L(P)∗,whereL(P) is a simple convex polytope called the double ofP(see [17] for the construction ofL(P)).

•L(∂Δk)=∂Δ2k+1.

The following are some easy or well known facts on products of simplices. We list them here and give a simple proof for reference.

Proposition 2.1Let P be an n-dimensional simple polytope with m facets and let K be the boundary of the simplicial polytope dual to P. Then the following statements are all equivalent:

(a)P is combinatorially equivalent to a product of simplices.

(b)K is simplicially isomorphic to ∂Δn1∗···∗∂Δnqfor some n1···,nq∈N.

(c)The vertex sets of all the minimal missing faces of K form a partition of V(K).

(d)L(K)is simplicially isomorphic to ∂Δl1∗···∗∂Δlqfor some l1···,lq∈N.

(e)There exists some fieldkso thathrk(RZK;k)=2m−dim(K)−1, or equivalentlyhrk(RZP;k)=2m−n.

(f)There exists some fieldkso thathrk(ZK;k)=2m−dim(K)−1,or equivalentlyhrk(ZP;k)=2m−n.

ProofThe equivalences of (a)⇔(b) and (b)⇔(c) are easy to see.

(b)⇒(d) IfK=∂Δn1∗···∗∂Δnq, then

(d) ⇒(c) SupposeL(K) =∂Δl1∗···∗∂Δlq. Notice that for each 1 ≤j≤q, Δljis a minimal missing simplex ofL(K). So Δljmust have even number of vertices, which implies thatljis an odd integer. Then by (b)⇔(c),the vertex sets of all the minimal missing faces ofL(K)form a partition ofV(L(K)). This forces the vertex sets of all the minimal missing faces ofKto form a partition ofV(K) as well, which is (c).

(a)⇒(e) and (f) IfP=Δn1×···×Δnq,n1+···+nq=n, then

The number of facets ofPism=n+q. It is clear that for any field k,

(e)⇒(a) For any vertexvofK, letmvbe the number of vertices in linkKv. According to the proof of [18, Theorem 3.2] (note that the argument there works for any coefficient), there is a subspaceXof RZKso that

whereXis the disjoint union of 2m−mv−1copies of RZlinkKv. So we have

Then hrk(RZlinkKv;k) ≤ 2mv−n+1. On the other hand, [18, Theorem 3.2] tells us that hrk(RZlinkKv;k)≥2mv−n+1(since dim(linkKv)=n−2). So we obtain

Note that ifvis the vertex corresponding to a facetFofP, then RZlinkKv=RZF. Therefore, we have shown that if the condition (e) holds forP, it should hold for any facet ofPas well.

By iterating the above argument, we deduce that the condition (e) holds for all the two dimensional faces ofP. It is easy to show that the real moment-angle manifold of ak-gon is a closed connected orientable surface with genus 1+(k−4)2k−3(see [4, Proposition 4.1.8]). So any 2-dimensional face ofPis either a 3-gon or a 4-gon. Then by Theorem 2.1(b),the polytopePis combinatorially equivalent to a product of simplices.

(f)⇒(a) First of all, [18, Lemma 2.2] says that there is a homeomorphism ZK～=RZL(K).SinceKhasmvertices, dim(L(K))=m+dim(K)=m+n−1. If hrk(ZK;k)= 2m−n, then we have

So (e) holds forL(K). Since we have already shown (e) ⇒(a) ⇔(b),L(K) is simplicially isomorphic to∂Δn1∗···∗∂Δnqfor somen1···,nq∈N. Then we finish the proof by the equivalence of (d) and (a).

Remark 2.2The equivalences of(b),(e)and(f)in Proposition 2.1 are stated in[4,Section 4.8] as an exercise.

Moreover, we can judge whether a simple polytopePis combinatorially equivalent to a specific product of simplices via some combinatorial invariants called bigraded Betti numbers,which are derived from the Stanley-Reisner ring ofP(see [4, Section 3.2] for the definition).Indeed,it is shown in[9]that a simple polytopePis combinatorially equivalent to Δn1×···×Δnqif and only ifPhas the same bigraded Betti numbers as Δn1×···×Δnq. Simple polytopes with this kind of property are called combinatorially rigid (see [6, 8]).

Next, we give a new combinatorial criterion to judge whether a simple polytope is combinatorialy equivalent to a product of simplices.

Theorem 2.2Let K be the boundary of the simplicial polytope dual to a simple polytope P. Then P is combinatorially equivalent to a product of simplices if and only if the following conditions hold for K:For any maximal simplex σ in K and any vertex v of σ, the full subcomplex of K by restricting to V(K)−V(σ)is a simplex of K, denoted by ξσ, and moreover the intersection of ξσandlinkKv is a simplex(could be empty)as well.

ProofSuppose thatPis a product of simplices. ThenK=∂Δn1∗···∗∂Δnqfor somen1,···,nq∈N. Denote the vertices of∂Δnkbyvk0,vk1,···,vknkfor eachk=1,···,q. Then for a maximal simplexσinK, there exists 0 ≤lk≤nk,k=1,···,q, so that

Note that whennk=1,is empty. Then the intersection ofand linkKvkiis exactly the simplexifnk=1. The necessity of these conditions is proved.

For the sufficiency, we first show that if these conditions hold forK, then they also hold for the link of any vertex ofK. When dim(K) ≤1, the theorem is obviously true. So we assume dim(K) ≥2 below. Letube an arbitrary vertex ofK. Letσbe a maximal simplex ofKcontaininguand letvbe an arbitrary vertex ofσdifferent fromu. By our assumption,the intersectionξσ∩linkKuandξσ∩linkKvare both simplices. Letτbe the simplex withV(τ)=V(σ)−{u}. Thenτis a maximal simplex in linkKu. SinceV(ξσ)=V(K)−V(σ), we have

Sinceξσ∩linkKuis a simplex,the full subcomplex of linkKuby restricting toV(linkKu)−V(τ)must agree withξσ∩linkKu. Moreover,sincevcould be any vertex ofτ, we need to show that the intersection of the simplexξσ∩linkKuwith linklinkKuvis also a simplex. Observe that linklinkKuv=linkKu∩linkKv. So we have

The intersection of the two simplicesξσ∩linkKuandξσ∩linkKvhas to be a simplex(could be empty) by the definition of simplicial complex. Moreover, whenσranges over all the maximal simplices ofKcontainingu, the vertexvwill range over all the vertices in linkKu. So our argument shows that these conditions hold for linkKu.

By iterating the above argument, we can prove that for any codimension-two simplexηofK, the link ofηinKis a simplicial circle which satisfies the conditions. This forces the link ofηto be either∂Δ2or∂Δ1∗∂Δ1. Dually it means that any 2-dimensional face ofPis either a 3-gon or a 4-gon. Then by Theorem 2.1(b), the polytopePis combinatorially equivalent to a product of simplices.

Next,we give some new descriptions of products of simplices in terms of geometric conditions on real moment-angle manifolds of simple convex polytopes. We first recall a concept in metric geometry (see [5, Definition 3.1.12]).

Definition 2.1(Quotient Metric Space)Let(X,d)be a metric space and letRbe an equivalence relation on X. The quotient semi-metric dRis defined as

where the infimum is taken over all choices of{pi}and{qi}such that the point qiisR-equivalent to pi+1for all i=1,···,k−1. Moreover, by identifying points with zero dR-distance, we obtain a metric space(X/R,d)called the quotient metric space of(X,d).

Suppose thatPis a simple convex polytope in a Euclidean space Rd. ConsiderPto be equipped with the intrinsic metric. More precisely,the intrinsic metric onPdefines the distance between any two pointsxandyinPto be the infimum of lengths of piecewise smooth paths inPthat connectxandy. Note that the intrinsic metric onPcoincides with the subspace metric onP,sincePis convex.

By the construction in (1.2), RZP=M(P,λ) is a closed manifold obtained by gluing 2mcopies ofPalong their facets. We can assume that the 2mcopies ofPare congruent convex polytopes inside the same Euclidean space and the gluings of their facets are all isometries.Then by Definition 2.1, we obtain a quotient metric on RZP, denoted bydP. It is clear that the metricdPis invariant with respect to the canonical action of (Z2)mon RZP(see (1.3)).

Remark 2.3We can also call (RZP,dP) a Euclidean polyhedral space, which just means that it is built from Euclidean polyhedra (see [5, Definition 3.2.4]).

Note that ifP′is another simple convex polytope combinatorially equivalent toPbut not congruent toP, the two metric spaces (RZP′,dP′) and (RZP,dP) are not isometric in general(though RZP′is homeomorphic to RZP).

Theorem 2.3Let P be an n-dimensional simple convex polytope with m facets, n≥2.Then the following statements are all equivalent:

(a)P is combinatorially equivalent to a product of simplices.

(b)There exists a non-negatively curved Riemannian metric onRZPthat is invariant under the canonical(Z2)m-action onRZP.

(c)There exists a simple convex polytope P′combinatorially equivalent to P so that the metric space(RZP′,dP′)is non-negatively curved.

(d)There exists a simple convex polytope P′combinatorially equivalent to P so that all the dihedral angles of P′are non-obtuse.

Note that a Riemannian metric on a manifold is non-negatively curved means that its sectional curvature is everywhere non-negative, while a metric space being non-negatively curved is defined via comparison of triangles (see [5, Section 4]).

Proof(a)⇒(b)The real moment-angle manifold of a product of simplices Δn1×···×Δnqis diffeomorphic to a product of standard spheresSn1×···×Snq,whereSk={(x1,···,xk+1)∈for anyk∈N. LetSkbe equipped with the induced Riemannian metric from Rk+1. Then it is easy to check thatSn1×···×Snqis a nonnegatively curved Riemannian manifold with respect to the product of the Riemannian metrics onSn1,···,Snq.

(b)⇒(a) Recall the definition ofπλ:M(P,λ)=RZP→Pin (1.2). For any proper facefofP, letMf=It is easy to see the following.

•Mfis an embedded closed submanifold of RZPwhich has 2m+dim(f)−n−mfconnected components, wheremfis the number of facets off.

• Each connected component ofMfis diffeomorphic to RZf.

Note thatMfis the fixed point set of a rankn−dim(f) subgroup of (Z2)munder the canonical action of (Z2)mon RZP. Morever, since the Riemannian metric is (Z2)m-invariant,each component ofMfis a totally geodesic submanifold of RZP(see [15, Theorem 5.1]), and so is non-negatively curved with respect to the induced Riemannian metric from RZP. This implies that the condition (b) holds for RZfas well.

In particular when dim(f) = 2, the RZfis a closed connected surface with non-negatively curved Riemannian metric. Then by Gauss-Bonnet theorem,the Euler characteristicχ(RZf)≥0, which implies thatfhas to be a 3-gon or a 4-gon. Then by Theorem 2.1(b), the polytopePis combinatorially equivalent to a product of simplices.

(a) ⇒(c) Suppose thatPis combinatorially equivalent to Δn1×···×Δnq,wheren1+···+nq=n. Consider the standard simplex Δkas a metric subspace of Rk+1with the intrinsic metric. LetP′= Δn1×···×Δnqbe the product of theqmetric spaces Δn1,···,Δnq. For each 1 ≤i≤q, let {vi0,···,vini} be the set of vertices of Δni. Then all the facets ofP′are (see[7])

wherefikiis the codimension-one face of the simplex Δni,which is opposite to the vertexviki.The total number of facets ofP′ism=n+q.

ClaimAs a metric space, (RZP′,dP′) is isometric to the product of theqmetric spaces(RZΔn1,dΔn1),···,(RZΔnq,dΔnq).

Indeed if we glue two copies ofP′along the facetFiki, we obtain

We can decompose the gluing procedure in the construction (1.2) for RZP′intoqsteps. Thei-th step only glues those facets of the form {Fiki,0 ≤ki≤ni} in the 2mcopies ofP′, which gives us the factor (RZΔni,dΔni), while fixing all other factors in the product. After the first step, we obtain 2m−n1−1copies of RZΔn1×Δn2×···×Δnq. After the second step, we obtain 2m−n1−n2−2copies of RZΔn1×RZΔn2×Δn3×···×Δnqand so on. Then our claim follows.

Moreover, observe that for anyk∈N, (RZΔk,dΔk) is isometric to the boundary of the(k+1)-dimensional cross-polytopeQk+1whose vertices are

Recall that then-dimensional cross-polytope is the simplicial polytope dual to then-dimensional cube (see Figure 1 for the casesn=2,3).

Figure 1 Cross-polytopes of dimension 2 and 3.

It is well known that the intrinsic metric on any convex hypersurface(i.e.,the boundary of a compact convex set with nonempty interior)in a Euclidean space Rn(n≥3) is non-negatively curved(see[5,p.359]). SinceQk+1is a convex polytope in Rk+1,(RZΔk,dΔk)is non-negatively curved for anyk≥2. Whenk= 1, the boundary ofQ2is a piecewise smooth simple curve in R2. By definition (see [5, Definition 4.1.9]), the intrinsic metric on any piecewise smooth simple curve is non-negatively curved because any geodesic triangle on the curve is degenerate.Therefore, we can conclude that (RZP′,dP′) is non-negatively curved because the product of non-negatively curved Alexandrov spaces is still non-negatively curved (see [5, Chapter 10]).

(c) ⇒(d) If the metricdP′on RZP′is non-negatively curved, we want to show that the dihedral angle between any two adjacent facetsF1andF2ofP′is non-obtuse. Otherwise, we assume that the dihedral angleθbetweenF1andF2is obtuse. Choose a pointOin the relative interior ofF1∩F2, a pointA∈F1andB∈F2so that the line segmentsare perpendicular toF1∩F2. Then ∠AOB=θ. Suppose that the lengths of the line segmentsare

In the gluing construction(1.2)for RZP′,consider two copies ofP′glued along the facetF1. We then have an isosceles triangle △AB1B2in RZP′(see Figure 2). Whenais small enough, the distance betweenB1andB2in(RZP′,dP′)is 2aby the definition of the quotient metric becauseis the shortest path betweenB1andB2in (RZP′,dP′). Moreover, letbe a triangle in the Euclidean plane R2which has the same lengths of sides as △AB1B2. Sinceθis obtuse, it is clear that △AB1B2is strictly thinner thani.e.,

Figure 2 Comparison of triangles.

But this contradicts our assumption that the metricdP′on RZP′is non-negatively curved(see[5, Section 4.1.5]). Therefore,θhas to be non-obtuse.

(d)⇒(a) Suppose thatF1,F2andF3are three facets ofP′withF1∩F2∩F3/=∅. ThenF1∩F2andF1∩F3are codimension-one faces ofF1. By our assumption, the dihedral angles of (F1,F2), (F1,F3) and (F2,F3) are all non-obtuse. We claim that the dihedral angle betweenF1∩F2andF1∩F3inF1is non-obtuse as well.

Indeed, we can assume thatP′sits inside Rnand letηi∈Rn(i=1,2,3)be a normal vector ofFipointing to the interior ofP(see Figure 3). By choosing a proper coordinate system of Rn, we can assume thatη1= (0,···,0,1) ∈RnandF1lies in the coordinate hyperplane{xn=0}⊂Rn. Letη2=(a1,···,an−1,an),η3=(b1,···,bn−1,bn). Since the dihedral angles of (F1,F2), (F1,F3) and (F2,F3) are all non-obtuse, the inner products ofη1,η2,η3satisfy

Figure 3 Dihedral angles of a simple convex polytope.

Note that(a1,···,an−1,0)and(b1,···,bn−1,0)are normal vectors ofF1∩F2andF1∩F3insideF1respectively. So (2.1) implies that the dihedral angle betweenF1∩F2andF1∩F3inF1is non-obtuse. Our claim is proved.

By iterating the above arguments,we can show that for any 2-dimensional facefofP′, any interior angle offis non-obtuse. Sincefis a Euclidean polygon, it must be either a 3-gon or a 4-gon. SincePis combinatorially equivalent toP′, any 2-face ofPis either a 3-gon or a 4-gon,too. Then by Theorem 2.1(b), the polytopePis combinatorially equivalent to a product of simplices.

Remark 2.4The equivalence of Theorem 2.3(a)and(d)is also stated in[10, Lemma 2.8].

Remark 2.5In the statement of Theorem 2.3(b), if we do not require the Riemannian metric on RZPto be(Z2)m-invariant,it is still likely thatPhas to be combinatorially equivalent to a product of simplices (see [16, Section 5.2]). But we do not know how to prove this so far.

3 Appendix

Here we give another proof of Theorem 2.1 (a)⇔(b). For brevity, we say that a simplicial complex is a sphere join if it is isomorphic to∂Δn1∗···∗∂Δnqfor somen1,···,nq∈N. One dimensional sphere join is either∂Δ2(boundary of a triangle) or∂Δ1∗∂Δ1(boundary of a square). Let us first prove the following theorem.

Theorem 3.1Let K be a simplicial complex of dimension n. Suppose that K satisfies the following two conditions:

(a)K is a pseudomanifold,

(b)the link of any vertex of K is a sphere join of dimension n−1,Then K is a sphere join.

Recall thatKis ann-dimensional pseudomanifold if the following conditions hold:

(i) Every (n−1)-simplex ofKis a face of exactly twon-simplices forn>1.

(ii) For every pair ofn-simplicesσandσ′inK, there exists a sequence ofn-simplicesσ=σ0,σ1,···,σk=σ′such that the intersectionσi∩σi+1is an (n−1)-simplex for alli.The condition (ii) means thatKis a strongly connected simplicial complex.

ProofFirst of all, assumption (b) implies that the link of anyk-simplex inKis a sphere join of dimensionn−k−1. This is because for anyk-simplexσwith a vertexw, the link ofσinKis the link of the (k−1)-simplexσ∩linkKwin linkKw. Then by the assumption that linkKwis a sphere join of dimensionn−1, the link of any (k−1)-simplex in linkKwmust be a sphere join of dimensionn−k−1 (corresponding to the fact that any face of a product of simplices is also a product of simplices).

Letwbe an arbitrary vertex ofK. By assumption (b), the link linkKwis of the form linkKw=∂Δn1∗··· ∗∂Δnq,wheren1+ ··· +nq=n. Denote the vertices of∂Δnkbyvk0,vk1,···,vknkfork=1,2,···,q, so that

LetIbe the set of verticesv11,···,v1n1,···,vq1,···,vqnq. Then[I]is a maximal simplex in linkKwand the simplex [I,w] spanned byIandwis of dimensionn. SinceKis a pseudomanifold by assumption(a),there is a unique vertexvinKsuch that[I,v]∩[I,w] =[I]. We have two cases below.

Case 1The case wherev /∈linkKw. In this case, we claimK=∂[v,w]∗linkKw. The proof is as follows. Choose an element fromIarbitrarily, sayvij(1 ≤i≤q, 1 ≤j≤ni). SetI= (I{vij})∪{vi0}. Thenis an (n−1)-simplex of linkKwby (3.1), so there is a unique vertexvofKsuch thatas before sinceKis a pseudomanifold. Now we shall observe the link of an (n−2)-simplexinK. By our construction, fourn-simplices inKcontainingare as follows:

Therefore the verticesvij,w,vi0,v,vare in the link of the (n−2)-simplex [I∩I]. But by assumption(b), this link is a sphere join of dimension one which can have at most four vertices.Note thatvij,w,vi0are mutually distinct andv,vare different fromvij,w,vi0. So we must havev=v. Now letvijrun over all elements ofI, thenIruns over all (n−1)-simplices in linkKwthat share a(n−2)-simplex withI. Moreover by(3.1), linkKwis a strongly connected simplicial complex. By applying our argument to [I] and all other (n−1)-simplices inK, we can show that∂[v,w]∗linkKwis a subcomplex ofK. However,∂[v,w]∗linkKwandKare both pseudomanifolds and have the same dimension,so they must agree. This proves the claim.

Case 2The case wherev∈linkKw, sovis one ofv10,v20,···,vq0. We may assumev=v10without loss of generality. Then

We look at linkKv. Sincev=v10, it follows from (3.1) that linkKvcontains as a subcomplex. This together with assumption (b) implies that there is a vertexw′different from vertices in (3.3) such that linkKvis one of the following:

However, the fact (3.2) implies that none of the above occurs except the first one. So we have

The simplex [I] is in linkKvby (3.2) and then-simplices [I,v] and [I,w] share [I]. Note thatw∈linkKvin this case butw/=vijfor alliandj. From (3.4), we can concludew=w′.Then

Remember thatv=v10. We claim thatKcontains

as a subcomplex. Indeed, anyn-simplex in (3.6) is spanned byn+1 vertices which consist ofn1+1 vertices from∂[w,v10,v11,···,v1n1]andnivertices from∂[vi0,vi1,···,vini]fori=2,3,···,q.Sincev10=v, eitherworvis in then1+1 vertices from∂[w,v10,v11,···,v1n1]. Ifw(resp.v) is in then1+1 vertices from∂[w,v10,v11,···,v1n1], then anyn-simplex formed this way is inKby(3.5) (resp. (3.1)). This proves the claim.

Finally, sinceKand the subcomplex (3.6) are both pseudomanifolds and have the same dimension, they must agree. So we finish the proof of the theorem.

Proof of Theorem 2.1(a) ⇔(b) Suppose that any 2-dimensional face ofPis either a 3-gon or a 4-gon. We want to show thatPis combinatorially equivalent to a product of simplices, or equivalently∂P∗is a sphere join. Let us do induction on the dimension ofP.When dimP= 2, the proof is trivial. If dimP≥3, we can show that∂P∗satisfies the two conditions in Theorem 3.1. Indeed, condition (a) is obvious. By induction assumption, all facets ofPare product of simplices which means that∂P∗satisfies condition (b). So we finish the induction by Theorem 3.1.

AcknowledgementsThe authors want to thank Hanchul Park and Suyoung Choi for some helpful comments and thank Shicheng Xu and Jiaqiang Mei for some valuable discussions on the geometry of Alexandrov spaces.