Hanchul PARK

1 Introduction

The real topological toric manifold is a topological generalization of the real toric manifold.Every n-dimensional real topological toric manifold M is characterized by a pure(n−1)-dimensional simplicial complex K,called the base complex of M,and a map λ:V(K) → Zn2,satisfying certain conditions,called a characteristic map when V(K)is the set of vertices of K.The pair(K,λ)is called a characteristic pair and we write M=M(K,λ).

For a real topological toric manifold M,one can define new real topological toric manifolds called canonical extensions of M.This concept is originally due to Ewald(the namer of“canonical extension”)[6]for toric manifolds and Bahri-Bendersky-Cohen-Gitler[1]for(real)topological toric manifolds(their original work is for quasitoric manifolds and small covers,but their method extends well without change for topological toric manifolds and real topological toric manifolds).We remark that the canonical extension is deeply related with a classical operation on simplicial complexes called the simplicial wedge operation,which is performed at the base complex.

The integral cohomology ring of a real topological toric manifold M is difficult to compute in general unlike that of a topological toric manifold.Recently Suciu-Trevisan[9–10]have found a formula to compute the homology H∗(M;Q),and Choi and the author[5]have strengthened their result for the cohomology H∗(M;G),where G is a commutative ring in which 2 is a unit.Using their result,we provide a method to compute the cohomology of canonical extensions of a real topological toric manifold(see Section 3).

There is a special kind of canonical extension called doubling.Roughly speaking,the doubling is the canonical extension performed at each vertex of the base complex.The doubling operation is especially intriguing because it has numerous interesting topological and geometric properties(see[1,11]).Some of them,in the category of real topological toric manifolds,are covered in this paper.The main result is as follows.

Theorem 1.1For a real topological toric manifoldM=M(K,λ),its doubleM′satisfies the following properties:

(1)There is a graded ring isomorphism

whereZKis the moment-angle complex ofK,andGis a commutative ring such that2is a unit.In particular,M′is orientable.

(2)IfM′has even dimension,then there is a complex structure onM′.

(3)If there exists a characteristic map:V(K)→Znsuch that itsmod 2reduction isλ,thenM′admits a freeTm−naction whose orbit space is the topological toric manifoldM(K,.

Recall that the manifold M(K,)is called a lifting of M.An immediate corollary is as follows.This gives an easier way to construct in finitely many real topological toric manifolds with arbitrary odd torsion,which was previously accomplished in[5].

Corollary 1.1LetM=M(K,λ)be a real topological toric manifold such thatH∗(ZK)has aq-torsion element for some odd integerq.ThenH∗(M′)has aq-torsion element.

Moreover,we present questions about real topological toric manifolds related to Halperin’s toral rank conjecture.

This paper is organized as follows.In Section 2,we review the definition of topological toric manifolds and real topological toric manifolds,and characteristic pairs corresponding to them,and simplicial wedge construction and canonical extensions.In Section 3,we compute cohomology of canonical extensions of a real topological toric manifold.In Section 4,we present various properties of doublings of real topological toric manifolds.Finally,in Section 5,we give some questions and conjectures related to Halperin’s toral rank conjecture and the sum of Betti numbers of real topological toric manifolds.

2 Topological Toric Manifolds and Characteristic Pairs

A topological toric manifold defined in[7]is a closed smooth 2n-manifold M with an effective smooth(C∗)n-action such that there is an open and dense orbit and M is covered by finitely many invariant open subsets each of which is equivariantly diffeomorphic to a smooth representation space of(C∗)n.Similarly,we say that a closed smooth manifold M of dimension n with an effective smooth action of(R∗)nhaving an open dense orbit is a real topological toric manifold if it is covered by finitely many invariant open subsets each of which is equivariantly diffeomorphic to a direct sum of real one-dimensional smooth representation spaces of(R∗)n.

For a simplicial complex K,the vertex set is denoted by V(K).A pure simplicial complex K of dimension n−1 is called star-shaped if there is a geometric realization|K|⊂Rnsuch that any ray in Rnfrom the origin intersects|K|once and only once.A star-shaped simplicial complex is automatically a triangulated sphere.The following is implied by[7,Theorem 7.2].As a manifold with the restricted action of the compact torus Tn=(S1)n⊂ (C∗)n,a topological toric manifold is characterized by the following information:

(1)a star-shaped simplicial complex K;

(2)a map λ:V(K)→ Znsuch that for any σ ∈ K,{λ(i)|i∈ σ}is a subset of an integral basis of Zn.

Similarly,as amanifold,any real topological toric manifold is determined by the pair(K,λ)such that

(1)a star-shaped simplicial complex K;

(2)a map λ:V(K)→such that for any σ ∈ K,{λ(i)|i∈ σ}is a linearly independent set in

In either case,the condition(2)is called the non-singularity condition,K is the base complex of M,and λ is called a characteristic map.The pair(K,λ)is called a characteristic pair.We denote by M(K,λ)the(real)topological toric manifold determined by(K,λ).In this paper,we regard topological toric manifolds as Tn-manifolds and real topological toric manifolds asmanifolds.

Now we define the notion of canonical extensions of M(K,λ).We briefly present the construction of[1]here. Let K be a simplicial complex of dimension n−1 on vertices V(K)=[m]:={1,2,···,m}.A subset of V(K)is called a non-face of K if it is not a face of K.A non-face is minimal if any proper subset is a face of K.Note that a simplicial complex is determined by its minimal non-faces.

In the setting above,let J=(j1,···,jm) ∈ Nmbe a vector of positive integers.For the vertex set V(K)={1,···,m},V(K(J))is a set

Denote by K(J)the simplicial complex with vertex set V(K(J))with minimal non-faces ω(J)for each minimal non-face ω of K.When J=(1,···,1,2,1,···,1)is the m-tuple with 2 as the i-th entry,it is evident that

where K{i}is the full subcomplex with m−1 vertices except i and LkK{i}is the link of K at i.We sometimes use the notation K(J)=wedi(K)and call it the simplicial wedge of K at i or simply the wedge.The operation itself is called the simplicial wedge construction or wedging.By consecutive application of wedgings to K,one can construct K(J)for any J.

Any characteristic map λ:[m]→ Zn2can be regarded as an(n × m)-matrix written as Λ again,

which is called a characteristic matrix.For convenience of notation,we write

for k≥1.If k=1 then Skis an empty matrix.The following definition is the mod 2 version of that introduced in[1].

definition 2.1LetKbe an(n−1)-dimensional star-shaped simplicial complex on[m].For a given real topological toric manifoldM=M(K,λ)and anm-tuple of positive integersJ=(j1,···,jm),the canonical extensionM(J)is the real topological toric manifold determined by the characteristic pair(K(J),λ(J)),whereλ(J)is given by the block matrix

whose columns are indexed by11,12,···,1j1,21,22,···,2j2,···,m1,···,mjm.

One can check the non-singularity condition for λ(J),For details see[1].Refer to[3–4]for general topological toric manifolds and real topological toric manifolds over K(J).

3 Cohomology of Real Topological Toric Manifolds and Wedge Operations

First of all,we need some notations of[2,5]which will be used throughout this paper.Let K be a star-shaped simplicial complex of dimension n−1 with vertex set[m]and M(K,λ)a real topological toric manifold over K.For ω⊆[m],denote by Kω={σ∈K|σ⊆ω}the full subcomplex of K with respect to ω.Let Λ =(λ(1) ···λ(m))be the characteristic matrix of λ.The(n× m)-matrix Λ can be regarded as the linear map

by matrix multiplication when the vector spaces are understood as row vector spaces.

Let us use the notation 2Xfor the power set of X.For any integer k≥0,there is a natural identification φ:2[k]→determined by the following:

(1)φ({i})is the i-th coordinate vector of,1≤i≤ k,and

(2)for A,B ⊆[k],φ(AΔB)=φ(A)+φ(B),where AΔB=(A∪B)(A∩B)is the symmetric difference.

In this paper,we identify the power set 2V(K)and the vector spaceis equipped with Z2-vector space structure.Note that,with this identification,one can define a map φ−1◦(·Λ)◦ φ:2[n]→ 2[m],written again as·Λ.

Throughout this paper,let us assume that G is a ring such that 2 is a unit.

Theorem 3.1(see[9]for G=Q and[5,Theorem 4.6])There is a graded group isomorphism

Remark 3.1 Actually,the ring structure of H∗(M(K,λ);G)is described in[5].Moreover,as a graded ring,it is isomorphic to a graded subring of the graded ring H∗(RZK;G),where RZKis the real moment-angle complex of K.For simplicity,we are going to focus only on the group structure of the cohomology in this paper.It is well-known that there is a graded group isomorphism:

See(2)of[2]for references.It is worthwhile to remark that,with coefficients in G,the cohomology of M(K,λ)is a subgroup of the cohomology of RZK.

Once we have a general formula for the rational cohomology of a real topological toric manifold M(K,λ),it looks natural to apply the formula for its canonical extensions.

For s⊆ [n],the set sΛ ⊆ [m]can be understood as the vector inwhich is the sum of i-th row vectors of Λ whenever i∈ s.

Let α⊆V(K(J))be a set of vertices of K(J).The following is a simple lemma to help to compute cohomology of real topological toric manifolds.Recall(2.1)for the notation ω(J).

Lemma 3.1For a given subsetα ⊆ V(K(J)),letK(J)αbe the full subcomplex ofK(J)with respect toα.IfK(J)αis not contractible,thenα = ω(J)for someω ⊆ [m].

ProofSuppose that αω(J)for any ω ⊆ [m].Then after a suitable relabeling and reindexing,11∈α but 12/∈α.Then,by the definition of K(J),for any face τ of K(J)α{11},τ∪{11}becomes a face of K(J)α.So,K(J)αis a cone with apex{11},which proves the lemma.

Theorem 3.1 and(3.1)tell us that within the vector space 2[m],only the subspace S(Λ):={sΛ ∈|s ∈},which is the image of·Λ,contributes to the cohomology H∗(M(K,λ);G).Although one could consider the set S(Λ(J))for the cohomology of the canonical extension M(J),we can use Lemma 3.1 to further reduce S(Λ(J))to

which is a linear subspace in 2V(K(J)),since it is the intersection of two subspaces.

Proposition 3.1LetJ=(j1,j2,···,jm)be anm-tuple of positive integers.Then the following vectors spanS(Λ,J):

(1){i}(J),when1≤i≤mandjiis even;

(2)sΛ(J)for somes⊆ [n],ifjiis odd whenever thei-th component ofsΛis nonzero.

Example 3.1 Let and J=(2,1,3,2,3).Then by definition 2.1,

To compute S(Λ,J),let us use the language of row vectors.Then by definition,S(Λ,J)is precisely the set of row vectors r which satisfy the following two conditions:

(1)r is in the row space of Λ(J),and

(2)r is a linear combination of

First,observe that{1}(J)and{4}(J)are in the row space of Λ(J)since the first and fourth entries of J are even.On the other hand,{i}(J)need not be in the row space of Λ(J)if the i-th entry of J is odd.Instead,one checks that

are in the row space of Λ(J).Next,consider

which is the second row of Λ and observe that non-zero entries ofΛ are second and fifth ones,and the second and fifth entries of J are all odd.The second row of the block matrix

and finally check that t+r2+r5={2}(J)+{5}(J)={2,5}(J)∈ S(Λ,J).This generally holds for any sΛ(J)satisfying Proposition 3.1(2).In fact,one can check that these are all:that is,S(Λ,J)=span{{1}(J),{4}(J),{2,5}(J)}.See the proof below.

Proof of Proposition 3.1 The proof is straightforward once one searches for sums of row vectors of Λ(J)of the form ω(J)for ω ⊆ V(K).Note that the sum of rows of the block

isif k is even andif k is odd.More precisely,S(Λ(J))=S⊕C,

Suppose that r∈S(Λ,J)is decomposed to r=rS+rCfor rS∈S and rC∈C.If rC=0,then r=rSis certainly a linear combination of vectors satisfying(1).If rC0,then one easily checks that rCcontributes exactly to a vector satisfying(2).

Since the wedge operation is topologically iterated suspensions(see for example the proof of Proposition 2.2 of[3]),eH∗(K(J)ω(J))is isomorphic toeH∗(Kω)as graded groups,up to degree shift,for any J.Combining this fact with Proposition 3.1 and Theorem 3.1,one concludes the following corollary.

Corollary 3.1Let us assume thatGis a ring such that2is a unit.For a given real topological toric manifoldM,letJandJ′bem-tuples of positive integers andM(J)andM(J′)corresponding real topological toric manifolds.IfJ ≡ J′mod 2,thenH∗(M(J);G)andH∗(M(J′);G)are isomorphic as ungraded groups.As two special cases,we have

(1)if every entry ofJis even,thenH∗(M(J);G)H∗(RZK;G)as ungraded groups,and

(2)if every entry ofJis odd,thenH∗(M(J);G)H∗(M;G)as ungraded groups.

Proof Let us write M=M(K,λ)and assume that J ≡ J′mod 2.A direct application of Theorem 3.1 and Lemma 3.1 leads to

Proposition 3.1 guarantees that S(Λ,J)and S(Λ,J′)are canonically isomorphic.Since the wedge operation is topologically iterated suspensions(see for example the proof of Proposition 2.2 of[3]),K(J)ω(J))is isomorphic to(Kω)as graded groups,up to degree shift,for any J.Hence the first assertion is proven.The two special cases can be shown again using Proposition 3.1.If every entry of J is odd,then S(Λ,J)is isomorphic to the image of ·Λ.If every entry of J is even,then{{i}(J)|1≤ i≤ m}spans S(Λ,J).

Even though our argument so far is for canonical extensions,in particular cases like the following example there is a good chance of computing the ungraded cohomology groups of real topological toric manifolds over K(J),not only canonical extensions.

Example 3.2 Think of the pentagon K=∂P5whose vertices are labeled by 1,2,3,4,5 in counterclockwise direction and consider real topological toric manifolds over the K(J)=K(a1,a2,···,a5).According to[3,Theorem 8.3],a real topological toric manifold M(Λ)over K(J)is given by the characteristic matrix

up to rotational symmetry of the pentagon.Here,the columns are labeled as 11,···,1a1,···,51,···,5a5and the block matrices are as follows:

where i=1,···,5 and viis the i-th column of

for d=0 or 1.Moreover,

and

for arbitrary choice of vectors n1and n4.Let us denote the number of nonzero entries of niby kifor i=1 or 4.

By an easy application of Theorem 3.1 and Lemma 3.1,one can see that the sum of QBetti numbers of M(Λ),written as trk(M(Λ)),is determined by a1,a2,···,a5,k1,k4,and d mod 2.Assume that d=0(the case d=1 goes similarly).If k1and k4are even,then we can treat Λ like a canonical extension for computing trk(M(Λ))and use Proposition 3.1 and Corollary 3.1,because even number of ones are added to be zero.Note that K has the following non-contractible full subcomplexes

all of which contibute 1 to trk(RZK)(the empty set was added for convention).Therefore trk(RZK(J))=12 for any J.In the following incomplete table,the first column is for some selected parities of a1,a2,···,a5and the second column indicates ω so that the corresponding K(J)ω(J)contributes to trk(M(Λ))in Theorem 3.1.

If either k1or k4is odd,then we cannot directly apply Proposition 3.1,but similar work can be done still and it is left to the reader.The point is even in these cases,only the parities of a1,···,a5,k1,k4,and d matter.The complete calculation will be done in future work.

Remark 3.2 The cohomology of canonical extensions of topological toric manifolds are well covered in[1].Its analogue for Z2-cohomology of real topological toric manifolds holds as well.Note that this paper deals with cohomology of real topological toric manifolds with coefficient G such that 2 is a unit in G.

4 Doubling Operation of Real Topological Toric Manifolds

In previous section,we have seen that the cohomology H∗(M(J);G)is determined by J mod 2.In this section,we are going to focus a specific kind of canonical extensions of real topological toric manifolds called doubles and study their basic properties.The terms“double”and “doubling” for the simplicial complex were originally used in the literature such as[11].

definition 4.1LetKbe a simplicial complex on[m],not necessarily star-shaped.LetJ=(2,2,···,2)be them-tuple every component of which is two.The simplicial complexK(J)is called the double ofKand is denoted byK′.Assume thatKis star-shaped with dimensionn − 1and we are given a characteristic mapλ:[m]→overK,defining a real topological toric manifoldM=M(K,λ).Then the canonical extensionM(J)is called the double ofMand is denoted byM′=M(K′,λ′).In either case,the operation itself is called the doubling.

Let us recall a definition of moment-angle complexes.Let(X,A)be a CW-pair of spaces and K a given simplicial complex on[m].For σ∈K,write

and define a polyhedral product or a generalized moment-angle complex as

Note that Z(K;(D2,S1))=ZKand Z(K;(D1,S0))=RZK.If(X,A)is equipped with a Γ-action for a group Γ,then there is a canonical Γm-action on Z(K;(X,A)).For example,there is a canonical Tm-action on ZKand a canonical-action on RZKrespectively.When J=(2,2,···,2),for ω ⊆ [m],we use the notation ω′:= ω(J)={i1,i2|i∈ ω}for simplicity.For RZK′,one has an additional natural action ofas a subgroup o,called the diagonal action(written as·dto avoid confusion),given by

for ω ∈=and ω′

We remark that RZK′⊂(D1)2mis equipped with a-action and there is an action of a subgroupwhich is induced by the-action of RZK.

An important implication of[1,Section 7]is the following theorem.

Theorem 4.1(see[1,Section 7])There is aTm-action onRZK′extending the diagonal action such that there is an equivariant homeomorphism betweenRZK′andZKfor this action.

To roughly see how the theorem works,note that the 2-disc D2⊂C has a natural S1-action and D1×D1⊂R×R has a naturalaction.Fix an identification D2→D1×D1so that the Z2-action on D1×D1given by

is compatible with the Z2-action on D2as the subgroup Z2≤S1.Hence we can equip D1×D1with the induced S1-action.Recalling(4.1),let us write D(σ;(D2,S1))=D(σ)for a cell of ZKand D(σ;(D1,S0))=RD(σ)for that of RZK′.For every simplex σ ∈ K,one can easily see that under our identification.

Let K be a simplicial complex on[m].We already know thatacts on RZK.If M=M(K,λ)is a real topological toric manifold with characteristic matrix Λ:→,then an important fact in toric topology states that kerΛfreely acts on RZKand the quotient RZK/kerΛ with the quotient action of/kerΛis equivariantly homeomorphic to M.Assume that we are given M′=M(K′,λ′),which is the double of a real topological toric manifold M,and denote by Λ′its characteristic matrix.The matrix Λ′can be regarded as a linear mapand we have M′RZK′/kerΛ′ZK/kerΛ′.The diagonal action of≤ Tmon ZKinduces a quotient action of/kerΛ′on M′ZK/kerΛ′,which is again called the diagonal action.

Proposition 4.1Every vector of the subspacekerΛ′has the formω′forω ⊂ [m].More precisely,

ProofThis is done by direct calculation.First of all,for a linear map defined by the following matrix(Inis the identity matrix of size n)

note that kerL is given by the column space

By the non-singularity condition,we can assume that Λ has the form of(4.2)for a suitable order of columns and up to row operations.In other words,Λ=?InA?n×m.One observes that the double Λ′can be written as

when the columns are labeled as 11,···,m1,12,···,m2.To have the form of(4.2),we perform some row operations.Namely,one adds the(i+m)-th row to the i-th row for 1≤i≤n.After that,we obtain

or

Applying(4.3),we conclude

and so,recalling that

completes the proof.

The following definition is due to Zhi L¨u.

definition 4.2LetKbe a star-shaped simplicial complex of dimensionn−1withmvertices.LetMbe a real topological toric manifold overK.If there is a topological toric manifoldNsuch thatMis the fixed point set of the conjugation onN,thenNis called a lifting ofM.Equivalently,for a characteristic mapλ:V(K)→ Zn2overZ2,if there is a characteristic mapeλoverZsuch that the following diagram

commutes,is called a lifting ofλ.

Proof of Theorem 1.1 Observe that(1)is a stronger form of part(2)of Corollary 3.1 when M(J)=M′.The equation(3.2)becomes

when every entry of J is even.On the other hand,by(3.1)and Lemma 3.1,

Therefore H∗(M′;G)H∗(RZK′;G)as graded groups.For the cup product,an application of[5,Remark 4.7]gives

where the second identity is because of the fact all entries of J are even,and the third one is due to[2,Theorem 5.1].See[5,Theorem 3.1]for details of the formula.The last step to prove(1)is to recall RZK′is diffeomorphic to ZK.

By Theorem 4.1,ZKand RZK′are Tm-equivariantly homeomorphic and the diagonal action can be considered to be an action on ZKas a subgroup⊂Tm.Moreover,by Proposition 4.1,keracts freely on ZKas a subgroup of.Since K is a star-shaped simplicial complex,there is a Tm-invariant complex structure on ZKby[8,Theorem 3.3].Since kerΛ′is a discrete group and its free action on ZKpreserves the complex structure,(2)is proved.We remark that M′is invariant under the diagonal action.For(3),for the free Tm−n-action such that ZK/Tm−n=M(K,),observe that its subgroupacts freely on ZKand it coincides with kerΛ′.Therefore,the quotient space ZK/kerΛ′has the quotient group action ofwhich is still free,proving(3).

One can apply(1)of Theorem 1.1 to construct real topological toric manifolds whose cohomology rings have odd torsion elements(see also[5]).

Proof of Corollary 1.1 Apply of Theorem 1.1(1)for G=Q and G=Zqand use the universal coefficient formula.See the proof of Theorem 5.10 of[5]for details.

5 Further Questions and Open Problems

Motivated with(3)of Theorem 1.1,we present the following question.It is a version of the famous Halperin’s toral rank conjecture,applied for real topological toric manifolds.Related work for moment-angle complexes can be found in[12].

Question 5.1 Let us assume that K is a star-shaped simplicial complex of dimension n−1 with m vertices and M=M(K,λ)is a real topological toric manifold.Suppose that a compact torus Tkfreely acts on M.Then the following is true:k≤m−n and the equality holds if and only if M=N′×Tℓ,where N′is the double of a real topological toric manifold N which has a lifting and Tℓis the compact torus of rank ℓ.

For a real topological toric manifold M(K,λ),observe that

by Theorem 3.1 and(3.1).Moreover,it is easy to see that trkRZK(J)is independent of J.We say that M(K,λ)is maximal if trk(M(K,λ))=trkRZK.Note that Corollary 3.1(2)implies that M′is maximal for any real topological toric manifold M.Also note that M(j1,···,jm)is a double if jiis even for 1≤i≤m.

Question 5.2 Does every maximal real topological toric manifold have the form N′×Tℓfor another real topological toric manifold N and a torus Tℓ?

In a way similar to the case of maximal real topological toric manifolds,one can also consider“minimal” real topological toric manifolds.By definition,trkM is minimal if and only if M is(Q-)acyclic.Besides acyclic ones,we can also consider rational homology spheres.Note that a real topological toric manifold can neither be Z-acyclic nor be an integral homology sphere because of 2-torsions.

Question 5.3 Describe real topological toric manifolds which are Q-acyclic or rational homology spheres.

Note that RPnis Q-acyclic if n is even and is a rational homology sphere if n is odd.So one could ask about the existence of odd-dimensional Q-acyclic manifolds or even-dimensional rational homology spheres in the realm of real topological toric manifolds.The following observation is due to Suyoung Choi.

Proposition 5.1There is no odd-dimensional real topological toric manifold which isQ-acyclic.

ProofThe alternating sum of Q-Betti numbers is one,and that of Z2-Betti numbers is zero because of Poincar´e duality with Z2coefficients.They are the same invarient called the Euler characteristic,so a contradiction.

On the other hand,there does exist an even dimensional rational homology sphere.Consider the cyclic polytope C4(7).It admits a unique real topological toric manifold M of dimension 4 up to weakly equivariant diffeomorphism as can be seen in[3,Section 8].Actually one can check M is orientable and furthermore it is a rational homology sphere.

A simple argument based on Corollary 3.1 implies the following fact.

Lemma 5.1LetMbe a real topological toric manifold which is a rational homology sphere.ThenM(J)is also a rational homology sphere for anyJ=(j1,···,jm)such that everyjiis odd.

In summary,we have the following proposition.

Proposition 5.2For everyn∈Z+other than2,there exists ann-dimensional real topological toric manifold which is a rational homology sphere.

AcknowledgementsThe author is thankful to Suyoung Choi for all of helpful conversations and comments.

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