TORIC ORIGAMI STRUCTURES ON QUASITORIC MANIFOLDSANTON AYZENBERG, MIKIYA MASUDA, SEONJEONG PARK, AND HAOZHI ZENG

Abstract. We construct quasitoric manifolds of dimension 6 and higher which
are not equivariantly homeomorphic to any toric origami manifold. All necessary topological definitions and combinatorial constructions are given andthe statement is reformulated in discrete geometrical terms. The problem reduces to existence of planar triangulations with certain coloring and metricproperties.

Introduction
Origami manifolds appeared in differential geometry recently as a generalization of symplectic manifolds [4]. Toric origami manifolds are in turn generalizations of symplectic toric manifolds. Toric origami manifolds are a special class of
2n-dimensional compact manifolds with an effective action of a half-dimensional
compact torus T n . In this paper we consider the following question. How large isthis class? Which manifolds with half-dimensional torus actions are toric origami
manifolds?
Since the notion of a manifold with an effective half-dimensional torus action is
too general to deal with, we restrict to quasitoric manifolds. This class of manifolds
is large enough to include many interesting examples, and small enough to keepstatements feasible. In [15] Masuda and Park proved
Theorem 1. Any simply connected compact smooth 4-manifold M with an effective
smooth action of T 2 is equivariantly diffeomorphic to a toric origami manifold.
In particular, any 4-dimensional quasitoric manifold is toric origami. The samequestion about higher dimensions was open. In this paper we give the negative
answer.
Theorem 2. For any n ě 3 there exist 2n-dimensional quasitoric manifolds, which
are not equivariantly homeomorphic to any toric origami manifold.
We will describe an obstruction for a quasitoric 6-manifold to be toric origami
and present a large series of examples, where such an obstruction appears. Existenceof such examples in higher dimensions follows from 6-dimensional case. In spite of
topological nature of the task, the proof is purely discrete geometrical: it relies
on metric and coloring properties of planar graphs. Thus we tried to separate thediscussion of established facts in toric topology which motivated this study, from theproof of the main theorem to keep things comprehensible for the broad audience.
Date: September 22, 2014.
2000 Mathematics Subject Classification. Primary 57S15, 53D20; Secondary 14M25, 52B20,
52B10, 05C10.
Key words and phrases. toric origami manifold, origami template, Delzant polytope, quasitoric
manifold, characteristic function, planar triangulation, coloring, discrete isoperimetric inequality.
The first author is supported by the JSPS postdoctoral fellowship program. The second author
was partially supported by Grant-in-Aid for Scientific Research 25400095.
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A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG

The paper is organized as follows. In section 1 we briefly review the necessary
topological objects, and describe the standard combinatorial and geometrical models which are used to classify them. The objects are: quasitoric manifolds, symplectic toric manifolds, and toric origami manifolds. The corresponding combinatorialmodels are: characteristic pairs, Delzant polytopes, and origami templates respectively. In section 2 we introduce the notion of a weighted simplicial sphere, which,in a certain sense, unifies all these combinatorial models. We define a connected
sum of weighted spheres along vertices. This operation is dual to the operation of
producing an origami template from Delzant polytopes. It plays an important rolein the proof of Theorem 2. Section 3 contains the combinatorial restatement ofTheorem 2 and the proof of this equivalent statement. The interaction of our study
with the study of the Brownian map allows to prove that asymptotically most simple 3-polytopes admit quasitoric manifolds which are not toric origami. We describe
this interaction as well as other adjacent questions in the last section 4.
1. Topological preliminaries
1.1. Quasitoric manifolds. The subject of this subsection originally appearedin the seminal work of Davis and Januszkiewicz [5]. The modern exposition andtechnical details can be found in [3, Ch.7].
Let T n be a compact n-dimensional torus. The standard representation of T n isa representation T n ñ Cn by coordinate-wise rotations, i.e.
pt1 , . . . , tn q ¨ pz1 , . . . , zn q “ pt1 z1 , . . . , tn zn q,
for zi , ti P C, |ti | “ 1.
The action of T n on a manifold M 2n is called locally standard, if M has an atlasof standard charts, each isomorphic to a subset of the standard representation. Moreprecisely, a standard chart on M is a triple pU, f, ψq, where U Ă M is a T n -invariant
open subset, ψ is an automorphism of T n , and f is a ψ-equivariant homeomorphism
f : U Ñ W onto a T n -invariant open subset W Ă Cn (i.e. f pt ¨ yq “ ψptq ¨ f pyq for
all t P T n , y P U ). In the following M is supposed to be compact.
Since the orbit space Cn {T n of the standard representation is a nonnegative conenRě “ tx P Rn | xi ě 0u, the orbit space of any locally standard action obtains the
structure of a compact manifold with corners. Recall that a manifold with corners is
a topological space locally modeled by open subsets of Rně with the combinatorial
stratification induced from the face structure of Rně . There are many technicaldetails about the formal definition, which we left beyond the scope of this work
(the exposition relevant to our study can be found in [3] or [18]).
The orbit space Q “ M {T of a locally standard action carries an additional
information, called a characteristic function. Let FpQq denote the set of facets of Q
(i.e. faces of codimension 1). For each facet F of Q consider a stabilizer subgroup
λpF q Ă T n , which preserves points over the interior of F . This subgroup is 1dimensional and connected, thus it has the form tptλ1 , . . . , tλn q | t P T 1 u Ă T n , forsome primitive integral vector pλ1 , . . . , λn q P Zn , defined uniquely up to a common
sign. Thus, a primitive integral vector (up to sign) ΛpF q P Zn {˘ is associated withany facet F of Q. This map Λ : FpQq Ñ Zn {˘ is called a characteristic function(or a characteristic map). It satisfies the following so called (˚)-condition:(˚)

If the facets F1 , . . . , Fs intersect, then the vectors
ΛpF1 q, . . . , ΛpFs q span a direct summand of Zn .

Here we actually take not a class ΛpFi q P Zn {˘, but one of its two particular
representatives in Zn . Obviously, the condition does not depend on the choice ofTORIC ORIGAMI STRUCTURES ON QUASITORIC MANIFOLDS

3

sign, thus (˚) is well defined. The same convention appears further in the text
without special mention.
It is useful to view the characteristic function Λ on Q “ M {T as a generalizedcoloring of facets. We assign primitive integral vectors to facets instead of simple
colors, and condition (˚) is the requirement for this “coloring” to be proper. The
general idea, which simplifies many considerations in toric topology is that the
combinatorial structure of the orbit space Q together with the assigned coloring(characteristic function) completely encodes the equivariant homeomorphism typeof M in many cases. The precise statement also involves the so called Euler classof the action, which is an element of H 2 pQ; Zn q, and allows to classify all compact
manifolds with locally standard torus actions. The reader may find this generalstatement in [18].
Anyway, we will work only with a special type of locally standard torus actions,
namely quasitoric manifolds. In this special case Euler class vanishes, so we will
not care about it.
Definition 1.1. A manifold M 2n with a locally standard action of T n is called quasitoric, if the orbit space M {T is homeomorphic to a simple polytope as a manifoldwith corners.
Recall that a convex polytope P of dimension n is called simple if any of its
vertices lies in exactly n facets. In other words, a simple polytope is a polytope
which is at the same time a manifold with corners. Considering manifolds withcorners, simple polytopes are the simplest geometrical examples one can imagine.
This makes the definition of quasitoric manifold very natural.
A pair pP, Λq, where P is a simple polytope and Λ is a characteristic function, iscalled a characteristic pair. According to [5], there is a one-to-one correspondence
tquasitoric manifoldsu ú tcharacteristic pairsu
up to equivariant homeomorphism on the left-hand side and combinatorial equivalence on the right-hand side. Given a characteristic pair, one can construct thecorresponding quasitoric manifold explicitly.
Construction 1.2 (Model of quasitoric manifold). Let P be an n-dimensional simple polytope, FpP q the set of its facets, and Λ : FpP q Ñ Zn {˘ any map satisfying(˚). Consider a topological space(1.1)

MpP,Λq “ P n ˆ T n { „ .

The equivalence „ is generated by relations pp, t1 q „ pp, t2 q where p lies in a facet
F P FpP q and t1 t´1P λpF q. The torus T n acts on MpP,Λq by rotating second2coordinate and the orbit space MpP,Λq {T n is isomorphic to P . Condition (˚) implies
that the action is locally standard. Therefore, MpP,Λq is a quasitoric manifold.
Let η denote the projection to the orbit space η : MpP,Λq Ñ P . Each facetdef

F P FpP q determines a characteristic submanifold NF “ η ´1 pF q Ă MpP,Λq ofdimension 2n ´ 2. On its own, the manifold NF is again a quasitoric manifold withthe orbit space F .
1.2. Toric origami manifolds. In the following subsections we recall the definitions and properties of toric origami manifolds and origami templates. Moredetailed exposition of this theory can be found in [4], [15] or [10].
A folded symplectic form on a 2n-dimensional smooth manifold M is a closed
2-form ω whose top power ω n vanishes transversally on a subset W and whose
restriction to points in W has maximal rank. Then W is a codimension-one submanifold of M called the fold. If W is empty, ω is a genuine symplectic form. The
pair pM, ωq is called a folded symplectic manifold.

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A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG

The reader may get a feeling of this notion by working ř
locally. Darboux’s theorem
says that any symplectic form can be written locally as i dxi ^ dyi in appropriate
coordinates. The folded forms are exactly the forms written asÿ
x1 dx1 ^ dy1 `dxi ^ dyi
ią1

in appropriate coordinates (for this analogue of Darboux’s theorem see [4] andreferences therein). The fold W is thus a hypersurface given locally by x1 “ 0.
Since the restriction of ω to W has maximal rank, it has a one-dimensional kernel
at each point of W . This determines a line field on W called the null foliation. If
the null foliation is the vertical bundle of some principal S 1 -fibration W Ñ X overa compact base X, then the folded symplectic form ω is called an origami form andthe pair pM, ωq is called an origami manifold.
The action of a torus T on an origami manifold pM, ωq is called Hamiltonian if
it admits a moment map µ : M Ñ t˚ to the dual Lie algebra of the torus, which
satisfies the conditions: (1) µ is equivariant with respect to the given action of T
on M and the coadjoint action of T on the vector space t˚ (this action is trivial forthe torus); (2) µ collects Hamiltonian functions, that is, dxµ, V y “ ıV # ω for any
V P t, where V # is the vector field on M generated by V .
Definition 1.3. A toric origami manifold pM, ω, T, µq, abbreviated as M , is acompact connected origami manifold pM, ωq equipped with an effective Hamiltonian
action of a torus T with dim T “ 21 dim M and with a choice of a corresponding
moment map µ.
1.3. Symplectic toric manifolds. When the fold W is empty, a toric origami
manifold is a symplectic toric manifold. In this case the image µpM q of the momentmap is a Delzant polytope in t˚ , and the map µ itself can be identified with themap to the orbit space η : M Ñ M {T . A classical theorem of Delzant [7] says that
symplectic toric manifolds are classified by the images of their moment maps int˚ – Rn . In other words, there is a one-to-one correspondence
tsymplectic toric manifoldsu ú tDelzant polytopesu
up to equivariant symplectomorphism on the left-hand side, and affine equivalenceon the right-hand side. Let us recall the notion of Delzant polytope.
Definition 1.4. A simple convex polytope P Ă Rn is called Delzant, if its normal
fan is smooth (with respect to a given lattice Zn Ă Rn ). In other words, all normal
vectors to facets of P have rational coordinates, and, whenever facets F1 , . . . , Fnmeet in a vertex of P , the primitive normal vectors νpF1 q, . . . , νpFn q form a basisof the lattice Zn .
We do not need the construction of Delzant correspondence in full generality,
but we need to review the topological construction of symplectic toric manifold
corresponding to a given Delzant polytope. Forgetting symplectic and smooth
structures, any symplectic toric manifold, as a topological manifold with T n -action,
becomes an example of quasitoric manifold.
Construction 1.5 (Topological model of symplectic toric manifold). Let P be a
Delzant polytope in Rn . For a facet F P FpP q consider its outward primitive normal
vector ν̃pF q P Zn . Consider the corresponding vector modulo sign: νpF q P Zn {˘.
By the definition of Delzant polytope, ν : FpP q Ñ Zn {˘ satisfies (˚), thus provides
an example of a characteristic function. The quasitoric manifold
def

MP “ MpP,νq .
is exactly the symplectic toric manifold corresponding to P (up to equivariant
homeomorphism).

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1.4. Origami templates. Delzant theorem provides a one-to-one correspondence
between symplectic toric manifolds and Delzant polytopes. To generalize this correspondence to toric origami manifolds we need a notion of an origami template,which we review next.
Let Dn denote the set of all (full-dimensional) Delzant polytopes in Rn (w.r.t. agiven lattice) and Fn the set of all their facets.
Definition 1.6. An origami template is a triple pΓ, ΨV , ΨE q, where‚ Γ is a connected finite graph (loops and multiple edges are allowed) with thevertex set V and edge set E;‚ ΨV : V Ñ Dn ;‚ ΨE : E Ñ Fn ;subject to the following conditions:
1. If e P E is an edge of Γ with endpoints v1 , v2 P V , then ΨE peq is a facet ofboth polytopes ΨV pv1 q and ΨV pv2 q, and these polytopes coincide near ΨE peq(this means there exists an open neighborhood U of ΨE peq in Rn such thatU X ΨV pv1 q “ U X ΨV pv2 q).
2. If e1 , e2 P E are two edges of Γ adjacent to v P V , then ΨE pe1 q and ΨE pe2 q
are disjoint facets of ΨV pvq.
The facets of the form ΨE peq for e P E are called the fold facets of the origami
template.
For convenience in the following we call the vertices of graph Γ the nodes.
One can simply view an origami template as a collection of (possibly overlapping)
Delzant polytopes tΨV pvq | v P V u in the same ambient space, with some gluing
data, encoded by a template graph Γ. When n “ 2, the picture looks like afolded sheet of paper on a flat plane, which is one of the explanations for the term
“origami” (see Fig. 1). Nevertheless, to avoid the confusion, we should mention thatmost flat origami models in a common sense are not origami templates in the senseof Definition 1.6.

Figure 1. Examples of origami templates in dim “ 2. Fold facets
are shown in red.
The following is a generalization of Delzant’s theorem to toric origami manifolds.
Theorem 3 ([4]). Assigning the moment data of a toric origami manifold induces
a one-to-one correspondence
ttoric origami manifoldsu ú torigami templatesu
up to equivariant origami symplectomorphism on the left-hand side, and affine
equivalence on the right-hand side.

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A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG

As in the case of symplectic toric manifolds, we do not need the construction ofthis correspondence in full generality. But we give a topological construction of thetoric origami manifold from a given origami template.
Construction 1.7 (Topological model of toric origami manifold). Consider anorigami template O “ pΓ, ΨV , ΨE q, Γ “ pV, Eq. For each node v P V the Delzant
polytope ΨV pvq P Pn gives rise to a symplectic toric manifold MΨV pvq , see construction 1.5. Now do the following procedure:
1 Take a disjoint union of all manifolds MΨV pvq for v P V ;2 For each edge e P E with distinct endpoints v1 and v2 take an equivariant
connected sum of MΨV pv1 q and MΨV pv2 q along the characteristic submanifold NΨE peq (which is embedded in both manifolds by construction 1.2);
3 For each loop e P E based at v P V take a real blow up of normal bundle
to the submanifold NΨE peq inside MΨV pvq .
Step 2 makes sense because of pt.1 of Definition 1.6. Indeed, the polytopes
ΨV pv1 q and ΨV pv2 q agree near ΨE peq, thus MΨV pv1 q and MΨV pv2 q have equivariantly homeomorphic neighborhoods around NΨE peq , so the connected sum is welldefined. Pt. 2 of Definition 1.6 ensures that surgeries do not touch each other, so
all the connected sums and blow ups can be taken simultaneously.
Denote the resulting manifold of this construction by MO “ MpΓ,ΨV ,ΨE q . Since
all the connected sums were equivariant, there is a natural action of T n on MO . The
manifold MO is exactly the toric origami manifold associated with O via Theorem 3.
Example 1.8. Let us construct a toric origami manifold X, corresponding to theorigami template, made of two triangles (Fig. 1, left). The symplectic toric 4manifold corresponding to a triangle is known to be the complex projective plane
CP 2 . The characteristic submanifold corresponding to the fold facet is a projective
line CP 1 Ă CP 2 . Thus, X is a connected sum of two copies of CP 2 along theline CP 1 , which lies in both. This has a simple geometrical interpretation. If
we consider CP 1 Ă CP 2 as a projective line at infinity, and denote the tubularneighborhood of this line by U pCP 1 q, then CP 2 zU pCP 1 q is a 4-disk D4 . Thus X
is a result of attaching two copies of D4 along the boundary. Thus X – S 4 . The
action of T 2 on S 4 is also easily described. Consider the space C2 ˆ R, and let
T 2 act on C2 by coordinate-wise rotations, and trivially on R. The unit sphereS 4 Ă C2 ˆ R is invariant under this action. This gives a required action of T 2 on
X – S4.
An origami template O “ pΓ, ΨV , ΨE q is called orientable if the template graphΓ is bipartite, or, equivalently, 2-colorable. It is not hard to prove that the origami
template O is orientable whenever MO is an orientable manifold [2].
An origami template O “ pΓ, ΨV , ΨE q (and the corresponding manifold MO ) is
called coörientable if Γ has no loops (i.e. edges based at one point). Any orientable
template (resp. toric origami manifold) is coörientable, because a graph with loops
is not 2-colorable. If MO is coörientable, then the action of T n on MO is locallystandard [10, lemma 5.1]. The converse is also true. If the template graph has aloop, then the real normal blow up in Step 3 of construction 1.7 implies existenceof Z2 -components in stabilizer subgroups. Therefore non-coörientable toric origami
manifolds are not locally standard. In the following we consider only coörientable
templates and toric origami manifolds.
Construction 1.9 (Orbit space of toric origami manifold). The orbit space Q “
MpΓ,ΨV ,ΨE q {T n of a (coörientable) toric origami manifold is a manifold with corners.
It can be described as a topological space obtained by gluing polytopes ΨV pvq along

TORIC ORIGAMI STRUCTURES ON QUASITORIC MANIFOLDS

7

fold facets. More precisely,
(1.2)

Q“

ğ

pv, ΨV pvqq{ „,vPV

where pu, xq „ pv, yq if there exists an edge e with endpoints u and v, and x “ y PΨE peq. Facets of Q are given by non-fold facets of polytopes ΨV pvq identified in
the same way. To make this precise, let us call non-fold facets F1 P FpΨV pv1 qq and
F2 P FpΨV pv2 qq elementary neighboring w.r.t. to the edge e P E (with endpoints
v1 and v2 ) if F1 X ΨE peq “ F2 X ΨE peq. The relation of elementary neighborlinessgenerates an equivalence relation Ø on the set of all non-fold facets of all polytopes
ΨV pvq. Define the facet rF s of the orbit space Q as a union of facets in one
equivalence class:ğdef
rF s P FpQq,(1.3)
rF s “
pv, Gq{ „,vPV,G is a facet of ΨV pvq,
G is not fold,GØF

where „ is the same as in (1.2).
def
Let us define a primitive normal vector to the facet rF s of Q by νprF sq “ νpF q PnZ {˘. It is well defined since νpF q “ νpGq for F Ø G.
Note that the relation of elementary neighborliness determines a connected subgraph ΓrF s of Γ. All facets G Ø F are Delzant and lie in the same hyperplane HrF s .
Thus we obtain an induced origami template(1.4)

OrF s “ pΓrF s , ΨV |ΓrF s X HrF s , ΨE |ΓrF s X HrF s qof dimension n ´ 1. In particular, if η : MO Ñ Q denotes the projection to the orbit
space, then the characteristic submanifold η ´1 prF sq is the toric origami manifoldof dimension 2n ´ 2 generated by the origami template OrF s .
We had defined the facets of the orbit space Q “ MO {T n . All other faces aredefined as connected components of nonempty intersections of facets. On the other
hand, faces can be defined similarly to facets — by gluing faces of polytopes ΨV pvq
which are neighborly in the same sense as before.
Extending the origami analogy, we can think of the orbit space Q as “unfolding”
the origami template and then smoothening the angles adjacent to the former foldfacets (remember that we have to identify neighboring faces!).

“Figure 2. The orbit space of a manifold S 4 , corresponding to theorigami template shown on Fig. 1, left.
The orbit space Q “ MpΓ,ΨV ,ΨE q {T has the same homotopy type as the graphΓ, thus Q is either contractible (when Γ is a tree) or homotopy equivalent to awedge of circles. This observation shows that whenever the template graph Γ has
cycles, the corresponding toric origami manifold cannot be quasitoric (recall thatthe orbit space of quasitoric manifold is a polytope, which is contractible). As an
example, the origami template shown on Fig. 1, at the right corresponds to theorigami manifold which is not quasitoric.
Since we want to find a quasitoric manifold which is not toric origami, we need toconsider only the cases when the orbit space is contractible. Thus in the following
we suppose Γ is a tree.

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2. Weighted simplicial spheres
In the previous section we have seen that quasitoric manifolds are encoded by
the orbit spaces (which are simple polytopes) and characteristic functions (which
are colorings of facets by elements of Zn {˘). It will be easier, however, to workwith the dual objects, which we call weighted simplicial spheres. To some extent
this approach is equivalent to multi-fans, used to study origami manifolds in [15],but it is more suitable for our geometrical considerations.
Recall that a simplicial poset is a finite partially ordered set S such that:(1) There is a unique minimal element H P S,
def(2) For each I P S the interval subset rH, Is “ tJ P S | J ď Iu is isomorphic to
the poset of faces of pk ´ 1q-dimensional simplex (i.e. Boolean lattice of rank k).
In this case the element I is said to have rank k and dimension k ´ 1.
The elements of S are called simplices and elements of rank 1 are called vertices.
The set of vertices of S is denoted VertpSq.
A simplicial poset is called pure, if all maximal simplices have the same dimension. A simplicial poset S is called a simplicial complex, if for any subset of verticesσ Ď VertpSq, there exists at most one simplex whose vertex set is σ.
Construction 2.1. It is convenient to visualize simplicial posets using their geometrical realizations. To define the geometrical realization we assign the geometrical simplex ∆I of dimension rankpIq ´ 1 to each I P S and attach them together
according to the order relation in S. More formally, the geometric realization of S
is the topological space
ğ
def
|S| “
pI, ∆I q{ „,IPS

where pI1 , x1 q „ pI2 , x2 q if I1 ă I2 and x1 “ x2 P ∆I1 Ă ∆I2 . See details in [14]or [3].
A simplicial poset S is called a triangulated sphere if |S| is homeomorphic to asphere. S is called a PL-sphere if the barycentric subdivision S 1 (which is a simplicial complex) is PL-homeomorphic to the boundary of a simplex. In dimension 2,
which is the most important case for us, these two notions are equivalent. In thesequel we use the term simplicial sphere for either of them, and hope this will notlead to confusion.
Construction 2.2. We want to define a connected sum of two simplicial spheres
along their vertices. The topological meaning of this operation is clear: cut the small
open neighborhoods of vertices and attach the boundaries if possible. However, anattempt to define the connected sum combinatorially for the most general simplicial
posets leads to some technical problems. To keep things manageable, we exclude
certain degenerate situations.
For every I ă J in S there is a complementary simplex J r I P S, since theinterval r∅, Js is identified with the Boolean lattice. Define the link of a simplex
I P S as a partially ordered set linkS I “ tJ r I | J P S, J ě Iu with the orderrelation induced from S. Define the open star of a simplex I P S as a subset
def
star˝S I “ tJ P S | J ě Iu. There is a natural surjective map of sets DI : star˝S I ÑlinkS I sending J to J r I. We call a simplex I admissible if DI is injective.
Note that in a simplicial complex every simplex is admissible. An example ofnon-admissible simplex is shown on Fig. 3. There are two simplices containing thevertex a, and the complement of a in both of them is the same vertex b. Thus a is a
non-admissible vertex. One can view admissibility as the property of being “locallya simplicial complex”.
Let us define the connected sum of two simplicial posets S1 and S2 along admissible vertices. Let i1 P S1 and i2 P S2 be admissible vertices, and suppose there

TORIC ORIGAMI STRUCTURES ON QUASITORIC MANIFOLDS

9ba
Figure 3. Example of non-admissibility.
exists an isomorphism of posets ξ : linkS1 i1 Ñ linkS2 i2 (thus an isomorphism ofopen stars, by admissibility). Consider a poset
(2.1)

def

S1 i1#i2 S2 “ pS1 r star˝S1 i1 q \ pS2 r star˝S2 i2 q{ „,where I1 P linkS1 i1 Ă S1 is identified with I2 P linkS2 i2 Ă S2 whenever I2 “ ξpI1 q.
The order relation on S1 i1#i2 S2 is induced from S1 and S2 in a natural way. It canbe easily checked that the connected sum S1 i1#i2 S2 is again a simplicial poset.
If S1 , S2 are simplicial spheres, then so is S1 i1#i2 S2 . This statement would failif we do not impose the admissibility condition.
Remark 2.3. A connected sum of two simplicial complexes may not be a simplicial
complex (Fig. 4). This is the main reason why we consider a class of simplicial posets
instead of simplicial complexes.

#“

Figure 4. The class of simplicial complexes is not closed undertaking connected sums.
Definition 2.4. Let S be a pure simplicial poset of dimension n ´ 1. A map
Λ : VertpSq Ñ Zn {˘ is called a weighting if, for every simplex I P S with vertices
i1 , . . . , in , the vectors Λpi1 q, . . . , Λpin q span Zn . The pair pS, Λq is called a weighted
simplicial poset.
Definition 2.5. Let pS1 , Λ1 q and pS2 , Λ2 q be weighted simplicial posets. Let i1 , i2
be admissible vertices of S1 , S2 such that there exists an isomorphism ξ : linkS1 i1 Ñ
linkS2 i2 preserving weights: Λ2 ˝ ξ|linkS1 i1 “ Λ1 |linkS1 i1 . Then Λ1 , Λ2 induce theweight Λ on the connected sum S1 i1#i2 S2 . The weighted simplicial poset pS1 i1#i2
S2 , Λq is called a weighted connected sum of pS1 , Λ1 q and pS2 , Λ2 q.
Construction 2.6. Let pP, Λq be a characteristic pair (see section 1). Let KP “BP ˚ be the dual simplicial sphere to a simple polytope P . Since there is a natural
correspondence VertpKP q “ FpP q we get the weighting Λ : VertpKP q Ñ Zn {˘.
This defines a weighted sphere pKP , Λq. In particular, any Delzant polytope P
defines a weighted sphere pKP , νq, where νpF q is the normal vector to F P FpP q “
VertpKP q modulo sign (construction 1.5).
Construction 2.7. Let O “ pΓ, ΨV , ΨE q be an origami template and MO be thecorresponding toric origami manifold. Suppose that Γ is a tree. The orbit space
Q “ MO {T n is homeomorphic to an n-dimensional disc. The face structure of Q
defines a poset SQ , whose elements are faces of Q ordered by reversed inclusion

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A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG

(it is easy to show that such poset is simplicial). In particular, VertpSQ q “ FpQq.
Normal vectors to facets of Q (construction 1.9) determine the characteristic function ν : FpQq Ñ Zn {˘, νprF sq “ νpF q. Thus there is a weighted simplicial poset
pSQ , νq associated with a toric origami manifold MO .
Our next goal is to describe the weighted sphere pSQ , νq of a toric origami manifold as a connected sum of elementary pieces, corresponding to Delzant pieces ofthe origami template.
Construction 2.8. If Γ is a tree, then the simplicial poset SQ is the connected
sum of simplicial spheres KΨV pvq along vertices, corresponding to fold facets:
SQ – # KΨV pvq .

(2.2)

Γ

Let us introduce some notation to make this precise. Let e be an edge of Γ, andv be its endpoint. Let iv,e be the vertex of KΨV pvq corresponding to the facetΨE peq Ă ΨV pvq. Then (2.2) denotes the connected sum of all simplicial spheresKΨV pvq along vertices iv,e , iu,e for all edges e “ tv, uu of graph Γ. This simultaneousconnected sum is well defined. Indeed, if e1 ‰ e2 P E are two edges emanatingfrom v P V , then vertices iv,e1 and iv,e2 are not adjacent in KΨV pvq by pt.2 ofDefinition 1.6. Therefore, open stars star˝KΨ pvq iv,e1 and star˝KΨ pvq iv,e2 , which weVVremove in (2.1), do not intersect. Also note that all vertices iv,e are admissible,
since the spheres KΨV pvq are simplicial complexes.
Each sphere KΨV pvq comes equipped with a weighting νv : VertpKΨV pvq q Ñ
Zn {˘, since ΨV pvq is Delzant. By pt.1 of Definition 1.6 these weightings agree
on the links which we identify. Therefore we have an isomorphism of weighted
spheres(2.3)

pSQ , νq – #pKΨV pvq , νv q.
Γ

3. Proof of Theorem 2
Suppose that a quasitoric manifold MpP,Λq is equivariantly homeomorphic to theorigami manifold MpΓ,ΨV ,ΨE q . As was mentioned earlier, in this situation Γ is atree.
First, the orbit spaces should be isomorphic as manifolds with corners: P –T

Q “ MO {T . Second, MpP,Λq – MO implies that stabilizers of the torus actions
coincide for the corresponding faces of orbit spaces. Thus characteristic functions
on P and Q taking values in Zn {˘ are the same. Hence, the weighted simplicial
spheres pKP , Λq and pSQ , νq – #Γ pKΨV pvq , νq are isomorphic.
So far to prove Theorem 2 for n “ 3 it is sufficient to prove the following
statement.
Proposition 3.1. There exists a simple 3-dimensional polytope P and a characteristic function Λ : FpP q Ñ Z3 {˘ such that the dual weighted sphere pKP , Λq cannot
be represented as a connected sum, along a tree, of weighted spheres dual to Delzant
polytopes.
The proof of this proposition takes most part of this section. We proceed bysteps. At first notice that any simplicial 2-sphere, which is a simplicial complex,
is dual to some simple 3-polytope by Steinitz’s theorem (see e.g. [19]). So it issufficient to prove that there exists a weighted 2-dimensional simplicial complex K,
which cannot be represented as a connected sum, along a tree, of weighted spheres
dual to Delzant polytopes.

TORIC ORIGAMI STRUCTURES ON QUASITORIC MANIFOLDS

Sv

Rv

iv,e

iu,e

Cv,e

Cu,e

11

Su

Ce

K
Г
e
v

u

Figure 5. Connected sum of spheres along a tree
Construction 3.2. We introduce some notation in addition to that of construction 2.8, see Fig. 5. As before, let Γ “ pV, Eq be a tree. Suppose that a simplicial
pn ´ 1q-sphere Sv is associated with each node v P V , and for each edge e P E withan endpoint v P V there is an admissible vertex iv,e P Sv subject to the following
conditions: (1) linkSv iv,e is isomorphic to linkSu iu,e for any edge e with endpoints
v, u; (2) Vertices iv,e1 , iv,e2 are different and not adjacent in Sv for any two edges
e1 ‰ e2 emanating from v. Then we can form a connected sum along Γ as inconstruction 2.7: K “ #Γ Sv . For each v P V consider the simplicial subposetğ
star˝Sv iv,e .
(3.1)
Rv “ Sv z
ePE,vPe

This subposet will be called a region. Denote linkSv iv,e by Cv,e . By construction,
Cv,e is attached to Cu,e if e “ tv, uu. The resulting pn ´ 2q-dimensional simplicial
subposet of K is denoted by Ce . Since iv,e is admissible, the subposet Ce – Cv,e “linkSv iv,e is a simplicial pn ´ 2q-sphere.
We get a collection of pn ´ 2q-dimensional cycles Ce , e P E, dividing the pn ´ 1qsphere K into regions Rv , v P V . If e “ tv, uu, then Rv and Ru share a common
border Ce . Note that cycles Ce are mutually ordered, meaning that each Ce lies
at one side of any other cycle. Though the cycles may have common points (asschematically shown on Fig. 5) and even coincide (in this case the region between
them coincides with both of them).
On the other hand, any collection of mutually ordered pn´2q-dimensional spherical cycles in K defines the representation of K as a connected sum of smaller
simplicial spheres. A representation K “ #Γ Sv will be called a slicing.
Define the width of a slicing Θ to be the maximal number of vertices in its
regions:
(3.2)

def

widpΘq “ maxt| VertpRv q| | v P V u.

Define the fatness of a sphere K as the minimal width of all its possible slicings:
(3.3)

def

ftpKq “ mintwidpΘq | Θ is a slicing of Ku.

The essential idea in the proof of Proposition 3.1 is the following.
Lemma 3.3. Let K be an pn´1q-dimensional simplicial sphere and Λ : VertpKq Ñ
Zn {˘ a weighting. Let r denote the number of different values of this weighting,12

A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG

r “ |ΛpVertpKqq|. Suppose that ftpKq ą 2r. Then pK, Λq cannot be represented asa connected sum, along a tree, of simplicial spheres dual to Delzant polytopes.
Proof. Assume the converse. Then pK, Λq – p#Γ KΨV pvq , νq, where ΨV pvq are
Delzant polytopes. If we forget the weights, this defines a slicing Θ of K. The widthof every slicing of K is greater than 2r by the definition of fatness. In particular,
widpΘq ą 2r. Thus there exists a node v of Γ such that | VertpRv q| ą 2r.
The region Rv is a subcomplex of KΨV pvq . The restriction of Λ to the subset
VertpRv q coincides with the restriction of ν : VertpKΨV pvq q Ñ Zn {˘ to VertpRv q.
Recall, that ν̃pF q P Zn is the outward normal vector to the facet F P FpΨV pvqq “VertpKΨV pvq q, and νpF q P Zn {˘ is its class modulo sign. The outward normal
vectors to facets of a convex polytope are mutually distinct, thus |ν̃pVert Rv q| “
| VertpRv q| and, therefore, |νpVertpRv qq| ě | VertpRv q|{2. Thus |ΛpVertpRv qq| “|νpVertpRv qq| ą r, — the contradiction, since r is the total number of values of Λ.
So far we may find counterexamples to origami realizability among polytopes,
which are Zn -colored with a small number of colors, but whose dual simplicial
spheres have large fatness. Of course such examples do not appear when n “ 2 —this would contradict Theorem 1. A simplicial 1-sphere is a cycle graph Ck . By
considering diagonal triangulations of a k-gon, one can easily check that Ck canbe represented as a connected sum of several cycle graphs of the form C4 or C5 ,giving the slicing of width 3. Hence fatness of any 1-dimensional sphere is at most3, while any weighting takes at least 2 values, so the conditions of Lemma 3.3 arenot satisfied when n “ 2.
The existence of 2-spheres satisfying conditions of Lemma 3.3 is thus our next
and primary goal. At first, we prove that any 2-sphere admits a characteristic
function with few values. Then we construct spheres of arbitrarily large fatness.
Lemma 3.4. Any simplicial 2-sphere K admits a weighting Λ : VertpKq Ñ Z3 {˘such that |ΛpVertpKqq| ď 4.
Proof. Four color theorem states that there exists a proper vertex-coloring: VertpKq Ñ
tα1 , α2 , α3 , α4 u. Now replace colors by integral vectors α1 ÞÑ p1, 0, 0q, α2 ÞÑ p0, 1, 0q,
α3 ÞÑ p0, 0, 1q, α4 ÞÑ p1, 1, 1q. Any three of these four vectors span the lattice. Therefore the map Λ : VertpKq Ñ Z3 {˘ thus obtained satisfies condition(˚). Hence, Λ is a weighting, taking at most 4 values.

Remark 3.5. This is a standard trick in toric topology. Classically, it is applied
to prove that any simple 3-polytope admits a quasitoric manifold [5].
Though for our purpose we just need 2-spheres with ftpKP q “ 9, it seems intuitively clear that in dimension 2 and higher there exist spheres of arbitrarily large
fatness. But it is not a priori clear how to describe such spheres explicitly in combinatorial terms. We present one possible approach below, but some steps of ourconstruction do not generalize to dimensions ą 3.
Proposition 3.6. For any N ą 0 there exists a simplicial 2-sphere K such that
ftpKq ą N .
Proof. The underlying idea is the following. Suppose that a 2-sphere K is “thin”
i.e. ftpKq ! | VertpKq|. Then there exists a slicing Θ of K into pieces with small
numbers of vertices. In particular, the discrete length of any cycle Cv in a slicing Θshould be small. Then the sphere K is “tightened”, like the one shown on Fig. 6. It
has the feature that small cycles can bound large areas. To measure this property,
we introduce a natural metric on |K| in which all edges have length 1, and then

TORIC ORIGAMI STRUCTURES ON QUASITORIC MANIFOLDS

13

Figure 6. Metric features of a “thin” simplicial spherecompare the metric space |K| with a (2-dimensional) round sphere S of a constantradius. If there is a bijection |K| Ô S with close Lipschitz constants, then |K| is
not thin. The reason why S suits well for this consideration is that small curves on
S cannot bound large areas, which follows from the isoperimetric inequality.
Quite similar considerations and ideas are used in the theory of planar separators.
Some results of this theory can be used to prove Proposition 3.6 directly.
? If G is aplanar graph with k vertices, it is known that there exists a set of Op kq vertices
which separates G into two parts of roughly equal size (such separatingsets are?
called planar separators). It is also known that the asymptotic Op kq is the best
possible for planar separators [8]. If all 2-spheres were “thin”, then every planargraph would have a separator of size, bounded by some constant, which contradicts
the aforementioned asymptotic.
Now that we described the intuitive idea let us get to technical work.
Construction 3.7. Let K be a 2-dimensional simplicial complex. Define a (piecewise Riemannian) metric g and measure µ on |K| in such a way that each triangle|I| Ă |K| becomes equilateral Euclidian triangle with the standard metric and measure.
? We also assume that the length of each edge is 1, thus the area of each triangleis 3{4.
Let Lpγq denote the length of a piecewise smooth curve γ in |K|. If C Ă K is a
closed 1-dimensional cycle (simplicial subcomplex), then, obviously,(3.4)

Lp|C|q “ | VertpCq|.

A cycle C divides K into two subcomplexes K` and K´ , each homeomorphic toa closed 2-disc (we suppose C Ă K` , K´ ). Let us estimate the number of verticesin K´ in terms of its area (K` is similar). Let V´ , E´ , T´ denote the number ofvertices, edges and triangles in K´ . By the definition of measure, T´ “ ?43 µp|K´ |q.
We have V´ ´ E´ ` T´ “ 1 (Euler characteristic of K´ ) and E´ ă 3T´ (by counting
pairs e Ă t, where e is an edge and t is a triangle). Therefore,(3.5)

8V´ ď ? µp|K´ |q.
3

Let SR be a 2-dimensional round sphere of radius R, with the standard metric gs
and measure µs . A piecewise smooth closed curve γ Ă SR without self-intersections
divides SR into two regions A` , A´ . The isoperimetric inequality on a sphere (seee.g. [17, Ch.4]) has the form
(3.6)

R2 Ls pγq2 ě µs pA` qµs pA´ q,

where Ls pγq is the length of γ. Since µs pSR q “ 4πR2 we may assume that µs pA` q ě
2πR2 (otherwise consider A´ instead), thus

14

A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG

Ls pγq2.
2π
Notice that this inequality does not depend on the sphere radius.
Let K be a 2-dimensional simplicial sphere and R, c1 , c2 , c3 , c4 be positive real
numbers. Suppose there exists a bijective piecewise smooth map f : |K| Ñ SR such
that
(3.7)

µs pA´ q ď(3.8)

c1 Lpγq ď Ls pf pγqq ď c2 Lpγq,(3.9)

c3 µpΩq ď µs pf pΩqq ď c4 µpΩq,

for each piecewise smooth curve γ Ă |K| and measurable set Ω Ă |K|. Numbers
c1 , c2 , c3 , c4 will be called Lipschitz constants of the map f .
Lemma 3.8. Suppose that a simplicial 2-sphere K satisfies the conditions statedabove. If C is a cycle in K with at most N vertices, then either K` or K´ has at
4N 2 c2
most ?3πc2 vertices.
3

Proof. Among two regions f p|K´ |q, f p|K` |q Ă SR let f p|K´ |q be the one with thesmaller area. Combining (3.4), (3.5), (3.7), (3.8), and (3.9), we get
88µs pf p|K´ |qq
8Ls pf p|C|qq2
4N 2 c2?
?
V´ ď ? µp|K´ |q ďď
ď ? 2,
33c3
2 3πc33πc3which was to be proved.

(3.10)

Suppose ftpKq ď N . Then by definition there exists a slicing K “ #Γ Sv ,encoded by a tree Γ, such that each region Rv has at most N vertices (see construction 3.2). Let us show that the degree of each node v of Γ is bounded fromabove.
Lemma 3.9. If Θ is a slicing K “ #Γ Sv and widpΘq ď N , then deg v ď 2pN ´ 2q
for any node v of Γ.
Proof. Denote deg v by d. By construction, the region Rv is obtained from a sphere
Sv by removing d open stars which correspond to the edges of Γ emanating from v.
The complex Rv itself can be considered as a plane graph. Denote the numbers of
its vertices, edges and faces by V, E, R respectively. By the definition of the width,
we have V ď N . We also have V ´ E ` R “ 2, and 2E ě 3R (each region has atleast 3 edges). Thus, V ě 2 ` 12 R. Notice that each removed open star represents
a face of graph Rv , therefore, d ď R ď 2pV ´ 2q ď 2pN ´ 2q.
Lemma 3.10. Let K be a 2-dimensional simplicial sphere endowed with the map
f to the round sphere, satisfying Lipschitz bounds (3.8) and (3.9). For a natural
4N 2 c2number N set A “ ?3πc2 and B “ 2pN ´ 2q. If | VertpKq| ą maxpAB ` N, 2Aq,
3then ftpKq ą N .
Proof. Assume the contrary: ftpKq ď N . Then there is a slicing K “ #Γ Sv inwhich every region Rv has at most N vertices. Consequently, any cycle Ce , e P E
has at most N vertices. By Lemma 3.8, the cycle Ce divides K into two parts, oneof which has ď A vertices. Since | VertpKq| ą 2A, the other part has ą A vertices.
Assign a direction to each edge e of Γ in such a way that e points from the larger
component of K r Ce to the smaller, where the “size” means the number of vertices.
Γ is a tree, therefore there exists a source, i.e. a node u from which all adjacentedges emanate. Speaking informally, this node represents a “big sized bubble”,
meaning that the part of a sphere, lying across each border has a small size. Let

TORIC ORIGAMI STRUCTURES ON QUASITORIC MANIFOLDS

15

d denote the degree of the chosen node u. Denote by Γ1 , . . . , Γd the connectedcomponents of the graph Γ rˇ u. ByŮLemmaˇ 3.9 we have d ď B. By the constructionof the directions of edges, ˇVertp Γi Rv qˇ ď A for each Γi . Thus | VertpKq| ăˇřd ˇŮ| VertpRu q| ` i“1 ˇVertp Γi Rv qˇ ď N ` AB — the contradiction.

Lemma 3.11. For any N ą 0 there exists a 2-dimensional simplicial sphere K
such that:(1) There exists a piecewise smooth map f : |K| Ñ SR satisfying Lipschitz
bounds (3.8) and (3.9) for some constants c1 , c2 , c3 , c4 , R ą 0(2) | VertpKq| ą maxpAB `N, 2Aq, where A and B are defined in Lemma 3.10.
Proof. Start with the boundary of a regular tetrahedron with edge length 1: L “B∆3 . The projection from the center of L to the circumsphere f : L Ñ SR is obviously Lipschitz for some constants c1 , c2 , c3 , c4 ą 0. Now subdivide each triangle of|L| into q 2 smaller regular triangles as shown on Fig. 7.

Figure 7. Subdivision of a regular triangle.
This results in a simplicial complex Lpqq . As a space with metric and measure|Lpqq | is homothetic to |L| with a linear scaling factor q (recall that the metric onsimplicial complexes is introduced in such way that each edge has length 1). Thus
there exists a map f pqq : |Lpqq | Ñ SqR with the same Lipschitz constants as f . The
number of vertices | VertpLpqq q| can be made arbitrarily large.
Lemmas 3.11 and 3.10 conclude the proof of Proposition 3.6.



Remark 3.12. Actually, in the proof of Lemma 3.11 we could have started fromany simplicial sphere L, take any piecewise smooth map f : |L| Ñ SR , find Lipschitz
constants c2 , c3 ą 0 (they exist by the standard calculus arguments), and thenapply the same subdivision procedure. We used the boundary of a regular simplex,
because in this case Lipschitz map is constructed easily and allows an explicit
computation.
We give a concrete example of a quasitoric manifold which is not toric origami,
by performing this computation. The calculations themselves are elementary thus
omitted. It is sufficient to construct a simplicial sphere for N “ 8. For a projection
map from the boundary of a regular tetrahedron to the circumscribed sphere wehave Lipschitz constants c2 “ 3, c3 “ 13 . Thus maxpAB ` N, 2Aq « 15251.14. Subdivide each triangle in the boundary of a regular tetrahedron in q 2 small triangles
where q ě 88. This gives a simplicial sphere K with at least 15490 vertices andthe same Lipschitz constants as B∆3 . Thus ftpKq ą 8. Now take the dual simple
polytope P of K, consider any proper facet-coloring in four colors and assign acharacteristic function Λ, as described in Lemma 3.4. This gives a characteristic
pair pP, Λq, whose corresponding quasitoric manifold is not toric origami.
Of course, all our estimations are very rough, and, probably, there are betterways to construct fat spheres. For sure, there exist 2-spheres of fatness 9 with lessthan 15490 vertices.

16

A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG

Remark 3.13. Note that in dimension 3 and higher there is no simplicial subdivision of a regular simplex into smaller regular simplices. This is one of two places in
the proof, where the dimension restriction is crucial. The second place is the Fourcolor theorem in Lemma 3.4.
Proposition 3.1 proves Theorem 2 for n “ 3. Now we need to make the remainingcases n ą 3.
Proposition 3.14. There exist quasitoric manifolds of any dimension 2n, n ą 3,
which are not toric origami.
Proof. Let MpP,Λq be any quasitoric manifold, which is not toric origami. Take theproduct of MpP,Λq with S 2 (the circle T 1 acts on S 2 by axial rotations). On thelevel of orbit spaces, this corresponds to multiplying P with a closed interval I Ă R.
We claim that quasitoric manifold MpP,Λq ˆ S 2 is not toric origami. If MpP,Λq ˆ S 2
were a toric origami manifold, then all its characteristic submanifolds should betoric origami as well (see construction 1.9). But MpP,Λq is one of them. This gives acontradiction. Thus taking products with S 2 produces examples for all n ą 3. 
Remark 3.15. Sphere S 2 is the simplest example of a quasitoric manifold. In the
proof of Proposition 3.14 we could have used any other quasitoric manifold insteadof S 2 . If MpP,Λq and MpP 1 ,Λ1 q are quasitoric manifolds and one of them is not toricorigami, then the quasitoric manifold MpP,Λq ˆ MpP 1 ,Λ1 q “ MpP ˆP 1 ,Λ‘Λ1 q is not toricorigami as well.
Remark 3.16. On the other hand, new toric origami manifolds can be produced
from a given one in a similar way as we used for constructing non-examples. It iseasy to observe that if M is a toric origami manifold and M 1 is a toric symplectic
manifold, then M ˆ M 1 is again a toric origami manifold. We would also like tomention that projective bundles over toric origami manifolds are again toric origami
manifolds. More precisely, if M 2n is toric origami, and L1 , . . . , Lk are complex line
bundles over M , each having an S 1 action on fibers, then the projectivization¸˜kàM̃ “ P
Lj ‘ Cj“1n

1 kwith the induced action of T ˆ pS q is also a toric origami manifold.
4. Discussion and open questions
4.1. Asymptotically most of simplicial 2-spheres are fat. We already mentioned a relation of our study to the theory of planar separators in Section 3. We
also want to mention another connection to the theory of random infinite planar
maps. This rapidly developing part of probability theory aims, among other things,
to give a firm foundation for some facts in statistical physics and quantum gravity.
The basic idea of this study is the following [12, 13]. Fix a number k, a parameter of the whole construction. For a given n consider all possible (rooted) planek-angulations with n faces. For k “ 3, these are roughly the same as simplicial
spheres. Every plane graph has a standard metric, turning it into a metric space.
By letting the number of faces tend to infinity, and renormalizing the diameter ofgraphs in a correct way, one considers the limits of converging sequences of graphs.
The limits are taken with respect to the Gromov–Hausdorff metric defined on theset of isometry classes of metric spaces.
Since there is only a finite number of such graphs with a fixed number n of faces,
we can take a uniform distribution of such graphs. The uniform distributions on thesets of prelimit metric spaces give rise to a limiting distribution, which is viewed as a

TORIC ORIGAMI STRUCTURES ON QUASITORIC MANIFOLDS

17

random compact metric space (of course, here we omit a lot of technicalities, needed
to state everything precisely). This random metric space is called a Brownian map
and considered as a good 2-dimensional analogue of the Brownian motion.
A wonderful thing is that the Brownian map does not actually depend on theparameter k, if k is either 3 or even [13]. It is also known that the Brownian map
is almost surely homeomorphic to a 2-sphere [11]. This suggests the following
Claim 4.1. For each N ą 0 almost all simplicial 2-spheres K have ftpKq ą N .
More precisely, if An denotes the set of all simplicial 2-spheres with ď n triangles,
and Bn,N Ă An the subset of simplicial spheres having ftpKq ą N , then
lim

nÑ8

|Bn,N |“ 1.
|An |The reason is as follows (cf. [12, Cor.5.3]). If there were a lot of “thin” simplicial spheres, they all would have bottlenecks — small cycles, dividing them into
macroscopic regions. After taking a limit as n Ñ 8 and rescaling the metric, thesebottlenecks would collapse to points. Thus the limiting metric space would be
non-homeomorphic to a sphere with non-zero probability.
Therefore, for most of simple combinatorial 3-polytopes P there exists a characteristic function Λ such that MpP,Λq is not toric origami.
4.2. Orbit spaces of toric origami manifolds. We may ask a more intricate
question.
Problem 1. Find a simple polytope P such that any quasitoric manifold MpP,Λqover P is not equivariantly homeomorphic to a toric origami manifold.
This question is motivated by the following fact. There exist a simple 3-polytope
P such that any quasitoric manifold MpP,Λq over P is not equivariantly homeomorphic to a symplectic toric manifold. Stating shortly: there exist combinatorial typesof simple 3-polytopes which do not admit Delzant realizations. It was proved in [6]that any 3-dimensional Delzant polytope has at least one triangular or quadrangular face. Consequently, in particular, a dodecahedron does not admit a Delzant
realization.
Problem 1 can be restated in different terms: are there any combinatorial restrictions on the orbit spaces of toric origami manifolds?
4.3. Fat simplicial spheres in higher dimensions. The examples of non-origami
quasitoric manifolds in high dimensions were constructed from the 3-dimensionalcase. On the other hand, Lemma 3.3 applies for any dimension. The problem offinding higher-dimensional polytopes whose dual spheres have large fatness may beof independent interest.
Actually, even if we find such a fat polytope, to make use of the developed
technique we should also construct a characteristic function with a small rangeof values. This constitutes a certain problem, since characteristic function may
not even exist, if n ě 4 (this happens for dual neighborly polytopes, see [5]).
Nevertheless, there is a big class of simplicial pn ´ 1q-spheres, so called balanced
spheres, which admit a proper vertex-coloring in n colors. Such colorings giverise to characteristic functions, which have exactly n values, i.e. minimal possible.
Such characteristic functions and the corresponding quasitoric manifolds were called
linear models in [5]. Passing to a barycentric subdivision makes every simplicial
sphere into a balanced sphere. We suppose that passing to a barycentric subdivision
does not strongly affect the fatness. If so, given any fat sphere dual to a simple
polytope, one can pass to its barycentric subdivision, provide it with a linear modelcharacteristic function, and finally obtain a quasitoric manifold which is not toricorigami.

18

A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG

4.4. Minimizing the range of characteristic function. Another problem, which
naturally arises from Lemma 3.3 is to find, for a given polytope P , a characteristic
function Λ with the minimal possible range of values |ΛpFpP qq| (if at least one
characteristic function is known to exist). This minimal number seems to be ananalogue of Buchstaber invariant (see the definition in [9] or [1]), as was noted to us
recently by N. Erokhovets. It may happen that an interesting theory hides beyond
this subject.
4.5. Toric varieties. There exist obstructions to origami realizability, other thanthose described in section 3. If a weighted simplicial sphere K can be represented
as a connected sum, along a tree, of simplicial spheres dual to Delzant polytopes,
this does not mean automatically that K corresponds to an origami template. The
reason is that a convex polytope contains more information than its normal fan(or, in our terminology, dual weighted simplicial sphere). It can be impossible toassemble an origami template from a collection of Delzant polytopes, even if theirdual weighted spheres suit together well.
Such situations appeared when we tried to answer the following
Problem 2. Does there exist a compact smooth toric variety, which is not equivariantly homeomorphic to a toric origami manifold?
Any projective toric variety corresponds to a convex polytope. Thus any smoothprojective toric variety is a symplectic toric manifold, which is a particular case of
toric origami. Thus, to prove the conjecture, one should consider non-projective
examples. Translating the problem into combinatorial language, the task is to finda complete smooth fan, which is not a normal fan of any polytope and, moreover,
not dual to any origami template. The simplest non-polytopal fan is the fan corresponding to a famous non-projective Oda’s 3-fold [16, p.84]. So it is natural tostart with a more concrete question:
Problem 3. Is Oda’s 3-fold a toric origami manifold?
Even this question happens to be rather non-trivial and cannot be solved solely
by the method developed in this paper.
4.6. Origami manifolds which are not quasitoric. In section 1 we mentionedthat a toric origami manifold MO is not quasitoric if its template graph has cycles.
Even if the orbit space of MO is contractible, the manifold MO may not be quasitoric. The simplest example of this kind is the sphere S 4 (example 1.8). The orbit
space of S 4 is a 2-gon, shown on Fig. 2, which is not a convex polytope. Excluding
situations of these two kinds we may ask the following question.
Problem 4. Let MO be a simply connected toric origami manifold and suppose thatthe dual simplicial sphere of its orbit space is a simplicial complex. Is the manifold
MO quasitoric?
In other words, does the orbit space of a simply connected toric origami manifold
admit a convex realization, provided that its dual simplicial sphere is a simplicial
complex?
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TORIC ORIGAMI STRUCTURES ON QUASITORIC MANIFOLDS

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Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 5588585, Japan.
E-mail address: ayzenberga@gmail.com
Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 5588585, Japan.
E-mail address: masuda@sci.osaka-cu.ac.jp
Division of Mathematical Models, National Institute for Mathematical Sciences,463-1 Jeonmin-dong, Yuseong-gu, Daejeon 305-811, Korea
E-mail address: seonjeong1124@nims.re.kr
Department of Mathematics, Osaka City University, Sumiyoshi-ku, Osaka 5588585, Japan.
E-mail address: zenghaozhi@icloud.com