数理解析研究所講究録
第 1922 巻 2014 年 112-123

112

QUASITORIC MANIFOLDS WHICH ARE NOT TORICORIGAMIANTON AYZENBERG, MIKIYA MASUDA, SEONJEONG PARK, AND HAOZHI ZENG
ABSTRACT. We construct 6-dimensional quasitoric manifolds which are nottoric origami manifolds.

INTRODUCTION
Origami manifolds appeared in diﬀerential geometry recently as a generalizationof symplectic manifolds [3]. Toric origami manifolds are in turn generalizationsof symplectic toric manifolds. Toric origami manifolds are a special class of 2ndimensional compact manifolds with an eﬀective action of a half-dimensional com. In this paper we consider the following question. How large is thispact torus
class? Which manifolds with half-dimensional torus actions are toric origami manifolds? In particular, which quasitoric manifolds admit toric origami structures? In[7] Masuda and Park proved
Theorem 1. Any simply connected compact smooth 4-manifold $M$ with an eﬀective
is equivariantly diﬀeomorp $hic$ to a toric origami manifold.
smooth action of
In particular, any 4-dimensional quasitoric manifold is toric origami. The samequestion about higher dimensions was open. Here we prove the negative result.
Theorem 2. There exist 6-dimensional quasitoric manifolds, which are not equivariantly homeomorphic to any toric origami manifold.
We will describe an obstruction for a quasitoric 6-man 垣 old to be toric origami
and present a large series of examples, where such an obstruction appears. In spiteof topological nature of the task, the proof is purely discrete geometrical: it relies
on metric and coloring properties of planar graphs.
$T^{n}$$T^{2}$

1. TOPOLOGICAL PRELIMINARIES

1.1. Quasitoric manifolds. The subject of this subsection originally appearedin the seminal work of Davis and Januszkiewicz [4]. The modern exposition andtechnical details can be found in [2, Ch.7].
Let
be a compact -dimensional torus. The standard representation ofonby coordinate-wise rotations. The action ofis a representation
is called locally standard, if $M$ has an atlas of standard charts,
a manifold
each isomorphic to a subset of the standard representation. In the following $M$ issupposed to be compact.
of the standard representation is a nonnegative cone
Since the orbit space, the orbit space of any locally standard action has thedenote the set of facets ofstructure of a compact manifold with corners. Let(i.e. faces of codimension 1). For each facet $F$ of consider a stabilizer subgroup, which preserves points over the interior of $F$ . This subgroup is 1, fordimensional and connected, thus it has the form$T^{n}$$T^{n}$$n$$T^{n}$$\tau^{n}\sim \mathbb{C}^{n}$$M^{2n}$$\mathbb{C}^{n}/T^{n}$$\mathbb{R}_{\geq}^{n}=\{x\in \mathbb{R}^{n}|x_{i}\geq 0\}$$\mathcal{F}(Q)$

$Q$

$Q$

$\lambda(F)\subset T^{n}$$\{(t^{\lambda_{1}}, \ldots, t^{\lambda_{n}})|t\in T^{1}\}\subset T^{n}$

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

, defined uniquely up to a common
some primitive integral vector
associated with
sign)
up
to
integralThus,
primitive
vectorsign.
acharacteristicacalledof . This map
any facetfunction(or a characteristic map). It satisfies the following so called $(*)$ -condition:$(\lambda_{1}, \ldots, \lambda_{n})\in \mathbb{Z}^{n}$$\Lambda(F)\in \mathbb{Z}^{n}/\pm is$

$($

$F$

$\Lambda:\mathcal{F}(Q)arrow \mathbb{Z}^{n}/\pm is$$Q$intersect, then the set of vectors
is a part some basis of, but one of its two particular
Here we actually take not a classnot depend on the choice ofdoescondition
Obviously,thein.
representatives$(*)$
is well defined. The same convention appears further in the text
sign, thus
without special mention.

If facets

$(*)$

,...,
,...,

$F_{s}$$F_{1}$$\Lambda(F_{1})$$\mathbb{Z}^{n}.$$\Lambda(F_{s})$$\Lambda(F_{i})\in \mathbb{Z}^{n}/\pm$$\mathbb{Z}^{n}$is calledwith a locally standard action ofDefinition 1.1. A manifold$M/T^{n}$polytope
as a
simpleais homeomorphic toquasitoric, if the orbit space
manifold with corners.
Recall that a convex polytope $P$ of dimension is called simple if any of its
vertices lies in exactly 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.
be a characteristic function, i.e. any map
be a simple polytope andLetis called a characteristic
satisfying ( -condition. The pairpair. According to [4], there is one to one correspondence{quasitoric manifolds} $-\{$ characteristic pairs
up to equivariant homeomorphism on the left-hand side and combinatorial equivalence on the right-hand side. The quasitoric manifold associated with a character. Let denote the projection to the orbit
will be denoted
istic pairdetermines a characteristic submanispace $\eta:M_{(P,\Lambda)}arrow P$ . Each facet
isof dimension $2n-2$ . On its own, the manifold
fold
again a quasitoric manifold with the orbit space $F.$
$M^{2n}$$T^{n}$

$n$

$n$

$P$

$\Lambda$$(P, \Lambda)$$*)$$\Lambda:\mathcal{F}(P)arrow \mathbb{Z}^{n}/\pm$

$a$

$\}$

$(P, \Lambda)$$M_{(P,\Lambda)}$$\eta$$F\in \mathcal{F}(P)$$N_{F}$$N_{F^{d}}=^{ef}\eta^{-1}(F)\subset M_{(P,\Lambda)}$1.2. Toric origami manifolds. In the following subsections we recall the definitions and properties of toric origami manifolds and origami templates. Moredetailed exposition of this theory can be found in [3], [7] or [6].
A folded symplectic form on a $2n$-dimensional smooth manifold $M$ is a closed
and whosevanishes transversally on a subset
whose top power
2-formrestriction to points in has maximal rank. Then is a codimension-one submanis called a folded symplectic manifold.
ifold of $M$ called the fold. The pairis a genuine symplectic
If is empty, is a genuine symplectic form andmanifold according to classical definition.
Since the restriction of to $Z$ has maximal rank, it has a one-dimensional kernel
at each point of . This determines a line field on called the null foliation. If
the null foliation is the vertical bundle of some principal -fibration $Zarrow Y$ over acompact base , then the folded symplectic form is called an origami form andis called an origami manifold.
the pairis(of any dimension) on an origami manifold
The action of a torus
called Hamiltonian if it admits a moment map $\mu:Marrow t^{*}$ to the dual Lie algebraof the torus, which satisfies the conditions: (1) is equivariant with respect to thegiven action of on $M$ and the trivial action of on $t^{*};(2)\mu$ collects Hamiltonian$V\rangle=\omega(V\#$ ,is the function on $M$ , taking
wherefunctions, that is,$V\#$is a vector flow on $M$ , generated by $V\in t,$,
at a point $x\in M,$
the value
is its dual 1-form.
and$Z$$\omega^{n}$$\omega$$Z$$Z$$(M, \omega)$$Z$$(M, \omega)$$\omega$$\omega$$Z$$Z$$S^{1}$$Y$$\omega$$(M, \omega)$$(M, \omega)$

$T$

$\mu$

$T$

$T$

$d\langle\mu,$$\langle\mu(x)$$\omega(V\#, \cdot)$$V\rangle$$\langle\mu,$$V\rangle$

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QUASITORIC MANIFOLDS WHICH ARE NOT TORIC ORIGAMI

Definition 1.2. A toric origami manifold $(M, \omega, T, \mu)$ , abbreviated as $M$ , is aequipped with an eﬀective Hamiltoniancompact connected origami manifold
pondingwith $\dim T=\frac{1}{2}\dim M$ and with a choice of aaction of a torus
map
moment$(M, \omega)$$T$$corre\mathcal{S}$$\mu.$

is empty, a toric origami
1.3. Symplectic toric manifolds. When the foldmanifold is a symplectic toric manifold. In this case the image $\mu(M)$ of the momentmap is a Delzant polytope in , and the map $\mu:Marrow\mu(M)$ itself can be identified
with the map to the orbit space $\eta:Marrow M/T^{n}$ . A classical theorem of Delzant [5]says that symplectic toric manifolds are classified by the images of their moment
maps in. In other words, there is a one-to-one correspondence$Z$$t^{*}$$t^{*}\cong \mathbb{R}^{n}${symplectic toric manifolds} $-$ { Delzant polytopes}
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 1.3. A simple convex polytope
fan is smooth (with respect to a fixed lattice$P\subset \mathbb{R}^{n}$is called Delzant, if its normal).

$\mathbb{Z}^{n}\subset \mathbb{R}^{n}$Construction 1.4 (Topological model of symplectic toric manifold). Let be aconsider its outward primitive normal
Delzant polytope in . For a facet. Consider the corresponding vector modulo $sign:\nu(F)\in \mathbb{Z}^{n}/\pm.$vectorsatisfies ( , thus provides
By the definition of Delzant polytope,
quasitoric
manifold
an example of a characteristic function. The$P$$F\in \mathcal{F}(P)$$\mathbb{R}^{n}$$\tilde{\nu}(F)\in \mathbb{Z}^{n}$$*)$$\nu:\mathcal{F}(P)arrow \mathbb{Z}^{n}/\pm$$M_{P}^{d}=^{ef}M_{(P,\nu)}.$is the symplectic toric manifold corresponding to $P$ (up to equivariant homeomorphism).

1.4. Origami templates. To generalize Delzant correspondence to toric origami
manifolds we need a notion of an origami template, which we review next.
(w.r.t.
denote the set of all (full-dimensional) Delzant polytopes inLetthe set of all their facets.
fixed lattice) and$\mathbb{R}^{n}$$\mathcal{D}_{n}$$a$$\mathcal{F}_{n}$, whereDefinition 1.5. An origami template is a triple
is a connected finite graph (loops and multiple edges are allowed) with thevertex set and edge set$(\Gamma, \Psi_{V}, \Psi_{E})$$\bullet$$\Gamma$$V$$\bullet\Psi_{V}:Varrow \mathcal{D}_{n}$$\bullet\Psi_{E}:Earrow \mathcal{F}_{n}$$E_{j}$;;

subject to the following conditions:$v_{2}\in V$ , thenis a facet of
1. If $e\in E$ is an edge of with endpoints, and these polytopes coincide nearandboth polytopes
insuch that(this means there exists an open neighborhoodof$\Psi_{E}(e)$$\Gamma$$v_{1},$$\Psi_{E}(e)$$\Psi_{V}(v_{2})$$\Psi_{V}(v_{1})$$U$$U\cap\Psi_{V}(v_{1})=U\cap\Psi_{V}(v_{2}))$$\mathbb{R}^{n}$

.

2. If
are two edges of.
are disjoint facets of$e_{1},$ $e_{2}\in E$$\Psi_{E}(e)$$\Gamma$adjacent to $v\in V$ , then$\Psi_{E}(e_{1})$and$\Psi_{E}(e_{2})$$\Psi_{V}(v)$The facets of the form
template.

$\Psi_{E}(e)$for $e\in E$ are called the fold facets of the origami

the nodes.
For convenience in the following we call the vertices of graph
One can simply view an origami template as a collection of (possibly overlapping)
Delzant polytopes $\{\Psi_{V}(v)|v\in V\}$ in the same ambient space, with some gluing
data, encoded by a template graph (see Fig. 1).
$\Gamma$$\Gamma$

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

FIGURE 1. Examples of origami templates in $\dim=2$ . Fold facets
are shown in red.

Theorem 3 ([3]). Assigning the moment data of a toric origami manifold induces
one-to-one correspondence$a${toric origami manifolds} $-$ { origami templates}
up to equivariant origami symplectomorphism on the left-hand side, and aﬃneequivalence on the right-hand side.

Construction 1.6 (Topological construction of toric origami manifold). Consider, $\Gamma=(V, E)$ . For each node $v\in V$ the Delzant
an origami template, see construction 1.4.
polytope
gives a symplectic toric manifold
Now do the following procedure:
for $v\in V$ ;1 Take a disjoint union of all manifolds
take an equivariant
and2 For each edge $e\in E$ with distinct endpoints
along the characteristic submaniand
connected sum of(which is embedded in both manifolds);
fold
3 For each loop $e\in E$ based at $v\in V$ take a real blow up of normal bundle
insideto the submanifold
Step 2 makes sense because of pt.1 of Definition 1.5. Indeed, the polytopes
have equivari, thus
andagree nearand, so the connected sum is wellantly homeomorphic neighborhoods arounddefined. Pt. 2 of Definition 1.5 ensures that surgeries do not touch each other, so
all the connected sums and blow ups can be taken simultaneously.
. ThisDenote the resulting manifold of this construction by
is exactly the toric origami manifold associated with via Theorem 3.
$O=(\Gamma, \Psi_{V}, \Psi_{E})$$\Psi_{V}(v)\in \mathcal{P}_{n}$$M_{\Psi_{V}(v)}$$M_{\Psi_{V}(v)}$$v_{1}$$M_{\Psi_{V}(v_{1})}$$v_{2}$$M_{\Psi_{V}(v_{2})}$$N_{\Psi_{E}(e)}$$N_{\Psi_{E}(e)}$$\Psi_{V}(v_{1})$$M_{\Psi_{V}(v)}.$$\Psi_{E}(e)$$\Psi_{V}(v_{2})$$M_{\Psi_{V}(v_{2})}$$M_{\Psi_{V}(v_{1})}$$N_{\Psi_{E}(e)}$$M_{O}=M_{(\Gamma,\Psi_{V},\Psi_{E})}$$O$

Example 1.7. 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. The characteristic submanifold corresponding to the fold facet is a projective
along the line
. Thus, $X$ is a connected sum of two copies ofline, which lies in both. It is easily seen that this manifold is$\mathbb{C}P^{2}$$\mathbb{C}P^{2}$$\mathbb{C}P^{1}\subset \mathbb{C}P^{2}$$S^{4}.$$\mathbb{C}P^{1}$)
An origami template $O=(\Gamma, \Psi_{V}, \Psi_{E})$ (and the corresponding manifold
isis called co\"orientable if has no loops (i.e. edges based at one point). Ifon
is locally standard [6, lemma 5.1]. The
co\"orientable, then the action of
converse is also true. In the following we consider only co\"orientable templates andtoric origami manifolds.
$M_{O}$$M_{O}$$\Gamma$$T^{n}$$M_{O}$

Construction 1.8 (Orbit space of toric origami manifold). The orbit space $Q=$of (co\"orientable) toric origami manifold is a manifold with corners.
along
It can be described as a topological space obtained by gluing polytopes$M_{(\Gamma,\Psi_{V},\Psi_{E})}/T^{n}$$a$$\Psi_{V}(v)$116
QUASITORIC MANIFOLDS WHICH ARE NOT TORIC ORIGAMI

fold facets. More precisely,(1.1)$Q=v\in V\sqcup(v, \Psi_{V}(v))/\sim,$

where $(u, x)\sim(v, y)$ if there exists an edge with endpoints and , and $x=y\in$identified in. Facets of are given by non-fold facets of polytopes
andprecise,call
facetsletus
non-foldway.
To make thisthe same$e\inE$(with endpoints
elementary neighboring w.r. . to the edgeand ) if $F_{1}\cap\Psi_{E}(e)=F_{2}\cap\Psi_{E}(e)$ . The relation of elementary neighborlinessthe set of all non-fold facets of all polytopes
generates an equivalence relation
as a union of facets in one. Define the facet $[F]$ of the orbit spaceequivalence class:
$u$

$e$

$v$

$\Psi_{V}(v)$$Q$$\Psi_{E}(e)$$F_{1}\in \mathcal{F}(\Psi_{V}(v_{1}))$$t$$F_{2}\in \mathcal{F}(\Psi_{V}(v_{2}))$$v_{1}$$v_{2}$$rightarrow on$$Q$$\Psi_{V}(v)$$[F] def= \sqcup(v, G)/\sim, [F]\in \mathcal{F}(Q)$ ,(1.2)$Gisnotf\circ 1d,Grightarrow Fv\in V,G\in \mathcal{F}(\Psi_{V}(v)),$

where is the same as in (1.1).
Let us define a primitive normal vector to the facet $[F]$ of by $v([F])^{d}=^{ef}\nu(F)\in$. It is well defined since $\nu(F)=\nu(G)$ for $Frightarrow G.$
Note that the relation of elementary neighborliness determines a connected subare Delzant and lie in the same hyperplaneof . All facets
graph
Thus we obtain an induced origami template$\sim$$Q$$\mathbb{Z}^{n}/\pm$$Grightarrow F$$\Gamma$$\Gamma_{[F]}$(1.3)$H_{[F]}.$$O_{[F]}=(\Gamma_{[F]}, \Psi_{V}|_{\Gamma_{[F]}}\cap H_{[F]}, \Psi_{E}|r_{[F]}\cap H_{[F]})$of dimension $n-1$ . In particular, if $\eta:M_{O}arrow Q$ denotes the projection to the orbit
space, then the characteristic submanifold $\eta^{-1}([F])$ is the toric origami manifoldof dimension $2n-2$ generated by the origami template$O_{[F]}.$

Extending the origami analogy, we can think of the orbit space as“unfolding”
the origami template and then smoothening the angles adjacent to the former foldfacets (remember that we have to identify neighboring faces!).
$Q$

FIGURE 2. The orbit space of a manifold
origami template shown on Fig. 1, left.

$S^{4}$, corresponding to thehas the same homotopy
It is easy to see that the orbit space
type as the graph , thus is either contractible (when is a tree) or homotopy
equivalent to a wedge of circles. This observation shows that whenever the templategraph has cycles, the corresponding toric origami manifold cannot be quasitoric.
As an example, the origami template shown on Fig. 1, at the right corresponds tothe origami manifold which is not quasitoric. Since we want to find a quasitoric
manifold which is not toric origami, we need to consider only the cases when theorbit space is contractible. Thus in the following is supposed to be a tree.
$Q=M_{(\Gamma,\Psi_{V},\Psi_{E})}/T^{n}$$\Gamma$$Q$$\Gamma$$\Gamma$$\Gamma$

2. WEIGHTED SIMPLICIAL SPHERESIn the previous section we have seen that quasitoric manifolds are encoded by
the orbit spaces (which are simple polytopes) and characteristic functions (which

117
A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG). It will be easier, however, to work
are colorings of facets by elements ofwith the dual objects, which we call weighted simplicial spheres.
such that: (1)Recall that a simplicial poset is a finite partially ordered set, (2) For each $I\in S$ the interval subset
There is a unique minimal element$[\emptyset, I]def=\{J\in S|J\leq I\}$ is isomorphic to the poset of faces of $(k-1)$ -dimensional
simplex (i.e. Boolean lattice of rank ). In this case the element is said to haverank and dimension $k-1$ . The elements of are called simplices and elementsof rank 1 are called vertices. The set of vertices of is denoted Vert(S).
A simplicial poset is called pure, if all maximal simplices have the same dimension. A simplicial poset is called a simplicial complex, if for any subset of vertices, there exists at most one simplex whose vertex set is$\mathbb{Z}^{n}/\pm$$S$$\emptyset\in S$

$I$

$k$

$S$

$k$

$S$

$S$

$\sigma\subseteq Vert(S)$$\sigma.$

Construction 2.1. It is convenient to visualize simplicial posets using their geoof dimension rank $(I)-1$
metrical realizations. Assign the geometrical simplex$I\inS$
relation in . Moreaccording
the
order
to
together
each
and
them
to
attachspaceformally, the geometric realization of is the topological$\Delta_{I}$

$S$

$S$

$|S|^{d}=^{ef}\sqcup(I, \triangle_{I})/\sim I\in S$ ’

where

if $I_{1}<I_{2}$ and$(I_{1}, x_{1})\sim(I_{2}, x_{2})$$x_{1}=x_{2}\in\Delta_{I_{1}}\subset\Delta_{I_{2}}$. See details in [2].

is homeomorphic to a
A simplicial poset is called a triangulated sphere if
sphere.
is called a PL-sphere if the barycentric subdivision $S’$ (which is a simplicial complex) is PL-homeomorphic to the boundary of a simplex. In dimension
2, which is the only important case for us, these two notions are equivalent. In thesequel we call either of them simplicial spheres.
Construction 2.2. Let us define a connected sum of two simplicial spheres along
their vertices. At first we should exclude certain degenerate situations.
. In otherFor every $I<J$ in
there is a complementary simplex
words,
complementary to the face . Define a link of ais the face ofsimplex $I\in S$ as a partially ordered set $1ink_{S}I=\{J\backslash I|J\in S, J\geq I\}$ withthe order relation induced from . Define an open star of a simplex $I\in S$ as$def=\{J\in S|J\geq I\}$ . There is a natural surjective map of setsa subsetis$D_{I}:star_{S}^{o}Iarrow 1ink_{S}$. We call a simplex I admissible if
sending to
$S$

$|S|$

$S$

$J\backslash I\in S$

$S$

$I$

$J$

$J\backslash I$$S$$star_{S}^{o}$ $I$

$J$

$I$

$D_{I}$$J\backslash I$

injective.
Note that in a simplicial complex every simplex is admissible. One can view
admissibility as the property of being “locally a simplicial complex
along admisand
Let us define the connected sum of two simplicial posets
be admissible vertices, and suppose there
sible vertices. Let
andexists an isomorphism of posets $\xi:1ink_{S_{1}}i_{1}arrow 1ink_{S_{2}}i_{2}$ (thus an isomorphism ofopen stars, by admissibility). Consider a poset$S_{1}$$i_{1}\in S_{1}$(2.1)$S_{2}$$i_{2}\in S_{2}$$S_{1i_{1}}\#_{i_{2}}S_{2}^{d}=^{ef}(S_{1}\backslash star_{\mathring{S}_{1}}i_{1})\sqcup(S_{2}\backslash star_{S_{2}}^{o}i_{2})/\sim,$

whenever $I_{2}=\xi(I_{1})$ .
is identified withwhere
in a natural way. The
andThe order relation on
is induced fromposet
is simplicial. If
are simplicial spheres, then so is(this property may break for non-admissible vertices).
Definition 2.3. Let
be a pure simplicial poset of dimension $n-1$ . A map$\Lambda:Vert(S)arrow \mathbb{Z}^{n}/\pm is$ called a weighting if,
for every simplex $I\in S$ with vertices
is called a weighted
span . The pair
, . . . , , the vectors,...,
simplicial poset.
$I_{2}\in 1ink_{S_{2}}i_{2}\subset S_{2}$$I_{1}\in 1ink_{S_{1}}i_{1}\subset S_{1}$$S_{1}$$S_{1i_{1}}\#_{i_{2}}S_{2}$$S_{2}$$S_{1i_{1}}\#_{i_{2}}S_{2}$$S_{1},$ $S_{2}$$S_{1i_{1}}\# i_{2}S_{2}$$S$$i_{1}$$i_{n}$$\Lambda(i_{1})$Definition 2.4. Let
be admissible vertices of$(S_{1}, \Lambda_{1})$$S_{1},$ $S_{2}$$\Lambda(i_{n})$$\mathbb{Z}^{n}$$(S, \Lambda)$

be weighted simplicial posets. Let
andexists an isomorphism
that
there
such
$(S_{2}, \Lambda_{2})$$i_{1},$$i_{2}$$\xi:1ink_{S_{1}}i_{1}arrow$

118
QUASITORIC MANIFOLDS WHICH ARE NOT TORIC ORIGAMI. Thenpreserving weights:
induce the. The weighted simplicial poset
weight on the connected sum.
is called a weighted connected sum of
and$1ink_{S_{2}}i_{2}$$S_{2},$$\Lambda_{1},$$(\Lambda_{2}\circ\xi)|_{1inks_{1}^{i_{1}}}=\Lambda_{1}|_{1ink_{S_{1}}i_{1}}$$\Lambda$$\Lambda_{2}$$(S_{1i_{1}}\#_{i_{2}}$$S_{1i_{1}}\# i_{2}S_{2}$$(S_{2}, \Lambda_{2})$$(S_{1}, \Lambda_{1})$$\Lambda)$

Construction 2.5. Let
be a characteristic pair (see section 1). Let $K_{P}=$be the dual simplicial sphere to a simple polytope . Since there is a natural
correspondence Vert $(K_{P})=\mathcal{F}(P)$ we get the weighting. In particular, any Delzant polytope
This defines a weighted sphere, where $\nu(F)$ is the normal vector to $F\in \mathcal{F}(P)=$defines a weighted sphere$Vert(K_{P})$ modulo sign (construction 1.4).
$(P, \Lambda)$$P$$\partial P^{*}$$\Lambda:Vert(K_{P})arrow \mathbb{Z}^{n}/\pm.$$P$$(K_{P}, \Lambda)$$(K_{P}, \nu)$Construction 2.6. Let
be an origami template andbe thecorresponding toric origami manifold. Suppose that is a tree. The orbit space$Q=M_{O}/T^{n}$ is homeomorphic to an-dimensional disc. The face structure of, whose elements are faces ofdefines a poset
ordered by reversed inclusion(it is easy to show that such poset is simplicial). In particular, Vert $(S_{Q})=\mathcal{F}(Q)$ .
Normal vectors to facets of (construction 1.8) determine the characteristic func$\nu([F])=\nu(F)$ . Thus there is a weighted simplicial poset
tion
associated with a toric origami manifold$O=(\Gamma, \Psi_{V}, \Psi_{E})$$M_{O}$$\Gamma$

$Q$

$n$

$Q$

$S_{Q}$$Q$$\nu:\mathcal{F}(Q)arrow \mathbb{Z}^{n}/\pm,$$(S_{Q}, \nu)$$M_{O}.$

Construction 2.7. If is a tree, then the simplicial poset
is the connectedalong vertices, corresponding to fold facets:
sum of simplicial spheres$\Gamma$$S_{Q}$$K_{\Psi_{V}(v)}$(2.2)$S_{Q}\cong\# K_{\Psi_{V}(v)}\Gamma^{\cdot}$

Let us introduce a notation to make this precise. Let be an edge of , andbe its endpoint. Let
be the vertex ofcorresponding to the facet. Then (2.2) denotes the connected sum of all simplicial spheres
along vertices
for all edges $e=\{v, u\}$ of graph . This simultaneousconnected sum is well defined. Indeed, if $e_{1}\neq e_{2}\in E$ are two edges emanatingfrom $v\in V$ , then vertices
by pt.2 ofandare not adjacent in
Therefore,
open stars, which we
Definition 1.5.
andadmissible,
remove in (2.1), do not intersect. Also note that all vertices
aresince the spheres
are simplicial complexes.
Each spherecomes equipped with a weighting $v_{v}:Vert(K_{\Psi_{V}(v)})arrow$, sinceis Delzant. By pt.1 of Definition 1.5 these weightings agree
on identified links. Therefore we have an isomorphism of weighted spheres$\Gamma$$e$$v$$i_{v,e}$$K_{\Psi_{V}(v)}$$\Psi_{E}(e)\subset\Psi_{V}(v)$$i_{v,e},$$K_{\Psi_{V}(v)}$$\Gamma$$i_{u,e}$$i_{v,e_{1}}$$i_{v,e_{2}}$$star_{K_{\Psi_{V}(v)}}^{o}i_{v,e_{1}}$$K_{\Psi_{V}(v)}$$star_{K_{\Psi_{V}(v)}}^{o}i_{v,e_{2}}$$i_{v,e}$$K_{\Psi_{V}(v)}$$K_{\Psi_{V}(v)}$$\mathbb{Z}^{n}/\pm$$\Psi_{V}(v)$(2.3)$(S_{Q}, v)\cong\#(K_{\Psi_{V}(v)}, \nu_{v})\Gamma^{\cdot}$

3. PROOF OF THEOREM 2
Suppose that a quasitoric manifold
is equivariantly homeomorphic tothe origami manifold
is a tree. First, the orbit spaces should be$M_{(P,\Lambda)}$$M_{(\Gamma,\Psi_{V},\Psi_{E})},$$\Gamma$isomorphic as manifolds with corners: $P\cong Q=M_{O}/T^{n}$ . Second,
implies that stabilizers of the torus actions coincide for the corresponding
faces of orbit spaces. Thus characteristic functions on $P$ andtaking values inthe same. Hence, the weighted simplicial spheres
andare isomorphic. So far to prove Theorem 2 it is suficient to prove$M_{(P,\Lambda)}\cong T$$M_{O}$$Q$$\mathbb{Z}^{n}/\pm are$$(K_{P}, \Lambda)$$(S_{Q}, \nu)\cong$$\#_{\Gamma}(K_{\Psi_{V}(v)}, \nu)$Proposition 3.1. There exists a simple 3-dimensional polytope $P$ and a characteristic functionthat the dual weighted spherecannotbe represented as a connected sum, along a tree, of weighted spheres dual to Delzant
polytopes.
$\Lambda:\mathcal{F}(P)arrow \mathbb{Z}^{3}/\pm such$$(K_{P}, \Lambda)$

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

We proceed by steps. At first notice that any simplicia12-sphere, which is a
simplicial complex, is dual to some simple 3-polytope by Steinitz’s theorem (seee.g. [10]). Thus it is suﬃcient to prove the statement for weighted simplicial
complexes-spheres of dimension 2.

FIGURE 3. Connected sum of spheres along a tree

Construction 3.2. We introduce some notation in addition to that of construction 2.7, see Fig. 3. As before, let $\Gamma=(V, E)$ be a tree. Suppose that a simplicial
is associated with each node $v\in V$ , and for each edge $e\in E$ withsubject to the following
there is an admissible vertex
an endpoint
for any edge with endpoints
is isomorphic toconditions: (1)$u;(2)$ Vertices
for any two edges
are diﬀerent and not adjacent in
as inemanating from . Then we can form a connected sum along$v\inV$$K=\#_{\Gamma}S_{v}$subposet
simplicial. For eachconsider theconstruction 2.7:$(n-1)$ -sphere$S_{v}$$v\in V$$i_{v,e}\in S_{v}$$1ink_{S_{u}}i_{u,e}$$1ink_{S_{v}}i_{v,e}$$e$$S_{v}$$i_{v,e_{1}},$ $i_{v,e_{2}}$$v,$$\Gamma$$e_{1}\neq e_{2}$

$v$

(3.1)

$R_{v}=S_{v}\backslash _{e\in E}\sqcup_{v\in e}star_{S_{v}}^{o}i_{v,e}.$. By construction,
byThis subposet will be called a region. Denote
simplicial$e=\{v,u\}$$(n-2)$-dimensional
resulting.
if
Theis attached to$K$$C_{e}\cong
C_{v,e}=$. Sinceis admissible, the subposet
subposet of is denoted by
is a simplicial $(n-2)$ -sphere.
$e\in E$ , dividing the $(n-1)-$
We get a collection of $(n-2)$ -dimensional cycles$v\in V$ . If $e=\{v, u\}$ , thenshare a common
andsphere $K$ into regionslies. Note that cycles
borderare mutually ordered, meaning that each(aspointsat one side of any other cycle. Though the cycles may have common
schematically shown on Fig. 3) and even coincide (in this case the region between
them coincides with both of them).
On the other hand, any collection of mutually ordered $(n-2)$ -dimensional spherical cycles in $K$ defines the representation of $K$ as a connected sum of smaller
will be called a slicing.
simplicial spheres. A representation
Define the width of a slicing
to be the maximal number of vertices in its
regions:$1ink_{S_{v}}i_{v,e}$$C_{v,e}$$C_{u,e}$$C_{v,e}$$C_{e}$$i_{v,e}$$1ink_{S_{v}}i_{v,e}$$C_{e},$$R_{v}$$R_{v},$$C_{e}$$R_{u}$$C_{e}$$C_{e}$$K=\#{}_{\Gamma} S_{v}$$\Theta$

(3.2)

wid

$( \Theta)^{d}=^{ef}\max\{|$Vert $(R_{v})||v\in V\}.$Define the fatness of a sphere $K$ as the minimal width of all its possible slicings:(3.3)$ft(K)def=\min$ $wid(\Theta)|\Theta$ is a slicing of $K$

{

}.

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QUASITORIC MANIFOLDS WHICH ARE NOT TORIC ORIGAMI

The essential step in the proof of Proposition 3.1 is the following.
Lemma 3.3. Let $K$ be an $(n-1)$ -dimensional simplicial sphere and $\Lambda:Vert(K)arrow$weighting. Let denote the number of diﬀerent values of this weighting,$r=|\Lambda(Vert(K))|$ . Suppose that $ft(K)>2r$ . Thencannot be represented asa connected sum, along a tree, of simplicial spheres dual to Delzant polytopes.
$\mathbb{Z}^{n}/\pm a$$r$$(K, \Lambda)$, whereare$K$
slicing
.
width
a
of
The
forgetweights,
thisdefinestheDelzant polytopes. If weof every slicing of $K$ is greater than $2r$ by the definition of fatness. In particular,

Proof. Assume the converse. Then$\Psi_{V}(v)$$(K, \Lambda)\cong\#_{\Gamma}(K_{\Psi_{V}(v)}, \nu_{v})$$\Theta$. Thus there exists a node
The region
is a subcomplex ofofsuch that $|Vert(R_{v})|>2r.$. The restriction of to the subset
Vert
coincides with the restriction of $\nu:Vert(K_{\Psi_{V}(v)})arrow \mathbb{Z}^{n}/\pm toVert(R_{v})$ .
Recall, thatis the outward normal vector to the facet$wid(\Theta)>2r$$R_{v}$$v$$\Gamma$$\Lambda$$K_{\Psi_{V}(v)}$$(R_{v})$$F\in \mathcal{F}(\Psi_{V}(v))=$$\tilde{\nu}(F)\in \mathbb{Z}^{n}$

its class modulo sign. The outward normal
, and
Vert
Vert R $)|=$
vectors to facets of a convex polytope are mutually distinct, thus$|Vert(R_{v})|$ and, therefore, $|\nu(Vert(R_{v}))|\geq|Vert(R_{v})|/2$ .
Thus $|\Lambda(Vert(R_{v}))|=$$|\nu(Vert(R_{v}))|>r$ ,the contradiction, since is the total number of values of$\nu(F)\in \mathbb{Z}^{n}/\pm is$$(K_{\Psi_{V}(v)})$$|\tilde{\nu}($$\Lambda.$$r$–$\square$So far we may find counterexamples to origami realizability among polytopes,
which are-colored with a small number of colors, but whose dual simplicial
spheres have large fatness. Of course such examples do not exist for $n=2$
this would contradict Theorem 1. The existence of 2-spheres satisfying conditionsof Lemma 3.3 is thus our next and primary goal. At first, we prove that any 2sphere admits a characteristic function with few values; then construct 2-spheres ofarbitrarily large fatness.
$\mathbb{Z}^{n}$–Lemma 3.4. Any simplicial 2-sphere $K$ admits a weighting $\Lambda:Vert(K)arrow \mathbb{Z}^{3}/\pm$
such that $|\Lambda(Vert(K))|\leq 4.$

Proof. Four color theorem states that there exists a proper vertex-coloring: Vert $(K)arrow$. Now replace colors by integral vectors, $\alpha_{2}\mapsto(0,1,0)$ ,$\alpha_{3}\mapsto(0,0,1)$ , $\alpha_{4}\mapsto(1,1,1)$ . This gives the required characteristic function.
$\{\alpha_{1}, \alpha_{2}, \alpha_{3}, \alpha_{4}\}$$\alpha_{1}\mapsto(1,0,0)$$\square$Proposition 3.5. For any $N>0$ there exists a simplicial 2-sphere $K$ such that
$ft(K)>N.$

Proof
Construction 3.6. Let $K$ be a simplicia12-comp1ex. Define (piecewise Riemannian) metric and measure on $|K|$ in such a way that each triangle $|I|\subset|K|$ becomes an equilateral Euclidian triangle with the standard metric and edge length 1.
Thus the area of each triangle isLetdenote the length of a piecewise smooth curve in $|K|$ . If $C\subset K$ is a
closed 1-dimensional cycle (simplicial subcomplex), then, obviously,
$a$

$g$

$\mu$

$\sqrt{3}/4.$$L(\gamma)$$\gamma$(3.4)$L(|C|)=|Vert(C)|.$A cycle divides $K$ into two subcomplexes $K+andK_{-}$ , each homeomorphic to$K$
Let us estimate the number of vertices
a closed 2-disc (we suppose
similar).
-denote the number ofisLetin terms of its area (in.
. By the definition of measure,vertices, edges and triangles in(by)
countingof(Eulerandcharacteristic
We have, where is an edge and is a triangle). Therefore,
pairs
$C$

$C\subset K_{+},$$K_{-}$$K_{+}$$\mathcal{V}_{-},$$\mathcal{E}_{-},$$K_{-}$$\mathcal{T}_{-}=\frac{4}{\sqrt{3}}\mu(|K_{-}|)$$K_{-}$$\mathcal{V}_{-}-\mathcal{E}_{-}+\mathcal{T}_{-}=1$$e\subset t$

(3.5)

$e$

$\mathcal{T}$$t$$\mathcal{V}_{-}\leq\frac{8}{\sqrt{3}}\mu(|K_{-}|)$.

$\mathcal{E}_{-}<3\mathcal{T}_{-}$121
A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG

be a 2-dimensional round sphere of radius , with the standard metricwithout self-intersections
and measure . A piecewise smooth closed curve. The isoperimetric inequality on a sphere (seeinto two regions
divides, whereis the length ofe.g. [8, Ch.4]) has the form
(otherwise
considermay. Sinceassumethat
weinstead), thus

Let

$R$

$\mathbb{S}_{R}$$g_{s}$$\gamma\subset \mathbb{S}_{R}$$\mu_{s}$$A_{-}$$A_{+},$$\mathbb{S}_{R}$$R^{2}L_{s}(\gamma)^{2}\geq\mu_{s}(A_{+})\mu_{s}(A_{-})$$L_{s}(\gamma)$$\mu_{s}(A_{+})\geq 2\pi R^{2}$$\mu_{s}(\mathbb{S}_{R})=4\pi R^{2}$$\gamma$$A_{-}$(3.6)$\mu_{\mathcal{S}}(A_{-})\leq\frac{L_{s}(\gamma)^{2}}{2\pi}.$

Let $K$ be a 2-dimensional simplicial sphere and $R,$
numbers. Suppose there exists a bijective piecewise smooth mapthat$c_{1},$ $c_{2},$ $c_{3},$(3.7)$c_{1}L(\gamma)\leq L_{s}(f(\gamma))\leq c_{2}L(\gamma)$(3.8)$c_{3}\mu(\Omega)\leq\mu_{s}(f(\Omega))\leq c_{4}\mu(\Omega)$be positive real
such

$f:|K|arrow \mathbb{S}_{R}$,,and measurable set
for each piecewise smooth curve
will be called Lipschitz constants of the map $f.$$\gamma\subset|K|$$c_{1},$$c_{4}$$\Omega\subset|K|$. Numbers$c_{2},$ $c_{3},$ $c_{4}$Lemma 3.7. In the above setting, suppose the cycle $C\subset K$ contains at most $N$
vertices.
orcontains at mostvertices. Then either$\underline{4N^{2}c^{2}}$$K_{-}$$K_{+}$$\sqrt{3}\pi c_{3}$let $f(|K_{-}|)$ be the one with the
Proof. Among two regions $f(|K_{-}|)$ ,smaller area. Combine (3.4), (3.5), (3.6), (3.7), and (3.8):$f(|K+|)\subset \mathbb{S}_{R}$(3.9)$V_{-} \leq\frac{8}{\sqrt{3}}\mu(|K_{-}|)\leq\frac{8\mu_{s}(f(|K_{-}|))}{\sqrt{3}c_{3}}\leq\frac{8L_{s}(f(|C|))^{2}}{2\sqrt{3}\pi c_{3}}\leq\frac{4N^{2}c_{2}^{2}}{\sqrt{3}\pi c_{3}}.$$\square$Lemma 3.8. If is a slicing $K=\#_{\Gamma}S_{v}$ and $wid(\Theta)\leq N$ , then $\deg v\leq 2(N-2)$
for any node of$\Theta$$\Gamma.$$v$

is obtainedby removing open stars which correspond to the edges offrom a sphereemanating from . The complex
itself can be considered as a plane graph.
respectively. By thevertices,edges
theand faces by
numbers of its
Denote, and. We also havedefinition of the width, we have
. Notice that each removed(each region has at least 3 edges). Thus,, therefore, $d\leq \mathcal{R}\leq 2(\mathcal{V}-2)\leq 2(N-2)$ .
open star represents a face of graph

Proof. Denote the degree of

$v$

by

$d$

. By construction, the region$R_{v}$$d$$S_{v}$$\Gamma$$R_{v}$$v$$\mathcal{V},$$\mathcal{E},$$\mathcal{R}$$\mathcal{V}-\mathcal{E}+\mathcal{R}=2$$\mathcal{V}\leq N$$2\mathcal{E}\geq 3\mathcal{R}$$\mathcal{V}\geq 2+\frac{1}{2}\mathcal{R}$$\square$$R_{v}$

Lemma 3.9. Let $K$ be a 2-dimensional simplicial sphere endowed with the map
to the round sphere, satisfying Lipschitz bounds (3.7) and (3.8). For a natural
and $B=2(N-2)$ . If $| Vert(K)|>\max(AB+N, 2A)$ ,number $N$ set
then $ft(K)>N.$$f$$A= \frac{4N^{2}c^{2}}{\sqrt{3}\pi c_{3}}$Proof. Assume the contrary: $ft(K)\leq N$ . Then there is a slicing$K=\#_{\Gamma}S_{v}$invertices. Consequently, any cycle
which every region
has at mostdivides $K$ into two parts, one
has at most $N$ vertices. By Lemma 3.7, the cycleof which hasvertices. Since $|Vert(K)|>2A$ , the other part has $>A$ vertices.
Assign a direction to each edge of in such a way that points from the larger
component ofto the smaller, where the siz means the number of vertices,
is a tree, therefore there exists a source , i.e. a node from which all adja,...,
the connectedcent edges emanate. Let denote the degree of andcomponents of. By Lemma 3.8 we have $d\leq B$ . By the constructionof the directions of edges, $|Vert(\sqcup_{\Gamma_{i}}R_{v})|\leq A$ for each . Thus $|Vert(K)|<$$| Vert(R_{u})|+\sum_{i=1}^{d}|Vert(\sqcup_{\Gamma_{i}}R_{v})|\leq N+AB$
the contradiction.
$N$$R_{v}$$C_{e},$$e\in E$

$C_{e}$$\leq A$$e$$\Gamma$

$e$

$e’$

$K\backslash C_{e}$$\Gamma$

$u$

$d$

$u$

$\Gamma_{1}$$\Gamma_{d}$$\Gamma\backslash u$$\Gamma_{i}$$\square$–

122
QUASITORIC MANIFOLDS WHICH ARE NOT TORIC ORIGAMI

Lemma 3.10. For any $N>0$ there exists a 2-dimensional simplicial sphere $K$
such that:(1) There exists a piecewise smooth map $f:|K|arrow \mathbb{S}_{R}$ satisfying Lipschitz$R>0$bounds (3.7) and (3.8) for some constants
and are defined in Lemma 3.9.
(2) $| Vert(K)|>\max(AB+N, 2A)$ , where$c_{1},$ $c_{2},$ $c_{3},$ $c_{4},$

$A$

$B$

Proof. Start with the boundary of a regular tetrahedron with edge length 1: $L=$
. The projection from the center ofis obvito the circumsphere$c_{4}>0$
triangleofeach.
Now subdivide
ously Lipschitz for some constants
into smaller regular triangles as shown on Fig. 4.
$\partial\Delta^{3}$$c_{1},$$c_{2},$$L$$f:Larrow \mathbb{S}_{R}$$c_{3},$$q^{2}$$|L|$

FIGURE 4. Subdivision of a regular triangle.

. As a space with metric and measureThis results in a simplicial complex
with a linear scaling factor . Thus there exists a map
is homothetic towith the same Lipschitz constants as . The number of vertices$|Vert(L_{(q)})|$ can be made arbitrarily large.
$L_{(q)}$$|L|$$|L_{(q)}|$

$q$

$f$

$f_{(q)}:|L_{(q)}|arrow \mathbb{S}_{qR}$$\square$Lemmas 3.10 and 3.9 conclude the proof of Proposition 3.5.

$\square$Remark 3.11. Actually, in the proof of Lemma 3.10 we could have started with any, find Lipschitz
simplicial sphere , take any piecewise smooth map$c_{3}>0$ (they exist by the standard calculus arguments), and thenconstants
apply the same subdivision procedure. We used the boundary of a regular simplex,
because in this case Lipschitz map is constructed easily and admits 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 we. Thus $ma[x(AB+N, 2A)\approx 15251.14$ . Subhave Lipschitz constants $c_{2}=3,$small triangles
divide each triangle in the boundary of a regular tetrahedron inwhere $q\geq 88$ . This gives a simplicial sphere $K$ with at least 15490 vertices and. Thus $ft(K)>8$ . Now take the dual simple
the same Lipschitz constants asof $K$ , consider any proper facet-coloring in four colors and assign apolytopecharacteristic function , as described in Lemma 3.4. This gives a characteristic, whose corresponding quasitoric manifold is not toric origami.
pair
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.
$L$$f:|L|arrow \mathbb{S}_{R}$$c_{2},$$c_{3}= \frac{1}{3}$$q^{2}$$\partial\Delta^{3}$$P$$\Lambda$$(P, \Lambda)$

REFERENCES[1] A. Ayzenberg, M. Masuda, S. Park, H. Zeng, Cohomology of tomc omgami manifolds withacyclic proper faces, preprint arXiv:1407.0764.
[2] V. Buchstaber and T. Panov, Torzc Topology, preprint arXiv:1210.2368.
[3] A. Cannas da Silva, V. Guillemin and A. R. Pires, Symplectic Origami, IMRN 2011 (2011),, preprint arXiv:0909.4065.
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A. AYZENBERG, M. MASUDA, S. PARK, AND H. ZENG

[4] M. Davis, T. Januszkiewicz, Convex polytopes, Coxeter orbifolds and torus actions, Duke
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[6] T. Holm and A. R. Pires, The topology of toric oregami manifolds, Math. Research Letters 20(2013), 885-906, preprint arXiv:1211.6435.
[7] M. Masuda and S. Park, Torec $or\iota gami$ manifolds and multi-fans, to appear in Proc. ofSteklov Math. Institute dedicated to Victor Buchstaber’s 70th birthday, arXiv:1305.6347.
[8] R. Osserman, The isope mmetrec inequality, Bull. of AMS, Vol.84, N.6, 1978.
[9] T. Yoshida, Local torus actions modeled on the standard representation, Advances in Mathematics 227 (2011), pp. 1914-1955.
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DEPARTMENT OF MATHEMATICS, OSAKA CITY UNIVERSITY, SUMIYOSH1-KU, OSAKA 5588585, JAPAN.
$E$-mail address: ayzenberga@gmail.com

DEPARTMENT 0F MATHEMATICS, OSAKA CITY UNIVERSITY, SUMIYOSH1-KU, OSAKA 5588585, JAPAN.
-mail address: masuda@sci.osaka cu. ac. jp$E$

DIVISION 0F MATHEMATICAL MODELS, NATIONAL INSTITUTE F0R MATHEMATICAL SCIENCES,
463-1 JEONMIN-D0NG, YUSEONG-GU, DAEJEON 305-811, KOREA$E$-mail address: seonjeong1124@nims.re.kr

DEPARTMENT 0F MATHEMATICS, OSAKA CITY UNIVERSITY, SUMIYOSH1-KU, OSAKA 5588585, JAPAN.
-mail address: zenghaozhi@icloud.com$E$