数理解析研究所講究録
第 1922 巻 2014 年 55-59

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COHOMOLOGY OF TORIC ORIGAMI MANIFOLDSHAOZHI ZENG

DEPARTMENT OF MATHEMATICS, OSAKA CITY UNIVERSITY
ABSTRACT. Toric origami manifolds, introduced in [2], are generalizations ofsymplectic toric manifolds. In this note, we study the topology of orientable
toric origami manifolds with acyclic proper faces. This note is based on thejoint work with Anton Ayzenberg, Mikiya Masuda and Seonjeong Park, andmore details can be found in our paper [1].

1. TORIC ORIGAMI MANIFOLDSIn this section we recall the definitions and properties of toric origami manifolds
and origami templates. Details can be found in [2], [8] or [5].
A folded symplectic form on a $2n$-dimensional manifold $M$ is a closed 2-form
whose top power
vanishes transversally on a subset $W$ and whose restriction$W$
to points in
has maximal rank. Then $W$ is a codimension-one submanifold of$M$ and is called the fold.
is a genuine symplectic form. The
If $W$ is empty,pair
is called a folded symplectic manifold. Since the restriction of to$W$ has maximal rank, it has a one-dimensional kernel at each point of $W$ .
Thisdetermines a line field on $W$ called the null foliation. If the null foliation is thevertical bundle of some principal -fibration $Warrow X$ over a compact base $X$ , thenthe folded symplectic form is called an origami form and the pair $(M,\omega)$ is calledisan origami manifold. The action of a torus on an origami manifold
Hamiltonian if it admits a moment map $\mu:Marrow t^{*}$ to the dual Lie algebra of thetorus, which satisfies the conditions: (1) is equivariant with respect to the given
action of on $M$ and the coadjoint action of on the vector space (this action istrivial for the torus); (2) collects Hamiltonian functions, that is,for any $V\in t$ , where $V\#$ is the vector field on $M$ generated by $V.$$\omega$$\omega^{n}$$\omega$$(M, \omega)$$\omega$$S^{1}$$\omega$$T$$(M, \omega)$

$\mu$

$T$

$T$

$t^{*}$$d\langle\mu,$$\mu$$V\rangle=\iota_{V\#}\omega$

Definition. A toric origami manifold $(M, \omega, T, \mu)$ , abbreviated as $M$ , is a compact
equipped with an eﬀective Hamiltonian action
connected origami manifoldof a torus with $\dim T=\frac{1}{2}\dim M$ and with a choice of a corresponding moment
map$(M, \omega)$$T$$\mu.$

When the fold $W$ is empty, a toric origami manifold is a symplectic toric manifold. A theorem of Delzant [3] says that symplectic toric manifolds are classified
by their moment images called Delzant polytopes. Recall that a Delzant polytope
in “ is a simple convex polytope, whose normal fan is smooth (with respect to). In other words, all normal vectors to facets of havesome given lattice,...,coordinates,
and, whenever facets
meet in a vertex of , therational. Let
primitive normal vectors,...,
form a basis of the latticebe the(w.r.t. a given lattice) anddenote the set of all Delzant polytopes in
set of all their facets.
The moment data of a toric origami manifold can be encoded into an origami, wheretemplateis a connected graph (loops and multiple edges are allowed) with thevertex set and edge set ;$\mathbb{R}$$P$$\mathbb{Z}^{n}\subset \mathbb{R}^{n}$$F_{1}$$\nu(F_{n})$$\nu(F_{1})$$\mathbb{R}^{n}$$(G, \Psi_{V}, \Psi_{E})$$\bullet$

$G$

$V$

$E$

$P$

$F_{n}$$\mathbb{Z}^{n}$$\mathcal{F}_{n}$$\mathcal{D}_{n}$56$\bullet\Psi_{V}:Varrow \mathcal{D}_{n\rangle}$$\bullet\Psi_{E}:Earrow \mathcal{F}_{n}$;subject to the following conditions:$v_{2}\in V$ , thenis a facet
If $e\in E$ is an edge of with endpoints, and these polytopes coincide near
andof both polytopes
inof(this means there exists an open neighborhood$U\cap\Psi_{V}(v_{1})=U\cap\Psi_{V}(v_{2})$).
such thatand
If $e_{2}\in E$ are two edges of adjacent to $v\in V$ , then.
are disjoint facets offor $e\in E$ are called the fold facets of the origami
The facets of the form
template.
$G$$\bullet$$\Psi_{E}(e)$$v_{1},$$\Psi_{V}(v_{2})$$\Psi_{V}(v_{1})$$U$$\Psi_{E}(e)$$\bullet$$G$$e_{1},$$\Psi_{E}(e)$$\Psi_{E}(e_{1})$$\mathbb{R}^{n}$$\Psi_{E}(e_{2})$$\Psi(v)$$\Psi_{E}(e)$Example. The following picture is an example of origami templates.

FIGURE 1. The origami template with two copies of isosceles right triangles

The following is a generalization of the theorem by Delzant to toric origami

manifolds.
Theorem 1.1 ([2]). Assigning the moment data of a toric origami manifold induces
one to one correspondence{ origami templates}{toric origami manifolds}$a$$\langlerightarrow$up to equivariant origami symplectomorphism on the left-hand side, and aﬃneequivalence on the right-hand side.

Denote byunion$|(G, \Psi_{V}, \Psi_{E})|$the topological space constructed from the disjoint

by identifying facets $\Psi_{E}(e)\subset\Psi_{V}(v_{1})$ and $\Psi_{E}(e)\subset\Psi_{V}(v_{2})$for any edge $e\in E$ with endpoints
An origami template $(G, \Psi_{V}, \Psi_{E})$ is called co\"orientable if the graph $G$ has noloops (this means all edges have diﬀerent endpoints). Then the corresponding
toric origami manifold is also called co\"orientable. If $M$ is orientable, then $M$ isco\"orientable [5]. If $M$ is co\"orientable, then the action of
on $M$ is locally standard[5, lemma 5.1].
Let $(G, \Psi_{V}, \Psi_{E})$ be an origami template and $M$ the associated toric origami
manifold which is supposed to be orientable in the following. The topological space$|(G, \Psi_{V}, \Psi_{E})|$ is a manifold with corners with the face structure induced from the, and $|(G, \Psi_{V}, \Psi_{E})|$ is homeomorphic to $M/T$
face structures on polytopes
as a manifold with corners. The space $|(G, \Psi_{V}, \Psi_{E})|$ has the same homotopy typeas the graph $G$ , thus $M/T\cong|(G, \Psi_{V}, \Psi_{E})|$ is either contractible or homotopy
equivalent to a wedge of circles.
Under the correspondence of Theorem 1.1 the fold facets of the origami template
correspond to the connected components of the fold $W$ of $M$ . If $F=\Psi_{E}(e)$ is a fold$\sqcup_{v\in V}\Psi_{V}(v)$$v_{1},$$v_{2}.$$T^{n}$$\Psi_{V}(v)$57, then the corresponding component $Z=\mu^{-1}(F)$facet of the template$W\subset
M$of the fold
is a principal -bundle over a compact space . The space $B$
is $a(2n-2)$ -dimensional symplectic toric manifold corresponding to the Delzant$(G, \Psi_{V}, \Psi_{E})$$B$$S^{1}$

polytope $F.$

2. BETTI NUMBERS OF TORIC ORIGAMI MANIFOLDSFYom the classifying Theorem 1.1, a natural question is how to describe thecohomology ring and -equivariant cohomology ring of a toric origami manifold $M$
in terms of its corresponding origami template. If $M$ is simply connected, i.e. theassociated graph is a tree, this question is answered by Masuda and Panov in [7]and Holm and Pires in [5]. However, if $M$ is non-simply connected, this question isstill open in general, even for the betti numbers unless the case where $\dim M=4$
is solved by Holm and Pires in [6]. In this section, we will give an explicit formulafor the betti numbers of $M$ when $M$ is orientable and every proper face of $M/T$ isacyclic. Our first main result is the following.
$T$$G$

Theorem 2.1. Let $M$ be an orientable toric origami manifold of dimension $2n$$(n\geq 2)$ such that every proper
face of $M/T$ is acyclic. Then$b_{2i+1}(M)=0$

for $1\leq i\leq n-2,$$b_{1}(M)=b_{2n-1}(M)=b_{1}(M/T)$ .

Moreover, $H^{*}(M)$ is torsion free.
$\mathcal{P}$We can describe $b_{2i}(M)$ in terms of the face numbers of $M/T$ and $b_{1}(M)$ . Let
be the simplicial poset dual to $\partial(M/T)$ . As usual, we define$f_{i}=$$=$and the -vector$h$(2.1)

the number of $(n-1-i)$ -faces of $M/T$
the number of -simplices infor $i=0$ , 1, . . . , $n-1$$i$$(h_{0}, h_{1}, \ldots, h_{n})$$\mathcal{P}$by$\sum_{i=0}^{n}h_{i}t^{n-i}=(t-1)^{n}+\sum_{i=0}^{n-1}f_{i}(t-1)^{n-1-i}.$

Theorem 2.2. Let $M$ be an orientable toric origami manifold of dimension $2n$such that every proper face of $M/T$ is acyclic. Let
be the j-th Betti number of$M$ andbe the -vector of $M/T$ . Then$b_{j}$$(h_{0}, h_{1}, \ldots, h_{n})$$h$$\sum_{i=0}^{n}b_{2i}t^{i}=\sum_{i=0}^{n}h_{i}t^{i}+b_{1}(1+t^{n}-(1-t)^{n})$,in other words, $b_{0}=h_{0}=1$ and$b_{2i}=h_{i}-(-1)^{i}(\begin{array}{l}ni\end{array})b_{1}$for $1\leq i\leq n-1,$$b_{2n}=h_{n}+(1-(-1)^{n})b_{1}.$Example. Let $M$ be the 4–dimensional toric origami manifold corresponding tothe origami template shown on fig.2 (Example 3.15 of [2]). It is easy to check that
$M$ satisfies the condition of our theorems from fig.2. The-vector $(f_{0}, f_{1})=(8,8)$ ,so the -vector $(h_{0}, h_{1}, h_{2})=(1,6,1)$ . Then applying Theorem 2.1 and Theorem
2.2, we have $b_{0}=b_{4}=1,$ $b_{1}=b_{3}=1$ and $b_{2}=8.$$f$$h$

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FIGURE 2. The origami template with four polygons

3. TOWARDS THE RING STRUCTURE
A torus manifold $M$ of dimension $2n$ is an orientable connected closed smoothmanifold with an eﬀective smooth action of an -dimensional torus having a fixedpoint ([4]). An orientable toric origami manifold with acyclic proper faces in theorbit space has a fixed point, so it is a torus manifold. The action of on $M$ iscalled locally standard if every point of $M$ has a -invariant open neighborhoodequivariantly diﬀeomorphic to a -invariant open set of a faithful representation
space of . Then the orbit space $M/T$ is a nice manifold with corners. The torus
action on an orientable toric origami manifold is locally standard. In this section,
we study the cohomology ring structure of an orientable toric origami manifoldwith acyclic proper faces of the orbit space.
Let be the poset dual to the face poset of $M/T$ as before.
$T$

$n$

$T$

$T$

$T$

$T$

$\mathcal{P}$Proposition 3.1. Let $M$ be a locally standard torus manifold such that every proper
face of $M/T$ is acyclic, and the free part of the action gives a trivial principal bundle$M^{o}arrow M^{o}/T$ . Thenas graded rings.
$H_{T}^{*}(M)\cong \mathbb{Z}[\mathcal{P}]\oplus\tilde{H}^{*}(M/T)$

Let $\pi:ET\cross\tau^{M}arrow BT$ be the projection. Since $\pi^{*}(H^{2}(BT))$ maps to zeroinduces a graded ringby the restriction homomorphism $\iota^{*}:H_{T}^{*}(M)arrow H^{*}(M)$ ,homomorphism$\iota^{*}$(3.1)$arrow\iota$: $H_{T}^{*}(M)/(\pi^{*}(H^{2}(BT)))arrow H^{*}(M)$ .

Proposition 3.2. Let $M$ be an orientable toric origami manifold of dimension $2n$(3.1) is an isomorphism
such that every proper face of $M/T$ is acyclic, thendegreeexcept in degrees 2, 4 and $2n-1$ . Moreover, the rank of the cokernel of.
in degree 4 is
2 is $nb_{1}(M)$ and the rank of the kernel$\iota^{arrow}in$$\iota^{arrow}in$$(\begin{array}{l}n2\end{array})b_{1}(M)$$ofarrow\iota$

Example. Let $M$ be the toric origami manifold corresponding to the origami tem, 1 and theplate shown on fig.2. Topologically $M/T$ is homeomorphic toboundary of $M/T$ as a manifold with corners consists of two boundaries of -gons.
withThe multi-fan of $M$ is the union of two copies of the fan ofare primitive edge vectors in the fan of
the product torus action. Indeed, ifwhich spans a 2-dimensional cone, then the other primitive edge vectors,...,
in the multi-fan of $M$ are$S^{1}\cross[0$$]$$\mathbb{C}P^{1}\cross \mathbb{C}P^{1}$$v_{1},$ $v_{2}$$\mathbb{C}P^{1}\cross \mathbb{C}P^{1}$$v_{3}$$v_{8}$$v_{3}=-v_{1},$$v_{4}=-v_{2}$,and$v_{i}=v_{i-4}$

for $i=5$ , . . . , 8

and the 2-dimensional cones in the multi-fan are$\angle v_{1}v_{2}, \angle v_{2}v_{3}, \angle v_{3}v_{4}, \angle v_{4}v_{1},$$\angle v_{5}v_{6}, \angle v_{6}v_{7}, \angle v_{7}v_{8}, \angle v_{8}v_{5},$

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where

$\angle vv’$

(3.2)

denotes the 2-dimensional cone spanned by vectors$\tau_{i}\tau_{j}=0$if$v_{i},$$v_{j}$

$v,$

$v’$

. Note that

do not span a 2-dimensional cone.

We have(3.3)
Let$v_{1}^{*},$$\pi^{*}(u)=\sum_{i=1}^{8}\langle u,$$v_{2}^{*}$(3.4)

for any $u\in H^{2}(BT)$ .

$v_{i}\rangle\tau_{i}$. Takingbe the dual basis of$v_{1},$ $v_{2}$$\tau_{1}+\tau_{5}=\tau_{3}+\tau_{7},$$\tau_{2}+\tau_{6}=\tau_{4}+\tau_{8}$Since we applied (3.3) for the basis
new linear relation among$v_{1}^{*},$$v_{2}^{*}$$u=v_{1}^{*}$

or$v_{2}^{*}$, we see thatin $H_{T}^{*}(M)/(\pi^{*}(H^{2}(BT)))$ .

of $H^{2}(BT)$ , there is no other essentially$\tau_{i}’ s.$Now, multiply the equations (3.4) by$\tau_{i}^{2}=0$for any$\tau_{i}$and use (3.2). Then we obtain$i,$$(\mu_{1}:=)\tau_{1}\tau_{2}=\tau_{2}\tau_{3}=\tau_{3}\tau_{4}=\tau_{4}\tau_{1},$$(\mu_{2}:=)\tau_{5}\tau_{6}=\tau_{6}\tau_{7}=\tau_{7}\tau_{8}=\tau_{8}\tau_{5}$in $H_{T}^{*}(M)/(\pi^{*}(H^{2}(BT)))$ .

Our argument shows that these together with (3.2) are the only degree two relations
among ’s in $H_{T}^{*}(M)/(\pi^{*}(H^{2}(BT)))$ . The kernel of$\tau_{i}$$\overline{\iota}^{*}:H_{T}^{even}(M;\mathbb{Q})/(\pi^{*}(H^{2}(BT;\mathbb{Q} arrow H^{even}(M;\mathbb{Q})$in degree 4 is spanned by$\mu_{1}-\mu_{2}.$REFERENCES[1] A. Ayzenberg, M. $Masuda_{\}}$ S. Park and H. Zeng, Cohomology of $tor\iota cor\iota gami$ manifolds withacyclic proper faces, arXiv: 1407.0764.
[2] A. Cannas da Silva, V. Guillemin and A. R. Pires, Symplectic Origami, IMRN 2011 (2011),4252-4293, arXiv:0909.4065.
odiques et image convex de l’application moment, Bull. Soc.
[3] T. Delzant, Hamiltoniens
Math. France 116 (1988), 315-339.
[4] A. Hattori and M. Masuda, Theory of multi-fans, Osaka J. Math. 40 (2003), 1-68.
[5] T. Holm and A. R. Pires, The topology of toric origami manifolds, Math. Research Letters
20 (2013), S85-906, arXiv: 1211.6435.
[6] T. Holm and A. R. Pires, The fundamental group and Betti numbers of tomc $or\iota gami$ manifolds, arXiv: 1407.4737.
[7] M. Masuda and T. Panov, On the cohomology of torus manifolds, Osaka J. Math. 43 (2006),$per\iota$711-746.
[8] M. Masuda and S. Park, Tomc origami manifolds and multi-fans, to appear in Proc. ofSteklov Math. Institute dedicated to Victor Buchstaber’s 70th birthday, arXiv:1305.6347.