quasitoric manifolds and toric origami manifoldsmasuda/toric2014_osaka/s. park(slide).pdf ·...

Quasitoric manifolds and toric origami manifolds

Seonjeong Park

Joint work with Mikiya Masuda(OCU)In progress work with M. Masuda & Z. Haozhi

Division of Mathematical Models,National Institute for Mathematical Sciences, South Korea

2014 Toric Topology in Osaka, January 21–24, 2014Osaka City University

Seonjeong Park (NIMS) Quasitoric and toric origami 1 / 34

Table of contents

1 Quasitoric manifolds

2 Toric origami manifolds

3 Relationship between quasitoric manifolds and toric origami manifolds

Table of contents

Quasitoric manifolds

A quasitoric manifold is a closed smooth manifold M of dim. 2n with asmooth Tn action such that

the action of Tn on M is locally standard;

the orbit space M/Tn can be identified with a simple polytope.

Example

1 projective non-singular toric varieties

2 CP 2#CP 2

Non-example

For n > 1, Tn-action on S2n ⊆ Cn ⊕ R given by

(t1, . . . , tn) · (z1, . . . , zn, r) = (t1z1, . . . , tnzn, r)

is locally standard, but it cannot give rise to a quasitoric manifold.

Example

2 CP 2#CP 2

Non-example

(t1, . . . , tn) · (z1, . . . , zn, r) = (t1z1, . . . , tnzn, r)

Example

2 CP 2#CP 2

Non-example

(t1, . . . , tn) · (z1, . . . , zn, r) = (t1z1, . . . , tnzn, r)

Properties of quasitoric manifolds

1 There is a one-to-one correspondence

quasitoric manifolds M 1:1←→ characteristic pairs (P,λ).

2 Every omniorientation of a quasitoric manifold M determines a stablyalmost complex structure.

CP 2#CP 2

(−1−1

(0−1

) CP 2#CP 2

Properties of quasitoric manifolds

1 There is a one-to-one correspondence

quasitoric manifolds M 1:1←→ characteristic pairs (P,λ).

2 Every omniorientation of a quasitoric manifold M determines a stablyalmost complex structure.

CP 2#CP 2

(−1−1

(0−1

) CP 2#CP 2

Symplectic manifolds

A symplectic manifold M is a manifold equipped with a symplectic formω ∈ Ω2(M) that is closed (dω = 0) and non-degenerate.

Example

The unit sphere S2 in R3 is a symplecticmanifold.

ωv(X,Y ) = v · (X × Y ) = det(v,X, Y )

Non-example

1 For n > 1, S2n cannot be a symplectic manifold.

2 A quasitoric manifold CP 2#CP 2 cannot admit a symplectic form.

Example

ωv(X,Y ) = v · (X × Y ) = det(v,X, Y )

Non-example

Example

ωv(X,Y ) = v · (X × Y ) = det(v,X, Y )

Non-example

Folded symplectic manifolds

A folded symplectic form on a 2n-dim’l manifold M is a closed 2-form ω

ωn vanishes transversally on a submanifold Z,

ωn|Z has maximal rank.

The submanifold Z is called the folding hypersurface or fold.

Example

1 (R2d, x1dx1 ∧ dy1 +∑d

i=2 dxi ∧ dyi), Z = x1 = 0.2 (S2n ⊂ Cn ⊕ R, ωCn ⊕ 0), Z = S2n−1 ⊂ Cn ⊕ 0

Theorem [Cannas da Silva, 2010]

Let M be a 2n-dim’l manifold with a stably almost complex structure J .Then M admits a folded symplectic form consistent with J in any degree2 cohomology class.

NOTE: Omnioriented quasitoric manifolds admit folded symplectic forms.

Example

1 (R2d, x1dx1 ∧ dy1 +∑d

Example

1 (R2d, x1dx1 ∧ dy1 +∑d

Example

1 (R2d, x1dx1 ∧ dy1 +∑d

Origami manifolds

Let (M,ω) be a folded symplectic manifold with a fold Z.Let i : Z →M be the inclusion.Then i∗ω has a 1-dimensional kernel on Z.

Definition [Cannas da Silva-Guillemin-Pires, IMRN 2011]

ω is called an origami form if this line field is the vertical bundle of anoriented S1-fiber bundle Z

π→ B over a compact base B.

Example (continued)

1 ω = x1dx1 ∧ dy1 +∑d

i=2 dxi ∧ dyi on R2d is not origami.

2 ωCn ⊕ 0 on S2n ⊂ Cn ⊕ R is origami because the line field is thevertical bundle of the Hopf bundle π : S2n−1 → CPn−1.

Origami manifolds

Example (continued)

1 ω = x1dx1 ∧ dy1 +∑d

Origami manifolds

Example (continued)

1 ω = x1dx1 ∧ dy1 +∑d

Toric origami manifolds

The action of a Lie group G on an origami manifold (M,ω) is Hamiltonianif it admits a moment map µ : M → g∗ satisfying the conditions:

µ collects Hamiltonian functions, i.e., d〈µ,X〉 = ιX#ω,∀X ∈ g := Lie(G), where X# is the vector field generated by X;

µ is equivariant with respect to the given action of G on M and thecoadjoint action of G on the dual vector space g∗.

A toric origami manifold is a compact connected origami manifold (M,ω)equipped with an effective Hamiltonian action of a torus T withdimT = 1

2 dimM .

NOTE: If Z = ∅, a toric origami manifold is a toric symplectic manifold.

Toric origami manifolds

The action of a Lie group G on an origami manifold (M,ω) is Hamiltonianif it admits a moment map µ : M → g∗ satisfying the conditions:

µ collects Hamiltonian functions, i.e., d〈µ,X〉 = ιX#ω,∀X ∈ g := Lie(G), where X# is the vector field generated by X;

µ is equivariant with respect to the given action of G on M and thecoadjoint action of G on the dual vector space g∗.

A toric origami manifold is a compact connected origami manifold (M,ω)equipped with an effective Hamiltonian action of a torus T withdimT = 1

2 dimM .

NOTE: If Z = ∅, a toric origami manifold is a toric symplectic manifold.

Examples1 T = (S1)2 acts on (CP 2, ω) by

(t1, t2) · [z1, z2, z3] = [t1z1, t2z2, z3]

with moment map

µ(z1, z2, z3) =

(|z1|2

|z1|2 + |z2|2 + |z3|2,

|z1|2 + |z2|2 + |z3|2

2 T = (S1)2 acts on (S4 ⊂ C2 ⊕ R, ωCn ⊕ 0) by

(t1, t2) · (z1, z2, r) = (t1z1, t2z2, r)

with moment map

µ(z1, z2, r) = (|z1|2, |z2|2).

CP 2µ

Origami templates

An origami template consists of a pair (P,F), where P is a collection ofDelzant polytopes in Rn and F is a collection of facets and pairs of facetsof polytopes in P s.t.

(O1) ∀ F, F ′ ∈ F , if F ⊂ P and F ′ ⊂ P ′, then ∃ U ⊂open

Rn s.t.

U ∩ P = U ∩ P ′;(O2) if F ∈ F or F, F ′ ∈ F , then neither F nor any of its neighboring

facets occur elsewhere in F ;

(O3) the topological space, denoted |(P,F)|, constructed from the disjointunion tPj , Pj ∈ P, by identifying facet pairs in F is connected.

Theorem (Cannas da Silva-Guillemin-Pires, 2011)

toric origami manifolds 1:1←→ origami templates

NOTE: M/T ∼=homeo

|(P,F)| as a manifold with corners.

Origami templates

An origami template consists of a pair (P,F), where P is a collection ofDelzant polytopes in Rn and F is a collection of facets and pairs of facetsof polytopes in P s.t.

(O1) ∀ F, F ′ ∈ F , if F ⊂ P and F ′ ⊂ P ′, then ∃ U ⊂open

Rn s.t.

U ∩ P = U ∩ P ′;(O2) if F ∈ F or F, F ′ ∈ F , then neither F nor any of its neighboring

facets occur elsewhere in F ;

(O3) the topological space, denoted |(P,F)|, constructed from the disjointunion tPj , Pj ∈ P, by identifying facet pairs in F is connected.

Theorem (Cannas da Silva-Guillemin-Pires, 2011)

toric origami manifolds 1:1←→ origami templates

NOTE: M/T ∼=homeo

|(P,F)| as a manifold with corners.

RemarksNon-orientable toric origami manifolds1

with F = a red dot

Klein bottle

with F = two red dots

The black dot is corresponding to afixed point, and each red dot iscorresponding to the point whoseisotropy subgroup is Z2.

Non-simply connected origami templates

P = two intervalsF = two pairs of blue dots

1An origami template is oriented if the polytopes in P come with an orientation andF consists solely of pairs of facets which belong to polytopes with opposite orientations.

Motivation

For an orientable toric origami manifold M ,

if MT 6= ∅, M is a torus manifold2.

if |(P,F)| is (combinatorially) a simple polytope, M is a quasitoricmanifold.

Question

What about the converse?

2A torus manifold is a 2n-dim’l oriented compact connected smooth manifold Mwith an effective action of T = Tn with MT 6= ∅.

Motivation

For an orientable toric origami manifold M ,

if MT 6= ∅, M is a torus manifold2.

if |(P,F)| is (combinatorially) a simple polytope, M is a quasitoricmanifold.

Question

What about the converse?

2A torus manifold is a 2n-dim’l oriented compact connected smooth manifold Mwith an effective action of T = Tn with MT 6= ∅.

projective

manifolds

quasitoric

manifolds

torus manifolds

toric origami manifolds

T 2, RP 2

Not all torus manifolds admit toric origami forms.

From the following facts

S4 × S4 is simply connected,

(S4 × S4)/T 4 = |(P,F)| has four facets,

(S4 × S4)/T 4 = |(P,F)| has four vertices,

we can show that S4 × S4 cannot be a toric origami manifold.

Theorem (Masuda-P)

Let S2ni (ni ≥ 1) be the standard 2ni-sphere with the standardTni-sphere with the standard Tni-action for i = 1, . . . , k. Then

∏ki=1 S

with the product action of∏ki=1 T

ni admits a toric origami form if andonly if ni = 1 except for one i.

Not all torus manifolds admit toric origami forms.

From the following facts

S4 × S4 is simply connected,

(S4 × S4)/T 4 = |(P,F)| has four facets,

(S4 × S4)/T 4 = |(P,F)| has four vertices,

we can show that S4 × S4 cannot be a toric origami manifold.

Theorem (Masuda-P)

Let S2ni (ni ≥ 1) be the standard 2ni-sphere with the standardTni-sphere with the standard Tni-action for i = 1, . . . , k. Then

∏ki=1 S

with the product action of∏ki=1 T

ni admits a toric origami form if andonly if ni = 1 except for one i.

projective

manifolds

quasitoric

manifolds

torus manifolds

S4 × S4

toric origami manifolds

T 2, RP 2

All 4-dimensional quasitoric manifolds are equivariantly diffeomorphic tosome toric origami manifolds.

There is a one-to-one correspondence

simply conn. cpt sm. 4-mfd M with effective T 2 − action1:1←→ unimodular sequences v1, . . . , vd,

where vi and vi+1 form a basis of Z2 for i = 1, . . . , d and vd+1 = v1.

For d ≥ 4, there is vj (1 ≤ j ≤ d) which satisfies

εj−1vj−1 + εjvj+1 + ajvj = 0 (with aj = ±1 or 0, εj = det(vj , vj+1)).

Use induction on d.

Theorem (Masuda-P)

Any simply connected compact smooth 4-manifold M with an effectivesmooth action of T 2 is equivariantly diffeomorphic to a toric origamimanifold.

Use induction on d.

Theorem (Masuda-P)

Use induction on d.

Theorem (Masuda-P)

Example CP 2#CP 2

|(P,F)|

Remarks

Not all omniorientations of a quasitoric manifold determine toric origamiforms.

Let Mb be an omnioriented quasitoric manifold which is corresponding to aunimodular sequence (1, 0), (0, 1), (1, b), (0, 1). Then, the omniorientationof Mb admits a toric origami form if and only if b = ±1 or b = ±2.

ObservationLet F1, . . . , Fn, Fn+1 be the facets of ∆n.

1 By cutting the face F1 ∩ · · · ∩ Fk, we get a new polytopeP ≈ ∆n−k+1 ×∆k−1 whose facets G,F ′1, . . . , F

′k, F

′k+1, . . . , F

′n+1,

where G ≈ ∆n−k ×∆k−1, F ′1, . . . , F′k ≈ ∆n−k+1 ×∆k−2, and

F ′k+1, . . . , F′n+1 ≈ ∆n−k ×∆k−1.

2 Let P1 be the polytope obtained from ∆n by cutting a vertexF1 ∩ · · · ∩ Fn of ∆n. Then facets of P1 are of the form

G1, F′n+1 ≈ ∆n−1 &F ′1, . . . , F

′n ≈ vc(∆n−1).

3 By cutting a face G1 ∩ F ′1 ∩ · · · ∩ F ′k from P1, we get a new polytopeP2 ≈ vc(∆n−k ×∆k) whose facets are

F ′′n+1 ≈ ∆n−1, G2 ≈ ∆n−k−1 ×∆k, G′1 ≈ ∆n−k ×∆k−1,

F ′′1 , . . . , F′′k ≈ vc(∆n−k ×∆k−1), F ′′k+1, . . . , F

′′n ≈ vc(∆n−k−1 ×∆k).

′k, F

′k+1, . . . , F

′n+1,

F ′k+1, . . . , F′n+1 ≈ ∆n−k ×∆k−1.

G1, F′n+1 ≈ ∆n−1 &F ′1, . . . , F

′n ≈ vc(∆n−1).

F ′′n+1 ≈ ∆n−1, G2 ≈ ∆n−k−1 ×∆k, G′1 ≈ ∆n−k ×∆k−1,

F ′′1 , . . . , F′′k ≈ vc(∆n−k ×∆k−1), F ′′k+1, . . . , F

′′n ≈ vc(∆n−k−1 ×∆k).

′k, F

′k+1, . . . , F

′n+1,

F ′k+1, . . . , F′n+1 ≈ ∆n−k ×∆k−1.

G1, F′n+1 ≈ ∆n−1 &F ′1, . . . , F

′n ≈ vc(∆n−1).

F ′′n+1 ≈ ∆n−1, G2 ≈ ∆n−k−1 ×∆k, G′1 ≈ ∆n−k ×∆k−1,

F ′′1 , . . . , F′′k ≈ vc(∆n−k ×∆k−1), F ′′k+1, . . . , F

′′n ≈ vc(∆n−k−1 ×∆k).

Now, let Mr,s be a quasitoric manifold over ∆n ×∆m whose characteristicmatrix is of the form Λr,s.

Λr,s =

A quasitoric manifold Mm,n over ∆n ×∆m is equivariantly homeomorphicto a toric origami manifold.

1 Let F1, . . . , Fn+m+1 be the facets of ∆n+m and let the correspondingoutward normal vectors are e1, . . . , en+m,−e1 − · · · − en+m.

2 Let P be the polytope obtained by cutting F1 ∩ · · · ∩ Fn+m and thenby cutting G ∩ F ′1 ∩ · · · ∩ F ′n, where G,F ′1, . . . , F

′n+m are new facets

obtained from the first cutting.

3 The origami template which we want is the pair of P = ∆n+m, Pand F = (Fn+m+1, F

′′n+m+1).

∆1 ×∆2

e3 e1 + e2 + e3

2e1 + 2e2 + e3

∆2 ×∆1

e12e1 + e2 + e3

e1 + e2 + e3

A quasitoric manifold Mm,n over ∆n ×∆m is equivariantly homeomorphicto a toric origami manifold.

2 Let P be the polytope obtained by cutting F1 ∩ · · · ∩ Fn+m and thenby cutting G ∩ F ′1 ∩ · · · ∩ F ′n, where G,F ′1, . . . , F

′n+m are new facets

obtained from the first cutting.

3 The origami template which we want is the pair of P = ∆n+m, Pand F = (Fn+m+1, F

′′n+m+1).

∆1 ×∆2

e3 e1 + e2 + e3

2e1 + 2e2 + e3

∆2 ×∆1

e12e1 + e2 + e3

e1 + e2 + e3

A quasitoric manifold Mm,1 over ∆n ×∆m is equivariantly homeomorphicto a toric origami manifold.

2 Let P be the polytope obtained from ∆n+m by cuttingF2 ∩ · · · ∩ Fn+m+1 and then by cutting G ∩ F ′2 ∩ · · · ∩ F ′n ∩ F ′n+m+1,where G and F ′1, . . . , F

′n are new facets obtained by the first cutting.

3 P = ∆n+m, P and F = (F1, F′′1 ).

−e1 − e2 − e3

e3 −e1

−2e1 − e3

A quasitoric manifold Mm,1 over ∆n ×∆m is equivariantly homeomorphicto a toric origami manifold.

2 Let P be the polytope obtained from ∆n+m by cuttingF2 ∩ · · · ∩ Fn+m+1 and then by cutting G ∩ F ′2 ∩ · · · ∩ F ′n ∩ F ′n+m+1,where G and F ′1, . . . , F

′n are new facets obtained by the first cutting.

3 P = ∆n+m, P and F = (F1, F′′1 ).

−e1 − e2 − e3

e3 −e1

−2e1 − e3

Two quasitoric manifolds Mr,s and Mm−r+1,n−s+1 over ∆n ×∆m areweakly equivariantly diffeomorphic.

Define an isomorphism Θ: Zn+m → Zn+m by

e1 7→ −∑n+1

i=1 ei −∑m

j=r+1 en+j

en+1 7→ 2e1 + 2∑n

i=s+1 ei +∑m

j=1 en+j

e2, . . . , es, en+r+1, . . . , en+m 7→ e2, . . . , es, en+r+1, . . . , en+m

es+1, . . . , en, en+2, . . . , en+r 7→ −es+1, . . . ,−en,−en+2, . . . ,−en+r

Proposition

Any quasitoric manifold over ∆n ×∆m with n,m ≤ 2 is weaklyequivariantly homeomorphic to a toric origami manifold.

Two quasitoric manifolds Mr,s and Mm−r+1,n−s+1 over ∆n ×∆m areweakly equivariantly diffeomorphic.

Define an isomorphism Θ: Zn+m → Zn+m by

e1 7→ −∑n+1

i=1 ei −∑m

j=r+1 en+j

en+1 7→ 2e1 + 2∑n

i=s+1 ei +∑m

j=1 en+j

e2, . . . , es, en+r+1, . . . , en+m 7→ e2, . . . , es, en+r+1, . . . , en+m

es+1, . . . , en, en+2, . . . , en+r 7→ −es+1, . . . ,−en,−en+2, . . . ,−en+r

Proposition

Any quasitoric manifold over ∆n ×∆m with n,m ≤ 2 is weaklyequivariantly homeomorphic to a toric origami manifold.

Theorem [Choi & P & Suh]

Any quasitoric manifold over ∆n ×∆1 is homeomorphic to

two-stage generalized Bott manifolds,

M1,1 over ∆n ×∆1, or

M1,1 over ∆1 ×∆n.

M1,1∼=eq M1,n over ∆n ×∆1 & M1,1

∼=eq Mn,1 over ∆1 ×∆n.

Proposition

Any quasitoric manifold over ∆n ×∆1 is homeomorphic to a toric origamimanifold.

Denote

, 2m =

Consider a quasitoric manifold M over∏hi=1 ∆ni whose reduced

characteristic matrix is 1n1 2n1 · · · 2n1

1n2 1n2 · · · 2n2

......

. . ....

1nh1nh

· · · 1nh

Proposition

M admits a toric origami form.

Let F1, . . . , Fn, Fn+1 be the facets of ∆n whose outward normal vectorsare e1, . . . , en,−e1 − · · · − en, where n =

∑hi=1 ni.

1 Let P1 be the polytope obtained from ∆n by cutting F1 ∩ · · · ∩ Fn.

Then, P1 ≈ vc(∆n) and we get new facets G1, F(1)1 , . . . , F

(1)n+1,

2 Let P2 be the polytope obtained from P1 by cutting

G1 ∩ F (1)1 ∩ · · ·F (1)

n1 . Then, P2 ≈ vc(∆n−n1 ×∆n1) and we get new

facets G2, G(1)1 , F

(2)1 , . . . , F

(2)n+1,

3 Let P3 be the polytope obtaind from P2 by cutting

G2 ∩ F (2)n1+1 ∩ · · ·F

(2)n1+n2

. Then P3 ≈ vc(∆n−n1−n2 ×∆n2 ×∆n1)

and we get new facets G3, G(2)1 , G

(1)2 , F

(3)1 , . . . , F

(3)n+1,

4 Let Ph be the polytope obtained from Ph−1 by cutting

Gh−1 ∩ F(h−1)n−nh+1 ∩ · · · ∩ F

(h−1)n . Then Ph ≈ vc(∆nh × · · · ×∆n1)

and we get new facets Gh, G(1)h−1, . . . , G

(h−1)1 , F

(h)1 , . . . , F

(h)n+1.

5 P = ∆n, Ph, F = (Fn+1, F(h)n+1).

Denote

, 0m =

20...0

, (2) =

20...0

Consider a quasitoric manifold M over∏hi=1 ∆ni whose reduced

characteristic matrix is 1n1 (2) · · · (2)1n2 1n2 · · · 0n2

......

. . ....

1nh1nh

· · · 1nh

Proposition

M admits a toric origami form.

Bundle structures on polytopes

Let ∆ = ∩Nj=1x ∈ t∗ | 〈ηj , x〉 ≤ κj and ∆ = ∩Ni=1x ∈ t∗ | 〈ηi, x〉 ≤ κibe simple polytopes. We say that a simple polytope ∆ ⊂ t∗ is a bundlewith fiber ∆ over the base ∆ if there exists a short exact sequence

0→ tι→ t

π→ t→ 0

so that

∆ is combinatorially equivalent to the product ∆× ∆.

If ηj′ denotes the outward conormal to the facet F ′j of ∆ which

corresponds to Fj × ∆ ⊂ ∆× ∆, then η′j = ι(ηj) for all 1 ≤ j ≤ N .

If ηi′ denotes the outward conormal to the facet F ′i of ∆ which

corresponds to ∆× Fi ⊂ ∆× ∆, then π(ηi′) = ηi for all 1 ≤ i ≤ N .

The facets F ′1, . . . , F′N

are called fiber facets and the facets F ′1, . . . , F′N

are called base facets.

Bundle structures on origami templates

Let (P,F), (P, F), and (P, F) be origami templates defined in t∗, t∗, andt∗, respectively.

Definition

An origami template (P,F) is a bundle with fiber (P, F) over the base ∆(respectively, a bundle with fiber ∆ over the base (P, F)) if there exists ashort exact sequence

0→ tι→ t

π→ t→ 0

so that

each P ∈ P is a bundle with fiber P ∈ P over the base ∆(respectively, a bundle with fiber ∆ over a base P ∈ P.

if F occurs in F , then there F (respectively, F ) must occur in F(respectively, F) such that F is a bundle with fiber F over the base∆ (respectively, a bundle with fiber ∆ over a base F ).

A quasitoric manifold over a cube (∆1)3 is homeomorphic to a toricorigami manifold.

Let M be a quasitoric manifold over (∆1)3. Up to homeomorphism, by S.Hasui, the reduced characteristic matrix of M is one of the following 1 0 0

a1 1 0a2 a3 1

1 2 01 1 0b1 b2 1

1 2 c1

1 1 c2

1 2 21 1 21 1 1

.(i) The first case gives rise to a Bott manifold which admits a symplectictoric structure.

−e1 − e2 + ae3

= ♦e1 + e2 + b1e3

2e1 + e2 + b2e3

A quasitoric manifold over a cube (∆1)3 is homeomorphic to a toricorigami manifold.

Let M be a quasitoric manifold over (∆1)3. Up to homeomorphism, by S.Hasui, the reduced characteristic matrix of M is one of the following 1 0 0

a1 1 0a2 a3 1

1 2 01 1 0b1 b2 1

1 2 c1

1 1 c2

1 2 21 1 21 1 1

.(i) The first case gives rise to a Bott manifold which admits a symplectictoric structure.

−e1 − e2 + ae3

= ♦e1 + e2 + b1e3

2e1 + e2 + b2e3

c1e1 + c2e2 + e3

−e1 − e2

♦e1 + e2

2e1 + e2

♦e1 + e2 + e3

2e1 + e2 + e3

2e1 + 2e2 + e3

Remarks

Up to weakly equivariant homeomorphism, for a quasitoric manifold Mover (∆1)3, if M has no bundle structure, then the reduced characteristicmatrix of M is one of the following1 2 2

1 1 21 1 1

1 2 10 1 12 2 1

1 1 10 1 22 1 1

1 2 20 1 21 1 1

Future

Prove or disprove the following

1 any quasitoric manifold with β2 = 2 is equivariantly homeomorphic toa toric origami manifold.

2 any quasitoric manifold over∏hi=1 ∆ni is equivariantly homeomorphic

to a toric origami manifold.

3 any toric manifold is equivariantly homeomorphic to a toric origamimanifold.

Thank you for your attention!

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