Betti numbers of toric origami manifolds - MathNet20140807).pdf · Betti numbers of toric origami manifolds Seonjeong Park 1 ... Daejeon Convention Center Seonjeong Park (NIMS) Betti

Betti numbers of toric origami manifolds

Seonjeong Park 1

Jointly with A. Ayzenberg 2, M. Masuda 2, H. Zeng 2

1National Institute for Mathematical Sciences

2Osaka City University

(2014 ICM Satellite Conference)Topology of torus actions and applications

to geometry and combinatorics,August 7–11, 2014

Daejeon Convention Center

Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 1 / 24

Table of contents

1 Toric origami manifolds

2 Betti numbers of toric origami manifolds


Table of contents

1 Toric origami manifolds

2 Betti numbers of toric origami manifolds


Symplectic manifolds

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

Example

θ

hThe unit sphere S2 in R3 is a symplecticmanifold with ω = dθ ∧ dh.

But for n > 1, S2n cannot admit a symplectic form.


Symplectic toric manifold

A symplectic toric manifold is a compact connected symplectic manifold(M2n, ω) equipped with an effective hamiltonian action of an n-torus Tn

and with a corresponding moment map µ : M → Rn.

Delzant’s Theoremcompact toric

symplectic manifolds

1:1←→ Delzant polytopes ,

(M,ω, Tn, µ)1:1←→ µ(M)

Example

h

−1

+1


Origami


Origami manifolds

An origami form on a 2n-dim’l manifold M is a closed 2-form ω

ωn vanishes transversally on a submanifold i : Z →M ;

i∗ω has maximal rank, i.e., (i∗ω)n−1 does not vanish;

the 1-dimensional kernel on Z is the vertical bundle of an oriented S1

fiber bundle Zπ→ B over a compact base B.

(M,ω) is called an origami manifold with a fold Z.

Example

For n ≥ 1, (S2n ⊂ Cn ⊕ R, ωCn ⊕ 0) is an origami manifold with the foldZ = S2n−1 ⊂ Cn ⊕ 0, where ωCn = i

2

∑nk=1 dzk ∧ dzk and the

1-dimensional kernel on Z is the vertical bundle of the Hopf bundleπ : S2n−1 → CPn−1.


Origami manifolds







Example


2




Origami manifolds







Example


2




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.


Example 1

T = (S1)2 acts on (S4, ωC2 ⊕ 0) by

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

with moment mapµ(z1, z2, r) = (|z1|2, |z2|2).

S4µ

Note that the fold is an equator S3 ∼= (z1, z2, 0) ∈ S4.


Example 2

Note that S4 ⊂ C2 ⊕ R has an origami form ωC2 ⊕ 0.The origami form on S4 is invariant under the involution

(z1, z2, r) 7→ −(z1, z2, r).

Hence, it induces an origami form on RP 4 = S4/Z2 whose fold isRP 3 ∼= [z1, z2, 0].Furthermore, RP 4 is a toric origami manifold with moment map

µ[z1, z2, r] = (|z1|2, |z2|2).

RP 4µ


Origami templates

Dn = the set of all Delzant polytopes in RnEn = the set of all subsets of Rn which are facets of elements of DnG = a graph (need not to be simple, i.e., ∃ multiple edges and loops)

An n-dimensional origami template consists of a graph G, called thetemplate graph, and a pair of maps ΨV : V → Dn and ΨE : E → En suchthat

1 if e is an edge of G with end vertices u and v, then ΨE(e) is a facetof a each of the polytopes ΨV (u) and ΨV (v), and these polytopescoincide near ΨE(e); and

2 if v is an end vertex of each of two distinct edges e and f , thenΨE(e) ∩ΨE(f) = ∅.


Examples

eu v

P

F1

F2

F3ΨV (u) = ΨV (v) = P

ΨE(e) = F3

e

u

P

F1

F2

F3 ΨV (u) = P

ΨE(e) = F3

e e′

u v w P1 F1

F2F3F4

F5

F6 F7F8

P2 F1

F2F ′3

F ′5

F6 F7F8

ΨV (u) = P1

ΨV (v) = ΨV (w) = P2

ΨE(e) = F2

ΨE(e′) = F6


Toric origami manifolds and origami templates

Theorem [Cannas da Silva-Guillemin-Pires]

There is a one-to-one correspondence

toric origami manifolds 1:1←→ origami templates ,

up to equivariant origami-symplectomorphism on the left-hand side, andaffine equivalence of the image of the template in Rn on the right-handside.


Properties

1 M is orientable ⇔ G is bipartite.

2 The orbit space M/T is realized as X =⊔v∈V (v,ΨV (v))/ ∼, where

(u, x) ∼ (v, y) if there exists an edge e with endpoints u and v.

facets of X:⊔v∈V

F facet of ΨV (v)F not a fold facet

(v, F )

/∼

3 X ' G.


Examples

eu v

R2

P

F1

F2

F3 ΨV (u) = ΨV (v) = P

ΨE(e) = F3

S4/T ≈ X = Facets

e

e′u v

R2

P1 F1

F2F3F4

F5

F6 F7F8

R2

P2 F1

F2F ′3

F ′5

F6 F7F8

ΨV (u) = P1, ΨV (v) = P2

ΨE(e) = F2

ΨE(e′) = F6

X ∼= ∼=


Motivation

Theorem [Jurkiewicz]

Let (M,ω) be a symplectic toric manifold corresponding to a Delzantpolytope P . Then Hodd = 0, b2i(M) = hi(P ), and H∗(M) = Z(P )/J .

Questions

Let M be a toric origami manifold whose origami template is(G,ΨV ,ΨE).

1 Compute the Betti numbers of M by using the orbit space X.

2 Describe the cohomology ring of M by using the origami template.


Motivation

Theorem [Jurkiewicz]

Let (M,ω) be a symplectic toric manifold corresponding to a Delzantpolytope P . Then Hodd = 0, b2i(M) = hi(P ), and H∗(M) = Z(P )/J .

Questions

Let M be a toric origami manifold whose origami template is(G,ΨV ,ΨE).

1 Compute the Betti numbers of M by using the orbit space X.

2 Describe the cohomology ring of M by using the origami template.


Goal of this talk

[Masuda-Panov, 2006]

If a torus manifold M has a face-acyclic orbit space M/T , thenHodd(M) = 0 and the even-degree Betti numbers of M can be computedby using the face numbers of the orbit space.

For a toric origami manifold M , if the template graph G is a tree, then Mis orientable, MT 6= ∅, and M/T is face-acyclic. Hence M is a locallystandard torus manifold whose orbit space is face-acyclic.

Goal

Let M be an orientable toric origami manifold such that every proper faceof M/T is acyclic. Compute the Betti numbers of M .


Goal of this talk

[Masuda-Panov, 2006]

If a torus manifold M has a face-acyclic orbit space M/T , thenHodd(M) = 0 and the even-degree Betti numbers of M can be computedby using the face numbers of the orbit space.

For a toric origami manifold M , if the template graph G is a tree, then Mis orientable, MT 6= ∅, and M/T is face-acyclic. Hence M is a locallystandard torus manifold whose orbit space is face-acyclic.

Goal

Let M be an orientable toric origami manifold such that every proper faceof M/T is acyclic. Compute the Betti numbers of M .


Example

e

e′u v

R2

P1 F1

F2F3F4

F5

F6 F7F8

R2

P2 F1

F2F ′3

F ′5

F6 F7F8

ΨV (u) = P1, ΨV (v) = P2

ΨE(e) = F2

ΨE(e′) = F6

X = = X is not acyclic but

each proper face is acyclic.

e

e′u v

P =

F1

F2

ΨV (u) = P, ΨV (v) = P

ΨE(e) = F1

ΨE(e′) = F2

X has a non-acyclic face of codim 1.

X = = X is not acyclic but

each proper face is acyclic.


Fundamental group and first homology group

Proposition [With Masuda, 2013]

If a toric origami manifold M has a fixed point and the template graph Ghas no loop, then the quotient map M →M/T induces an isomorphismq∗ : π1(M)→ π1(M/T ) and hence π1(M) is a free group.

Corollary

If G is bipartite and every proper face of M/T is acyclic, then

H1(M) = Zb1(G),

hence b1(M) = b1(G).


Fundamental group and first homology group

Proposition [With Masuda, 2013]

If a toric origami manifold M has a fixed point and the template graph Ghas no loop, then the quotient map M →M/T induces an isomorphismq∗ : π1(M)→ π1(M/T ) and hence π1(M) is a free group.

Corollary

If G is bipartite and every proper face of M/T is acyclic, then

H1(M) = Zb1(G),

hence b1(M) = b1(G).


Setting

Let M be an orientable toric origami manifold associated with(G,ΨV ,ΨE) with b1(G) > 1.Choose an edge e in G such that b1(G− e) = b1(G)− 1.

M M ′ B

l l l

(G,ΨV ,ΨE) (G− e,ΨV ,ΨE\e) ΨE(e)


Betti numbers of toric origami manifolds

Theorem

Let M be a toric origami manifold such that all proper faces of M/T areacyclic. Then

b2i+1(M) = 0, 1 ≤ i ≤ n− 2.

Moreover, if M ′ and B are as above, then

b1(M ′) = b1(M)− 1,

b2i(M′) = b2i(M) + b2i(B) + b2i−1(B), 1 ≤ i ≤ n− 1.


Face numbers of M/T

Let M be a toric origami manifold of dim 2n and P the simplicial posetdual to ∂(M/T ). We define

fi = the number of (n− 1− i)− faces of M/T,

= the number of i-simplices in P for i = 0, 1, . . . , n− 1

and the h-vector (h0, h1, . . . , hn) by

n∑i=1

hitn−i = (t− 1)n +

n−1∑i=0

fi(t− 1)n−1−i.

Lemma

Assume every proper face of M/T is acyclic.

fi(M′/T ) = fi(M/T ) + 2fi−1(F ) + fi(F ) for 0 ≤ i ≤ n− 1.


Relation between Betti numbers and face numbers

Theorem

Let M be a toric origami manifold such that all proper faces of M/T areacyclic. Let bj := bj(M). Then

n∑i=0

b2iti =

n∑i=0

hiti + b1(1 + tn − (1− t)n),

in other words, b0 = h0 = 1 and

b2i = hi − (−1)i(n

i

)b1, for 1 ≤ i ≤ n− 1

b2n = hn + (1− (−1)n)b1.


Remark

From the previous theorem, we get generalized Dehn-Sommerville relationsfor ∂(M/T )

hn−i − hi = (−1)i((−1)n − 1)b1

(n

i

)= (−1)i(χ(∂(M/T ))− χ(Sn−1))

(n

i

)for 0 ≤ i ≤ n.



Betti numbers of toric origami manifolds - MathNet20140807).pdf · Betti numbers of toric origami manifolds Seonjeong Park 1 ... Daejeon Convention Center Seonjeong Park (NIMS) Betti

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