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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
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 2 / 24
Table of contents
1 Toric origami manifolds
2 Betti numbers of toric origami manifolds
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 2 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 3 / 24
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
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 4 / 24
Origami
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 5 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 6 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 6 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 6 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 7 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 7 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 8 / 24
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
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 9 / 24
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) = ∅.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 10 / 24
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
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 11 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 12 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 13 / 24
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 ∼= ∼=
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 14 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 15 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 15 / 24
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 .
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 16 / 24
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 .
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 16 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 17 / 24
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).
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 18 / 24
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).
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 18 / 24
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)
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 19 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 20 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 21 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 22 / 24
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.
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 23 / 24
Seonjeong Park (NIMS) Betti numbers of toric origami manifolds 24 / 24