Joint work with Wei Wang - Osaka City Universitymasuda/toric2014_osaka/Lu(slide).pdf · —Joint work with Wei Wang Zhi Lu ... M2n is locally iso. to the standard Tn-repre. on Cn;

beamer-tu-logo

Notions and Background Motivation Main result Proof

Examples of quasitoric manifolds as special unitarymanifolds

—Joint work with Wei Wang

Zhi Lu (½)

School of Mathematical SciencesFudan University, Shanghai

Toric Topology 2014 in OsakaJanuary 25, 2014

Zhi Lu (½½½) Examples of quasitoric manifolds as special unitary manifolds —Joint work with Wei Wang

beamer-tu-logo


Outline

Notions and BackgroundMotivation and ProblemMain resultsProof


beamer-tu-logo


Outline



beamer-tu-logo


Outline



beamer-tu-logo


Outline



beamer-tu-logo


Notions–unitary manifold

Unitary manifoldA unitary manifold is an oriented closed smooth manifoldwhose tangent bundle admits a stably complex structure.Namely, there exists a bundle map

J : TM ⊕ Rk −→ TM ⊕ Rk

such that J2 = −1.A unitary manifold is said to be special if the first Chern classvanishes.

Milnor showed that

Theorem (Milnor)A unitary manifold M is cobordant to zero if and only if its allChern numbers are zero.


beamer-tu-logo


Notions–unitary manifold

Unitary manifoldA unitary manifold is an oriented closed smooth manifoldwhose tangent bundle admits a stably complex structure.Namely, there exists a bundle map

J : TM ⊕ Rk −→ TM ⊕ Rk

such that J2 = −1.A unitary manifold is said to be special if the first Chern classvanishes.

Milnor showed that

Theorem (Milnor)A unitary manifold M is cobordant to zero if and only if its allChern numbers are zero.


beamer-tu-logo


Notions–quasitoric manifold

Definition

A quasitoric manifold M2n is a closed smooth manifold withan effective action of T n such that

1) M2n is locally iso. to the standard T n-repre. on Cn;

2) its orbit space M2n/T n is a simple convex polytope.

RK. A quasitoric manifold is the topological version of anonsingular compact toric variety, introduced byDavis–Januszkiewicz [Duke Math. J., 1991]


beamer-tu-logo


Standard T n-representationStandard T n-representation on Cn defined by

(z1, ..., zn) 7−→ (g1z1, ...,gnzn)

whose orbit space is the positive cone in Rn.

Simple convex polytopesA convex n-polytope is the convex hull of some finite points inRn.A convex n-polytope Pn is said to be simple if the number offacets meeting at each vertex is exactly n


beamer-tu-logo


Standard T n-representationStandard T n-representation on Cn defined by

(z1, ..., zn) 7−→ (g1z1, ...,gnzn)

whose orbit space is the positive cone in Rn.

Simple convex polytopesA convex n-polytope is the convex hull of some finite points inRn.A convex n-polytope Pn is said to be simple if the number offacets meeting at each vertex is exactly n


beamer-tu-logo


Simple Not simple


beamer-tu-logo


Example

S1 y CP1 defined by [z0 : z1] 7−→ (z0 : gz1], gives the1-simplex as its orbit space.

More generally, T n y CPn defined by

[z0 : z1 : · · · : zn] 7−→ [z0 : g1z1 : · · · : gnzn]

with n-simplex as its orbit space.


beamer-tu-logo


Example

S1 y CP1 defined by [z0 : z1] 7−→ (z0 : gz1], gives the1-simplex as its orbit space.

More generally, T n y CPn defined by

[z0 : z1 : · · · : zn] 7−→ [z0 : g1z1 : · · · : gnzn]

with n-simplex as its orbit space.


beamer-tu-logo


Two key points for theory of quasitoric manifolds

–Geometric topologyCharacteristic function: Each quasitoric manifoldπ : M2n −→ Pn determines

λ : F(Pn) −→ Zn

mapping n facets meeting at each vertex to a basis of Zn,where F(Pn):=all facets of Pn.Reconstruction: Up to equivariant diffeomorphism, M2n

can be recovered by the pair (Pn, λ), denoted by M(Pn, λ).


beamer-tu-logo








beamer-tu-logo








beamer-tu-logo








beamer-tu-logo



π : M2n −→ Pn: a quasitoric manifold over Pn.

–Algebraic topologyEquivariant cohomology: H∗

T n (Mn) ∼= R(Pn) whereR(Pn) is the Reisner-Stanley face ring of Pn:

R(Pn) = Z[F1, ...,Fl ]/I

I = (Fi1 · · ·Fir |Fi1 ∩ · · · ∩ Fir = ∅) is an ideal, and each Fi isa facet (ie., codim-one face) of Pn.Betti numbers:(b0,b2, ...,b2n) = (h0,h1, ...,hn) where(h0,h1, ...,hn) is the h-vector of Pn

· · ·


beamer-tu-logo






R(Pn) = Z[F1, ...,Fl ]/I


· · ·


beamer-tu-logo






R(Pn) = Z[F1, ...,Fl ]/I


· · ·


beamer-tu-logo






R(Pn) = Z[F1, ...,Fl ]/I


· · ·


beamer-tu-logo


Quasitoric manifolds as examples of unitary toricmflds–Buchstaber–Ray’s work

In their 1998 paper [Russ. Math. Surv. 53 (1998),371–373], Buchstaber and Ray studied the cobordism ofquasitoric manifolds. They first showed

Buchstaber–RayEach omnioriented quasitoric manifold is a unitary manifold.

Remark: An omniorientation of a quasitoric manifoldπ : M2n −→ Pn is a collection of all orientations ofM2n, π−1(F ) = MF ,F ∈ F(Pn).

There are 2m+1 omniorientations where m is the number ofall facets of Pn.


beamer-tu-logo








beamer-tu-logo








beamer-tu-logo



Theorem (Buchstaber–Ray, 1998)The unitary cobordism class of each unitary manifold containsan omnioriented quasitoric manifold as its representative.

In other words, each class of ΩU2n is represented by an

omnioriented quasitoric manifold, where ΩU2n is the abelian

group formed by the unitary cobordism classes of all2n-dimensional unitary manifolds.


beamer-tu-logo



Theorem (Buchstaber–Ray, 1998)The unitary cobordism class of each unitary manifold containsan omnioriented quasitoric manifold as its representative.

In other words, each class of ΩU2n is represented by an

omnioriented quasitoric manifold, where ΩU2n is the abelian

group formed by the unitary cobordism classes of all2n-dimensional unitary manifolds.


beamer-tu-logo


Buchstaber-Panov-Ray’s work

Furthermore, in their paper [Toric Genera, Internat. Math.Res. Notices 2010, 3207–3262], Buchstaber-Panov-Rayinvestigated when an omnioriented quasitoric manifold is aspecial unitary manifold.

Proposition (Buchstaber–Panov–Ray, 2010)Let M(Pn, λ) be a quasitoric manifold. Then M(Pn, λ) with anomniorientation is a special unitary manifold if and only if foreach facet F ∈ F(Pn), the sum of all entries of λ(F ) is exactly1.

Then they showed

Proposition (Buchstaber–Panov–Ray, 2010)Suppose that M(Pn, λ) with an omniorientation is a specialunitary manifold. When n < 5, M(Pn, λ) represents the zeroelement in ΩU

2n.


beamer-tu-logo


Buchstaber-Panov-Ray’s work

Furthermore, in their paper [Toric Genera, Internat. Math.Res. Notices 2010, 3207–3262], Buchstaber-Panov-Rayinvestigated when an omnioriented quasitoric manifold is aspecial unitary manifold.

Proposition (Buchstaber–Panov–Ray, 2010)Let M(Pn, λ) be a quasitoric manifold. Then M(Pn, λ) with anomniorientation is a special unitary manifold if and only if foreach facet F ∈ F(Pn), the sum of all entries of λ(F ) is exactly1.

Then they showed

Proposition (Buchstaber–Panov–Ray, 2010)Suppose that M(Pn, λ) with an omniorientation is a specialunitary manifold. When n < 5, M(Pn, λ) represents the zeroelement in ΩU

2n.


beamer-tu-logo


Buchstaber-Panov-Ray’s conjecture

Buchstaber–Panov–Ray Conjecture (2010)Suppose that M(Pn, λ) with an omniorientation is a specialunitary manifold. Then M(Pn, λ) represents the zero element inΩU

2n.


beamer-tu-logo


Motivation

Motivation of this talkTo consider the Buchstaber–Panov–Ray conjecture.


beamer-tu-logo


Main result

We shall construct some examples of speciallyomnioriented quasitoric manifolds that are not cobordant tozero in ΩU

∗ , which give the negative answer to the aboveconjecture in almost all possible dimensional cases.

Our main result is stated as follows.

Main resultFor each n ≥ 5 with only n 6= 6, there exists a 2n-dimensionalspecially omnioriented quasitoric manifold M2n whichrepresents a nonzero element in ΩU

∗ .


beamer-tu-logo


Main result

We shall construct some examples of speciallyomnioriented quasitoric manifolds that are not cobordant tozero in ΩU

∗ , which give the negative answer to the aboveconjecture in almost all possible dimensional cases.

Our main result is stated as follows.

Main resultFor each n ≥ 5 with only n 6= 6, there exists a 2n-dimensionalspecially omnioriented quasitoric manifold M2n whichrepresents a nonzero element in ΩU

∗ .


beamer-tu-logo


Proof

Our strategy is as follows: Related to the unorientedcobordism theory

Milnor’s work tells us that there is an epimorphism

F∗ : ΩU∗ −−−−→ N2

∗

where N∗ denotes the ring produced by the unorientedcobordism classes of all smooth closed manifolds, andN2

∗ = α2|α ∈ N∗.This actually implies that there is a coveringhomomorphism

Hn : ΩU2n −−−−→ Nn

which is induced by θn Fn, where θn : N2n −→ Nn is

defined by mapping α2 7−→ α.


beamer-tu-logo


Proof

Our strategy is as follows: Related to the unorientedcobordism theory

Milnor’s work tells us that there is an epimorphism

F∗ : ΩU∗ −−−−→ N2

∗

where N∗ denotes the ring produced by the unorientedcobordism classes of all smooth closed manifolds, andN2

∗ = α2|α ∈ N∗.This actually implies that there is a coveringhomomorphism

Hn : ΩU2n −−−−→ Nn

which is induced by θn Fn, where θn : N2n −→ Nn is

defined by mapping α2 7−→ α.


beamer-tu-logo


An explanation for Hn : ΩU2n −→ Nn

Theorem (Buchstaber–Ray)Each class of Nn contains an n-dimensional small cover asits representative, where a small cover is also introduced byDavis and Januszkiewicz, and it is the real analogue of aquasitoric manifold.

Davis and Januszkiewicz tell us that each quasitoricmanifold M2n over a simple convex polytope Pn always admitsa natural conjugation involution τ whose fixed point set Mτ isjust a small cover over Pn. Thus, τ induces a homomorphismφτn : ΩU

2n −→ Nn, which exactly agrees with the abovehomomorphism Hn : ΩU

2n −→ Nn.An approach: to construct the examples of specially

omnioriented quasitoric manifolds whose images under φτ arenonzero in N∗.


beamer-tu-logo









beamer-tu-logo









beamer-tu-logo


Examples of specially omnioriented quasitoricmanifolds

Throughout the following

for a k -dimensional simplex ∆k , ∆(k)i , i = 1, ..., k + 1 mean

the k + 1 facets of ∆k ,

for a product P = ∆k1 × · · · ×∆kr of simplices, Fki ,j meansthat the facet

∆k1 × · · · ×∆ki−1 ×∆(ki )j ×∆ki+1 × · · · ×∆kr

of P.


beamer-tu-logo



Example I

Let P4l+5 = ∆2 ×∆4l+3 with l ≥ 0. Define a characteristicfunction λ(2,0,...,0) on P4l+5 in the following way. First let us fixan ordering of all facets of P4l+5 as follows

F2,1,F2,2,F2,3,F4l+3,1, ...,F4l+3,4l+3,F4l+3,4l+4.

Then we construct the characteristic matrix Λ(2,0,...,0) of therequired characteristic function λ(2,0,...,0) on the above orderedfacets as follows:

Λ(2,0,...,0) =

(I2 12

J1 I4l+3 14l+3

)

We obtain the special unitary mfd M(P4l+5, λ(2,0,...,0)).Zhi Lu (½½½) Examples of quasitoric manifolds as special unitary manifolds —Joint work with Wei Wang

beamer-tu-logo



Example II

Let P8l+11 = ∆4 ×∆2 ×∆8l+5 with l ≥ 0. In a similar way asabove, fix an ordering of all facets of P8l+11 as follows:

F4,1,F4,2,F4,3,F4,4,F4,5,F2,1,F2,2,F2,3,F8l+5,1, ...,F8l+5,8l+5,F8l+5,8l+6.

Then we define a characteristic function λ(4,2,0,...,0) on theabove ordered facets of P8l+11 by the following characteristicmatrix

Λ(4,2,0,...,0) =

I4 14

I2 12

J1 J2 I8l+5 18l+5

We obtain the special unitary mfd M(P8l+11, λ(4,2,0,...,0))


beamer-tu-logo


Stong manifolds

Lemma

The images of M(P4l+5, λ(2,0,...,0)2 ) and M(P8l+11, λ

(4,2,0,...,0)2 )

under the mapeΩU

2n −→ Nn

are exactly Stong manifolds RP(2,0, ...,0︸︷︷︸4l+3

) and RP(4,2,0, ...,0︸︷︷︸8l+4

).

DefinitionA Stong manifold is defined as the real projective spacebundle denoted by RP(n1, ...,nk ) of the bundle γ1 ⊕ · · · ⊕ γkover RPn1 × · · · × RPnk , where γi is the pullback of thecanonical bundle over the i-th factor RPni .The Stong manifold RP(n1, ...,nk ) has dimensionn1 + · · ·+ nk + k − 1.


beamer-tu-logo


Stong manifolds

Lemma

The images of M(P4l+5, λ(2,0,...,0)2 ) and M(P8l+11, λ

(4,2,0,...,0)2 )

under the mapeΩU

2n −→ Nn

are exactly Stong manifolds RP(2,0, ...,0︸︷︷︸4l+3

) and RP(4,2,0, ...,0︸︷︷︸8l+4

).

DefinitionA Stong manifold is defined as the real projective spacebundle denoted by RP(n1, ...,nk ) of the bundle γ1 ⊕ · · · ⊕ γkover RPn1 × · · · × RPnk , where γi is the pullback of thecanonical bundle over the i-th factor RPni .The Stong manifold RP(n1, ...,nk ) has dimensionn1 + · · ·+ nk + k − 1.


beamer-tu-logo


Indecomposable Stong manifolds

Theorem (Stong)A Stong manifold RP(n1, ...,nk ) is indecomposable if and only if(

n − 1n1

)+ · · ·+

(n − 1

nk

)≡ 1 mod 2

where n = n1 + · · ·+ nk + k − 1.

⇓

RP(2,0, ...,0︸︷︷︸4l+3

) and RP(4,2,0, ...,0︸︷︷︸8l+4

) are indecomposable

so they represent nonzero elements in N∗.


beamer-tu-logo




n − 1n1

)+ · · ·+

(n − 1

nk

)≡ 1 mod 2

where n = n1 + · · ·+ nk + k − 1.

⇓

RP(2,0, ...,0︸︷︷︸4l+3

) and RP(4,2,0, ...,0︸︷︷︸8l+4




beamer-tu-logo




n − 1n1

)+ · · ·+

(n − 1

nk

)≡ 1 mod 2

where n = n1 + · · ·+ nk + k − 1.

⇓

RP(2,0, ...,0︸︷︷︸4l+3

) and RP(4,2,0, ...,0︸︷︷︸8l+4




beamer-tu-logo


Lemmas

Let α8l+10 = M(P4l+5, λ(2,0,...,0)) andβ16l+22 = M(P8l+11, λ(4,2,0,...,0))

Lemma 1

α8l+10 and β16l+22 form a subring of ΩU∗

Z[α8l+10, β16l+22|l ≥ 0]

which contains nonzero classes ofdimension 6= 2,4,6,8,12,14,16,24.

It remains to consider 2n = 14,16,24 (i.e., n = 7,8,12).


beamer-tu-logo


The case n = 7

Consider the polytope P7 = ∆4 ×∆3. Then we may definea characteristic function λ<7> on the ordered facets of P7 bythe following characteristic matrix

1 11 −1

1 11 −1

1 1 11 −1

1 1

,

which gives a special unitary manifold M(P7, λ<7>). A directcalculation gives the Chern number〈c3c4, [M(P7, λ<7>)]〉 = −2 6= 0, which implies that thisspecially omnioriented quasitoric manifold M(P7, λ<7>) is notcobordant to zero in ΩU

∗ .Zhi Lu (½½½) Examples of quasitoric manifolds as special unitary manifolds —Joint work with Wei Wang

beamer-tu-logo


The case n = 8

Consider the polytope P8 = ∆3 ×∆5 with a characteristicfunction λ<8> on the ordered facets of P8 by

1 11 −1

1 1−1 1 11 1 −1

1 11 −1

1 1

,

which also gives a special unitary manifold M(P8, λ<8>). Onehas the Chern number 〈c2

4 , [M(P8, λ<8>)]〉 = 4 6= 0. SoM(P8, λ<8>) is not cobordant to zero in ΩU

∗ .


beamer-tu-logo


The case n = 12Consider the polytope P12 = ∆3 ×∆9 with a characteristic

function λ<12> on the ordered facets of P12 by the matrix

1 11 −1

1 1−1 1 11 1 −1

1 11 −1

1 11 −1

1 11 −1

1 1

,

from which one obtains a special unitary manifold M(P12, λ<12>).Then one has that the Chern number 〈c2

6 , [M(P12, λ<12>)]〉 = 64 6= 0.Thus M(P12, λ<12>) is not cobordant to zero in ΩU

∗ .Zhi Lu (½½½) Examples of quasitoric manifolds as special unitary manifolds —Joint work with Wei Wang

beamer-tu-logo


Remark

We have done many tries to find a counterexample in thecase n = 6, but failed.

It seems to be reasonable to the assertion as in theBuchstaber–Panov–Ray conjecture that each 12-dimensionalspecially omnioriented quasitoric manifold is cobordant tozero in ΩU

∗ since each 6-dimensional orientable smooth closedmanifold is always cobordant to zero in N∗.


beamer-tu-logo


Remark

We have done many tries to find a counterexample in thecase n = 6, but failed.

It seems to be reasonable to the assertion as in theBuchstaber–Panov–Ray conjecture that each 12-dimensionalspecially omnioriented quasitoric manifold is cobordant tozero in ΩU

∗ since each 6-dimensional orientable smooth closedmanifold is always cobordant to zero in N∗.


beamer-tu-logo


Thank You!


Documents

Joint work with Wei Wang - Osaka City Universitymasuda/toric2014_osaka/Lu(slide).pdf · —Joint work with Wei Wang Zhi Lu ... M2n is locally iso. to the standard Tn-repre. on Cn;