Positive Yuya

(1)

Positive

flow

^-

spines manifolds ^and ^contact ³

^-

Yuya ^Koda ^I ^Hiroshima

^Univ ^.

⁾

Joint work w/

Ippei

^Ishii

⁽

^Keio ^Univ ^. ⁾

Masaharu Ishikawa

I

^Keio ^Univ ^. ⁾

Hironobu Nao e ( Chuo Univ ^. )

Intelligence ^of

^Low ^-

dimensional Topology May

²³ ^,

²⁰¹⁹

RIMS

(2)

Main ^Theorem

M

: a closed oriented ³ ^- manifold

{ _positive flow

^- _spines ^of ^'M

)

^→ ^Cont ^I ^M ⁾

/ _isotopy

§ I

Positive

flow

^-

spines

§

² ^.

^Contact ³

^-

^manifolds

§

³ ^.

^Main ^results

§ 4

^.

Tabulation

(3)

§ ^I

Positive

flow

^-

spines

§

² ^.

^Contact

³ ^-

^manifolds

§

³ ^.

^Main

^results

§ ⁴

^.

^Tabulation

(4)

M

^: ^a closed ^oriented ³ ^-

manifold

.

¥

⁼

I

Yt

Ite

µ

: a non ^-

singular

^flow ^on ^M

( generated ^by

^a ^non ^-

^singular

^vector

^field

^X ^on ^M

⁾

^.

A

compact

^oriented

^surface

Î ^c ^M îs â ^normal

section ^Eff

^lil

^I

^In

^X ( positively

^transverse

⁾

ⁱ ^for

^E

.pe#Ty .gg?i.RqfirsttimetomeetE

^Cii ⁾

⁽ ^IT

^ht ^P^: ^e ^.^M

Ip ^Mu

^.

^I ^{

^Yt ⁾

⁾

^t ^, ^o ⁿ ^I ^F

⁴

ⁱ

₎

^I

p ^p ^p

I ⁱⁱⁱ ) _p ^E 2

-2

Tip

, ⁾ ^e

^2-2

^⇒ ÎT ^' Î^p⁾ ê Înt

^-2

ⁱ ^and

liv ⁾ P ^c- ² -2

Ttp

, ⁾ ^c-

^2E ^&

Tap ^Ttip Ftp Tp

^⇒

⁽

⁾ ^*

^2-2In ²²

^, ^. ^"^:^:^• p ^,

(5)

P ^.^. ⁼ ^the

discontinuity

of

set

^IT

^: ^M^p^y ^to^→

^I

Î Û^× ÎR

Tip tp

⁾ ^,

⁾

=

I

^u

{

ft

Cp

⁾ ê ^M Î ^p ^c- Î ^,

Tip

^or^T

^use

^:^!^:^:^:^.^: ^✓^{ ^!^:^:^i.ⁱ^. ^co^T

^Smoothing tp

^I ^T ^§

^:p

^Tⁱ ^!^:^!^:ⁱ^:^:î.^:⁾ ôrÊ

^ii.

^2-2^:ⁱⁱ^:^. ⁰^:^:^:^:ⁱⁱⁱ^:^:

^f

ⁱ^.^. ^. ^.^,^:^:ⁱ^.^:^'^. ^or

^smoothing ^→

^TÛs²^soÎ Î ^teⁱ^:ⁱ^.ⁱ^T Ê

^}

^.

.

÷..

^co^o

Jo

^, ^. ^- ^. ^. ^-⁰ ^O

→

(6)

@

P

^c M ^: ^a

branched polyhedron

^.

Q

M

^- P ^I Int -2 ^x ^10.1 ) .

In

particular

^,

^I

⁼ ^D ^' ^⇒ ^M ^- ^P ⁼ ^Int ^D

?

I

→

P

^is ^a

flow

^-

spine ^of ^I

^M ^, ^E ⁾ ^.

E

^is ^carried

by

^P ^.

(7)

Definition

P

^c ^M ^: ^a ^flow ^-

spine

^of ^{( M} ^. ^E

)

⇐ lil P îs â spine ^, î.e ^. ^M ^- ^P Î Înt ^D^' ⁱ

def

Iii )

Vp

e P ,

7- _a

positive

^chart Î Û îx. ^y^. ⁾ ôf ^M ^Zâround ^p ^s ^'t ^.

Z

^

, y l ^. Type ^vertex ^r ^- type ^vertex

> x

( U ^. Un P ) I

og 9 ^Or

/

where I ôn Û îs

generated by Zz

^; ^and

Z

> Y

where

I ⁱⁱⁱ ) M ^- P ⁼ ^> ^x

, E ^on ^M ^- ^P ^is

generated by Fz

^.

(8)

Proposition

n n

! ⁿ

^

.

. .

i

' ' ' ' ' ' ' ' '''

V M ^: ^a

closed

^oriented ³ ^- ^manifold^µ ^, ^i.^.^. ^•^. ^.^P^.

^y

^. ^.

V

§

^: ^a ^non ^.

singular

^flow ^on ^, ^. ^.

Bp

⇒

a

flow

^.

spine

^{P of} ^I ^M ^. ^Be ⁾ ^. section

local

^normal

Ip

Vertices of ^a

flow

^- _spine

2-^{^} ^-^-^-^.^.^-

f-

^.

!

^2- ^- ^- ^-

f ^-

^: ^. ^.^. ^- ^-

y ^•

"

I ^{^} y ^.

i

^' ^•

7 7

...''

> x

\

^> ^x

I ^-

type

^r ^-

type

A

flow

^- ^P ^is

spine ^positive

^if

type ^{

^.^. vertex

^has

^at

^Py

of P ^leastîsÎ of^vertexl ^- ^; ând^.

(9)

M ^: ^a ^closed ^oriented ³ ^- ^manifold

It

⁼

{

Yt } _ten ^: a non ^.

singular

^flow ^on ^M

E (

^I ^D2⁾ ^c ^M ^: ^a ^normal ^section

^{for Z}

→

P

^c ^M ^: ^a

flow

^- _{spine of} ⁽ ^M ^, ^E ⁾ ^.

•

"

÷

,

- is

.

i

.

I

^.

to

^! ^• ^. ^• ^•

^I

• .

graph

3 ^-

regular

i÷

.

I

\

^.

f

^u

M ^- P

§

^• •^. ^.

⁽ ^E. ^TIRE ^{) )}

(10)

Example

Positive

abalone P

c-

S3

•

. . B

if

:÷÷ ^.

^.

^P

^is ^a

^{Nbd I} ^flow ^l

^-^S^-

^type ^spine

^C^P⁾ ^;^P^of⁾ ^the

^flow

^on

^S3

"

i

. ^with ^a

regular

^orbits

whose

^form^fiber^. ^the^a

^positive

^Seifert ^trefoil^fibration^.

(11)

§ I

Positive

flow

^-

spines

§

² ^.

^Contact

³ ^-

^manifolds

§

³ ^.

^Main

^results

§ ⁴

^.

^Tabulation

(12)

M ^: ^an ^oriented ³ ^- manifold Definition

(Î) ^d ^: â Î ^- form on M .

L is a (

positive ⁾

^contact ^form ^on ^M ^⇐ ^d ⁿ ^old ^> ⁰ ^.

def (2) G ^: ^a ² ^-

plane

^field ^on ^M ^.

}

îs â Î

positive ⁾

^contact ^structure

⇐ ^I d ^: a

I positive ⁾

^contact ^form ^on ^M

def

Sit ^.

3

⁼ ^Ker ^a ^.

(13)

Z

Example

ⁿ

(^' I

dstd

:=

d Z ^t ^x

dy

^.

(

^IR

? kerdstd )

mmmm

!!

Bst

'd ^> y

( Esta )

, . .az ,

=

Span III.

^Ey ^- ^a

^IT ⁾

^.

µ

(2) S ^' ⁼

{ ^I

^x ^, ^, ^y^,, xz . Ya )

I

^x , 't _Y_, 't

aft

YE ⁼ _I

}

^c ^IR ^" ^I ⁼ ⁶²

⁾

^.

dstd

¥

_,

⁽

^{Kid Yi} ^- ^Yidxi

⁾

^I

^S3

^.

(

^S3^.

kerdstd ⁾

^. ^TP ^{Tps 's} ³ ^Tp ^S3

mmmm ;

!!

} _Std ^. ^. ^- ^- ^- ^- ^.^-^-^- ^.^. ^- ^. ^. ^.^. ^. ^- ^. ^. ^.

( Gst

^.

d)

_p ⁼

_Tps

^' ⁿ ^J ^Tp

^S3

^.

jeez

(14)

Definition

(

^M ^, ^, ⁵, ) . ( Mz _, 52

)

^: ^contact ³ ^- manifolds

(

Mi

,

es, ) ^I

(

Mz ,

}z

) ^contact

omorphic

⇐ ^I

f

^: ^M , → Mz ^: ^a

diffeomorphism

^s^. ^t ^. ^f-^* ⁽³^, ⁾ ⁼

⁵²

def ^-

contact

morphism

Theorem

( Darboux

1882 )

V-2

^: a contact

form

on an oriented 3 ^- manifold M ,

t P ^E ^M ^,

⇒ coordinates x. Y , Z on an open

neighborhood

Û â ^M ôf p ^st ^.

p ⁼ ( Ô^, ^0,0 ) ând ^d Î u ⁼ dz ^t ^x dy ^.

In

particular

^,

(

p^U^y , Ker^a

⁾

^I

⁽

^> ^V^u^o

Ltu

^.

^esstdlv ⁾

^, ^where ^Vc ¹¹²³ ^: ^an ^open ^set

(15)

Theorem

( Gray

¹⁹⁵⁹

⁾

M ^: ^a closed oriented ³ ^- manifold

{

^at^-

It

^c. go,, ,

: a smooth

family

^of ^contact

^forms

^on ^M

⇒ ^7- _an

isotopy { ^}

t.co ,,, ^s^. t ^.

htt

)

Yt

*

I

Go

)

⁼ est ^V te lo . I ]

.

Theorem

(

^Martinet ¹⁹⁷¹ ^, ^Thurston ^- ^Winkel ⁿ

Kemper

¹⁹⁷⁵ ⁾

Every

^closed ôriented ³ ^- ^manifold âdmits â ^contact

structure ^Problem

^.

14 Classify ^{ isotopy Tight structures

^IM^Cont⁾ ^!!¹¹^u

contact

⁽ ^MÔT⁾ ônÎM ^M⁾

^} ^/

^- ^- ^- ^or ^contact ^om

^orphism

^.

=

.

(16)

Theorem

I Eliashberg

¹⁹⁸⁹

⁾

M

: a closed oriented ³ ^- manifold ^.

+ OT I M

) / _isotopy ^{

^coori ^. ^2.

^plane field

^on

My

coori .

homotopy

^.

Theorem

Tight

^I ^M ⁾

/ _isotopy

^is ^classified ^when

or M ⁼

S3 I Eliashberg

¹⁹⁸⁹

⁾

• M ⁼

I3 I

^Kanda ¹⁹⁹⁷ ⁾

Or M ^=L ⁽ p^, 8 ⁾

I

²⁰⁰⁰

Giroux

, Honda ²⁰⁰⁰ )

@ M ⁼ ^- -2 ( 2. 3,5 ) ^I

Etnyre

^- ^Honda ²⁰⁰¹ ⁾ ^. ^etc ^.

Theorem

I

Giroux ²⁰⁰²

)

M

: a closed oriented ³ ^- manifold .

Cont ^I M )

/ _isotopy

^I ^-^: ^I

{

open book

decomp

^.^'s ^{of 'M}

} / positive Hopf plumbing

^.

(17)

Definition

Ii ) d ^: a contact ^on ^a closed formoriented ³ ^- manifold ^.

Ra

^: ^the ^Reeb ^vector ^field ^on ^M

⇐ I ⁱ)

dd ^I

^Ra ^, ^.

⁾

Î Ô ⁱ ând

def Iii ⁾ ^d I ^Ra ⁾ ⁼ ^I ^.

121 The Reeb flow

Ea of

^L ^is ^a ^flow

generated by ^Ra

^.

2-

Example

ⁿ

I il

I

¹¹²³ _,

_dstd

^:^-^- d z t xd Y

)

.

..

Rasta

⁼

IT

. .

:

: :

:

> Y

X

"

(18)

(2)

(

^S ^' ^, dstd

)

^{^}

:÷:::::÷÷i :: : ÷

^.

Lemma

M ^: a closed oriented ³ ^- manifold ^. do .

d _, : contact forms ^on M .

Rao

⁼

Ra

_, ^⇒ ^Ker ^do ⁼ ^Ker ^d ^, ^.

isotopic ( Proof )

Gray stability

^. ^A

(19)

§ I

Positive

flow

^-

spines

§

² ^.

^Contact

³ ^-

^manifolds

§

³ ^.

^Main

^results

§ ⁴

^.

^Tabulation

(20)

M

: a closed oriented ³ ^- manifold Definition

A

contact structure

}

^on ^M ^is

supported _by

^a ^flow ^-

_spine

^Pc ^M

if ^I a contact form ^d on M

St

^.

(ⁱ )

9

⁼

Ker

^a ⁱ ^and

Ciii P is a

flow

^-

spine ^of

^I ^M ^,

Ike

⁾ ^.

the Reeb flow of d

Remark

_We _consider

}

^modulo

isotopy

^.

(21)

Example

Positive

abalone P

c- ^S3

•

. . B

if

:÷÷

^" ^: ⁱ

^.

^P

^withîs^→ ââ

^regular ^{Nbd I}

^orbits

^flow P ^l ^whose supports

^-^-^form^S

^spine ^type

^C^fiber^P^.⁾ ^the^;^P^of^a

⁵

⁾ⁱ ^trefoil^Seifert^the^std ^.

^flow

^. ^fibration^on

^S3

(22)

Theorem

(

Îshii ^- Îshikawa ^- ^K^. ^- ^Nao ê

)

I

.

V

positive

^flow

spine

^P ^c ^M ^,

⇒ a contact structure

5 supported by

^P ^.

2

. P ^c M ^: ^a

positive

^flow ^- ^spine ^.

Go . 9_, ^: ^contact ^structures ^on ^M

supported by

^P ^.

⇒ Go ⁼ ^G_, .

isotopic 3

.

V _contact structures

}

on M ^.

7- a

positive

^flow ^-

spine

^P ^c ^M

supporting ⁵

^.

→ ^: ^a closed Moriented ³ ^- manifold ^.

{ _positive flow

^- _spines ^of ^M

)

^→ ^Cont ^l ^M ⁾

/ _isotopy

(23)

Theorem

(

)

I

.

V

positive

^flow

spine

^P ^c ^M ^.

⇒ a contact ^structure

5 supported ^by

^P ^.

I

Idea )

Constructive ^.

(

Similar to Thurston ^- Winkel ⁿ

Kemper

^'s

argument

^. ⁾

P ^c M ^: ^a

positive

^flow

spine

^.

•

Using

^the stratification ^V ^{CP )} ^c ^{SCP )} ^c P ^c M _,

we define ^a ^L ^- form

7

^on ^M ^s ^. ^t ^. ⁷ ^nd ⁷ ^I ⁰ ^.

÷

,

I

we use the

positivity

^{of P} ^here ^.

⁾

Y ^÷ dz ^. ^. :

. . Z

•

7-

p

^: ^a

"

I ^. form

"

on P w/

dp

^> ⁰ ^on ^P ^.

(24)

→ L

§

⁺ Ry ^: ^a

required

^contact ^form ^on ^M ^.

- - I

T I

extension of Sufficiently

P ^to ^M large

Why positive

^flow ^-

^{spine (}

^l ^-

type

⁾ ^?

vertex

t

region

w

I

✓ 't

⁷ ^÷ ^d

Y ^÷

dw

(25)

Theorem

(

)

2

. P ^c M ^: ^a

positive

^flow ^- ^spine ^.

Go . 9_, ^: ^contact ^structures ^on ^M

supported by

^P^.

⇒ Go ⁼ 9_,

isotopic

-

(

^Idea )

Go ⁼

Ker

^do _, ^es,

= Ker di .

Using

⁷ ^, ^we ^find ^a ^I ^-

parameter family ^{

^at

^}

_te _I _0,1 ₎ ^of

contact

forms

on M .

The conclusion

follows from Gray Stability

^. ^I

(26)

Theorem

(

)

3

.

V _contact structures

}

^on ^M ^.

7- a positive flow ^-

spine

^Pc ^M

supporting

⁹ ^.

I Idea )

}

^→ ^an _{open book} ^de _comp^. of ^M

supporting

^G ^.

[ Giroux ²⁰⁰⁰ ]

→ a

positive

^flow ^-

spine

supporting

^G ^. ^a

(27)

The

positivity

^condition ^for ^flow ^-

^spines

^is

really

^essential ^?

. . . Yes ^!

Theorem

(

)

M ^: a 3 ^- manifold

admitting

^a

^tight

^contact ^structure ^.

⇒ ^I a non ^-

positive

^flow ^-

spine

^P ^of ^M ^s^. ^t ^.

lil P does not

support any

^contact ^structure ^; ^or

Iii ) P

supports

^two ^non ^-

^isotopic

^contact

structures

.

(28)

§ ^I

Positive

flow

^-

spines

§

² ^.

^Contact

³ ^-

^manifolds

§

³ ^.

^Main

^results

§ 4

^.

Tabulation

(29)

M

}

^: ^a ^contact ^structure ^on ^M

Definition

C I M .

5)

f

min

f

^# ^of ^vertices ^{of P} ^I

Psuipsp.at?nogsitgive

^flow ^- ^spine

)

Recall

c ( M , 9) ⁼ ^I ^⇒ ^I ^M ^. 9) ⁼

(

^S3^, esstd )

(30)

CIM , } ⁾ ⁼ ^I ^CCM ^. ^es ⁾ ⁼ ²

a. .

( ^S? Esta ) double br^.

cover

•&

my

along ^a singular _•

f

•

fiber I ^{unknot )}

P

Be •

( ^S3^. esstd ⁾

I

E. ^IT ^' ^' ^{12-2 )}

) ^I

^E^, ^-1112-27

)

( double br ^.

cover

• •

along ^a regular

••

⁽ ^43,2 ⁾^, ^{} tight} ⁾

f.) ^&

^, ^fiber ^I ^trefoil ⁾

^⑨ ^g)

^• ^.

• . . ^#

⑧ •

g

Be •

(

E. ^IT ^'^' ^{12-2 )}

) ^I

^I ^, ^-111527

) I

E. ^IT ^'^' ^{12-2 )}

) ^I

^E^, ^-111527

)

I

_fiber

wise

•

^.

⁽

^{IRP ?} ^estight⁾

Seifert fibration of S3 ^" _coil ^"

surgery

•.

^•

with ^a

regular

^fiber ^a ^trefoil ^.

^⑧ µ #

^.^. _• ^.

(

E. ^IT ^'^' ^{12-2 )}

) ^I

^E^, ^-1112-27

)

(31)

CIM .es )

I

(

^S3^, ^{Esta )}

( ¹¹²¹³³,

Hight

)

Z

(

^L I 3,2 ) ^, ^estight

)

(

^L ^13.1 ⁾ , estight )

(

^L 14,3 ) , estight )

* to be confirmed

3

(

L

15,2 ^Remark

^# ^of

^§ ^tight

^IRP ^? ^contact^L ^13,1⁾⁽ ^I^113,2⁽^structures⁾ ⁾^L^,

^2.3.31

^14,1^est^,^ight⁾

^{Hight )}

^on^L⁾^14.31

^lens

^115.1^spaces⁾ ^45,2Î⁾ûp^L^to^15.4îsotopy⁾ ^- ⁾ ^:

I I 2 I 3 I 4 ² ^I

(32)

Theorem

(

)

The

complexity ^of

^the ^link ^of

singularity

^of

f-

I ^x ^, Y , 2- ) ⁼ ^x ^' ^t y ³ ⁺ ^zn

with the contact structure given

by

^the

complex tangency

^is ^at ^most ⁿ ^.

It ^is

exactly

ⁿ ^if ⁿ ^E ⁵ ^.

Positive Yuya

Positive

flow

spines manifolds and contact 3

Yuya Koda I Hiroshima

)

Ippei

(

I

Intelligence of

dimensional Topology May

2019

RIMS

{ positive flow

)

/ isotopy

§ I

Positive

flow

spines

§

Contact 3

manifolds

§

Main results

§ 4

Tabulation

§ I

Positive

flow

spines

§

Contact

manifolds

§

Main

§ 4

Tabulation

M

manifold

¥

I

Ite

singular

( generated by

singular

field

)

compact

surface

section Eff

I

X ( positively

)

E

.pe#Ty .gg?i.RqfirsttimetomeetE

( IT

Ip Mu

I {

)

4

)

-2

Tip

2-2

-2

Ttp

2E &

Tap Ttip Ftp Tp

(

2-2In 22

discontinuity

of

IT

I

Tip tp

)

I

{

Cp

spines manifolds ^and ^contact ³

Yuya ^Koda ^I ^Hiroshima

⁾

⁽

Intelligence ^of

²⁰¹⁹

{ _positive flow

/ _isotopy

^Contact ³

^manifolds

^Main ^results

§ ^I

^Contact

^manifolds

^Main

§ ⁴

^Tabulation

( generated ^by

^singular

^field

⁾

^surface

section ^Eff

^I

^X ( positively

⁾

^E

⁽ ^IT

Ip ^Mu

^I ^{

⁾

⁴

₎

^2-2

^-2

^2E ^&

Tap ^Ttip Ftp Tp

⁽

^2-2In ²²

^IT

^I

⁾

^use

^Smoothing tp

^:p

^ii.

^f

^smoothing ^→

^}

^I

spine ^of ^I

^y

f ^-

spine ^positive

type ^{

^has

^Py

^{for Z}

^I