3次元多様体から平面への安定写像の1次半局所不変量

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Kyushu University Institutional Repository

3次元多様体から平面への安定写像の1次半局所不変量

山本, 稔

http://hdl.handle.net/2324/454845

出版情報：Kyushu University, 2003, 博士（数理学）, 課程博士バージョン：

権利関係：

(2)

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FIGURE 1

f-1(S(J_i)) fo(S(Jo)) !1 (S(J1)) f-1(S(J_1)) · fo(S(Jo)) !1 (S(f1))

0

• <>

^{' - - /}

,,-..._ ^>-< ><

(1) (2)

f-1 (S(J_i)) fo(S(Jo)) !1 (S(Ji)) f-1 (s(f_i)) fo(S(Jo)) !1 (S(J1))

~ A >I ~

(3)

(4)

f-1 (S(J_i)) fo(S(Jo)) f1(S(J1)) f-1 (S(J_i)) fo(S(fo)) Ji (S(J1))

' - - /

>-< A * ^X

,,-..._

(5) (6)

FIGURE 2

(3)

>

~

(1)

<>

(4)

>t * :t-<

>I<

(7)

· (10)

*

(13)

>

~

(2)

(5)

+

(8)

(11)

(14)

.FIGURE 3

~ ^~

r~ ^-;:::-- ^~

V· <

(3)

(6)

(9)

(12)

(4)

•

(1)

0 c.· ) ^J(S(f))

0

(1) if q is a D fold point

•O O•

. l . : ~ O ..J

0 0

J(S(f)) ••

._J 0

L.

(2) FIGURE 4

u n

C x f(S(f))

)(

(2) if q is an I fold point

-~%sx

J ( S ( f ) ~ x

X _] )C l_ • )C

(4)'if q₁and

q

₂correspond to (5) if q₁and

q

2 correspond to a D & D nodefold a D & I nodefold

FIGURE

5

(3)

•(

C

< C

. O<

f(S(f))

(3) if q is a cusp point

X;=( ;=(X

..J

. f(S(f)

xx~ xx

(6) if q₁and q2 correspond to an I & I nodefold · .

(5)

<

(1)

f-1 (S(J_i))

• ( ~

oc

e x ~

C

f-1(SU-1))

k

f-:1(SU-1))

~

X >='.

111

(2)

FIGURE

6 fo(S(fo))

L

C

( 1) if

Yo

corresponds to lips

fo(S(fo))

C

• C ~ • C

o c ~ o c ex c < ex

(2) if

Yo

corresponds to beaks

fo(S(Jo))

0

.

( 3) if

y₀

corresponds to a D swallowtail

. fo(S(fo))

\ ) ( x

A

X X

( 4) if

y₀

corresponds to an I swallowtail

FIGURE

7. (a)

>-<

(3)

Ji (S(f1))

< <

fi(S(J1))

0 00

!1 (S(f1))

)OC

)<)

oc

X V ^r, X

(6)

71

f-1 (S(J_i))

fo(S(Jo )) fi(S(J1))

•( 0

i ^co

^C• ^•CO

co

^C^• ^{•( 0}

^co

^C^•

oco

^C

oco

<·

oco

_<

C

oc

0(0 <O O<O

O<O

C• C•

(5) if Yo

corresponds to a cusp-plus-D fold (type 1)

f

-

1(SU-1)) fo(S(Jo )) !1

(S(J1))

• ( i

^C ⁽^. ^•⁽ ^•⁽ ^C•

oc c o oc

_<_·

oc

<O

co oco

O< · < O<

C

co

O<

C• (•

( 6) if Y o

corresponds to a cusp-plus~D fold (type 2)

f-1 (S(J_i)

)

fo

(S(Jo))

!1 (S(J1)

)

•(;:: \

^C^X^{C X} ^{• C}^;:: ^'^{C ;:'.}

^ex

0(;:: C)( 0(;:: <X 0 C ;:'. < )(

C )(

0<;::

< ><

0<;::

_C_><

O<;:: ^{C )C}

ex

^{C X}

(7) if Y o

corresponds to a cusp-plus-I fold

r

.

FIGURE

7.

(b)

(7)

f-1 (S(J_i))

O ·G

0

~

f-1 (SU-1))

·G

0

- ~

f-1 (SU-1))

9 -G

00

, o ~

fo(S(Jo)) ·

O•k

⁰

O O

⁰

•

II (/J • • •

(8)

if

Yo

corresponds to a

D & D

tacnodefold (type 1)

fo(S(Jo))

~~~

⁰

(9) if y₀corresponds to a D & D tacnodefold (type 2)

fo(S(Jo))

o- ~o-

00 00

• 0 0 , • . • 0

Ji

(S(Ji))

O•

0

!1 (S(J1))

O• 0

!1 (S(J1))

O•

00

•O

(10)

if y₀corresponds to a D

&

D tacnodefold (type 3)

FIGURE 7. (c)

00

O• 0

0

O•

00

•.O

(8)

f-1(SU-1)) o x 0

o><

f-1(SU-1))

xG

V r,

v ~

• r,

f-1(SU-1))

)GXb

fo(S(Jo))

o x ~ o , c ox

o>< . o>;

• V V • V

r\ , n •X ^0.

( 11) if y₀corresponds to a D & I tacnodefold (type 1)

fo(S(Jo))

x ~ x

X

. ; O)< •X •

V .v

r, r,

r, V

• ><

( 12)

if ^y0 corresponds to a

D

&

I

tacnodefold (type 2)

fo(S(Jo))

xx,I"" ,xx

)C;;:; ~ ) C X

x>< XX xx x><

)(X

)( ><

X

><

( 13) if y₀corresponds to an I

&

I tacnodefold

FIGURE 7. (d)

O)C

f1(S(J1)) ox

X

V r,

• ><

)(X

X

><

(9)

O•

00

•O

0 O•

O•

0

O• ><

00 ><

0

f-1

(S(J_i)) fo(S(fo))

00

\ O• O• O•

00

•O •O

•O O• 00 •O

O• 00 ^•O

(14) if y₀corresponds to a D & D & D triplefold (type 1)

f-1(SU-1)) fo(S(fo))

O•

0

-'7-~,---'7-•00 00

0 •O

0

7"""c:r-""77-~"7" • 00

0

••O 00

•O

(15) if Yo corresponds to a D & D & D triplefold (type 2)

f-1(SU-1)) fo(S(fo)) !1 (S(f1))

ox

oox

• 0 >< --:>-,:r--.--'----,- • )(

.

^{) (} ^OX ^{0 ) (} ^{• ) (} ^OX_ox^•OX _{0 ) (} ^{••)( )(}_•)(

( 16)

if Yo corresponds to a D & D & I triplefold ( type 1)

FIGURE

7.

(e)

(10)

f-1 (SU-1))

O•X oo>.<

\ oox

^{O• )<} ^O•V

ox

00 ^.-,

•• v

•OX o><

r , ~ 0 ><

• >< ^•O)C

• >< • >< • O)C

V V

r, r,

X ^•)( X ^•)( X

(17)

if

y0 corresponds to a D & D

&

I triplefold (type 2)

ox;<

0 )( ><

• x><

f-1 (S(J_i))

O^{v v}r, r,

• )( ><

..::,..:::,.,.____._...:::,...,,e:::;_"

fo(S(Jo))

ox;<

\ o><x

oxx ox><

0 )( ><

• x ><--- • ><x

!1 (S(J1 ))

oox\

^OOX

O•X 00)(

• 0 )(

• •)( 0 )(

•)(

) (

)( >< ><x

-::i':::::r--.-r~-:f> • >< )C

XX X)(

X)( )( ><

)CX )( )(

(18) if

Yo corresponds to a D & I & I triplefold

><

^X

><

>< )( ><

vvv

r , r , r ,

X X __

><---'---''----,,1=---·

X )(

><

fo(S(Jo))

>< )( >< >< >< )(

)( )( X )( )C )( )( X )( )( )( ><

)(X

xv

r,

• X)C )( ) ( X )( XX

!1 (S(J1))

v~v\v~x·

r i r \ n r i n

>;xx

VX)C

r, )<)<)(

---:::,,1-'-'-'-"T="->.s::---X

>< )

^C

XX)(

)C )C X

)( >< )(

)C X )(

)( )( X _{)( )( )(}

)(X)C )(X)C

( 19)

if

^y0 corresponds to an I & I & I triple fold

FIGURE

7.

(f)

(11)

<

(1)

•(

oc ex

(2)

J(S(f))

(1) if yo corresponds to a goose

f(S(f))

l\x

)o(

)0(

)()(

:::>C

X X

\..J

"

( 4) if yo corresponds to I gulls

•

(3)

>-<

(4) (5) (6)

FIGURE 8

f(S(f))

•C7cl . ^-<

0(

ex

C

0

(2) if Yo corresponds to a butterfly (3) if Yo corresponds to D gulls

f(S(f))

*

^7':"--,,' ^"-J

- ""

^/',.

\._ *

^:::::,L^{/ ' , .}

^y_

^/',.

(5) if Yo corresponds to a Dt (6) if Yo corresponds to a D

4

FIGURE

9.

(a)

(12)

J(S(f))

i·<

• C ⁰⁽

(7) if yo corresponds to a lips-plus-D fold

J(S(f))

. re

vc

X•c$t·c

><

OC )CO(

XCX XC X< )(0<

X C )( C

(10) if yo corresponds to a beaks-plus-I fold

J(S(J))

. " I t~ox

~

X ("\ V ("\ ox

' n

(13) ifyo corresponds to an I swallowtail-plus-D fold·

f(S(j)) J(S(J))

1

^x<

^•

⁰⁽

⁽ ^$0·(

⁰⁰⁽

>< )( C O< C •< 00<

XC •( 0(

(8} if Yo corresponds to (9) if Yo corresponds to

a lips-plus-I fold a beaks-plus-D fold

J(S(f)) J(S(J)) .

•O

~

.

·

~

⁰ ^O^•

^><o

^><^•

^><

^X ⁾⁽

^co

⁾⁽^•

( 11) if Yo corresponds to (12) if Yo corresponds to a D swallowtail-plus-D fold a D swallowtail-plus-I fold

f(S(J))

xx

>< X

"

(14) ifyo corresponds to an I swallowtail-plus-I fold

FIGURE 9. (b)

(13)

f(S(f))

(15) if Yo corresponds to a cusp-plus-cusp (type 1)

f(S(f))

O•( C

. 00(

•( •0(

( oc

ex ( 18) if yo corresponds to

a cusp-plus-D fold tangency (type 1)

f(S(f))

\O •

0

•O •

0

(21) if yo corresponds to a D & D flecnodefold (type 1)

f(S(f))

xx

)CX

(24) if yo corresponds to an I & I flecnodefold

J (S(f))

C ex

( 16) if Yo corresponds to a cusp-plus-cusp (type 2)

f(S(f))

.

⁽ ^.

0(

•( •0(

( 00(

oex ( 19) if yo corresponds to

a cusp-plus-D fold tangency (type 2)

f(S(f))

#

⁰

•O •

00 O•

(22) if Yo corresponds to "

a D & D flecnodefold (type 2)

FIGURE

9.

(c)

f (S(f))

(i

7) if yo corresponds to a cusp-plus-cusp (type 3)

J(S(f))

X< . )<O(

>< ~ .

X•C

x·c xoc

)(OC )( C

)CCX

(20) if yo corresponds to a cusp-plus-I fold tangency

J(S(f))

/"\

#

Q)(

• >< . )(

·x

X

(23) if Yo corresponds to a D & I flecnodefold

(14)

J(S(f))

( ~ ( .

• C • C

OC. OC

O•C ••< •OC 00(

(25) if Yo corresponds to

a D & D nodefold-plus-cusp (type I)

f(S(f)) 1

000(

000< 00•(

0:¥:( 00(

•OC , , O• C

oc 0(

• C ••< • C

C

a D & D nodefold~plus-cusp (type 4)

J(S(f))

a:

^D^&I nodefold-plus-cusp (type 3)

f(S(f))

OC

(26) if yo corresponds to

a D & D nodefold-plus-cusp (type 2)

J(S(J))

J (S(J))

jooc

•CO~O< O•CO•C

C OC

•( < •OC

oc

a D & D nodefold-plus-cusp (type 3)

J(S(J))

)( (

(29) if Yo corresponds to (30) if yo corresponds to

a D & I nodefold-plus-cusp (type 1) a D & I nodefold-plus-cusp (type 2)

J(S(f)) J(S(J))

. I

^xoc

~;~ ~~:

•)CC •X< OX(

0)( ( .

) ( ) ((

(32) if Yo corresponds to (33) if yo corresponds to a D & I nodefold-plus-cusp (type 4) an I & I nodefold-plus-cusp

FIGURE 9. (d)

(15)

f(S(J))

(34) if ^yocorresponds to

a D & D tacnodefold-plus-D fold (type 1)

f(S(f))

•) ( ·>< V n .v .~

(37) ifyo corresponds to

a D & D tacnodefold-plus-1 fold (type 1)

f(S(f))

V r,

OX•

)CO XO )(.

a D & I tacnodefold-plus-D fold (type 1)

f(S(f))

. xx

_·_XX

\

•XX 0 XX O)()C OXX OX X

O)(X

(43) ifyo corresponds to

a D & I tacnodefold-plus-1 fold (type 2)

J(S(J))

O•

0 00 •O 0

a D & D tacnodefold,plus-D fold (type 2)

f(S(f))

•)(

.v r,

•)( 0)( _ox

ox ·><

(3 8) if yo corresponds to

an I & I tacnodefold-plus-D fold (type 2)

f(S(J})

>< >< 0

. ·><o

( 41) if Yo corresponds to

a D & I tacnodefold-plus-D fold (type 2)

J(S(f))

xx xx

)C X

an I & I tacnodefold-plus-D fold

FIGURE 9. (e)

J(S(f))

j

^O•

•O O 00 •00

·oo

0 00

O• _O_• _O•O

a D & D tacnodefold-plus-D fold (type 3)

f(S(J))

ox

0•)( 0)(

ox

^O^•X

ox

an I & ltacnodefold-plus-1 fold (type 3)

f(S(f))

X)( )()(

xx

^XX

. )(X

( 4 2) if Yo corresponds to

a D & I tacnodefold-plus-1 fold (type 1)

f(S(f))

·xxx

><><x x v v

XX)C X><X ""' n

)(X)C

an I & I tacnodefold-plus-1 fold

(16)

J(S(f))

0 00 •00

•O

( 46) if yo corresponds to

a D & D & D & D quadruplefold (type 1)

J(S(f))

(49) ify0 corresponds to

a D & D & D & I quadruplefold (type 2)

J(S(f))

OX)C O•)O O)oc o><x o><x

(52) ifyo corresponds to

a D & D & I & I quadruplefold (type 2)

J(S(f))

xxxx

>< XX)C ><><><x

>< ^X)O

)( ><>< ><

)()()( X )C XX X

)ooc

x i

xx><><

)OCXX

an I & I & I & I quadruplefold

J(S(J))

•O

( 47) if yo corresponds to

a D & D & D & D quadruplefold (type 2)

J(S(J))

O•)C oox

•O)(~ ~ ~ + - - •OX 0)(

ox

· (50) if Yo corresponds to

J(S(J))

FIGURE

9.

(f) ·

J(S(f))

X ^Vn •><.

ooox

oo><

•OX

( 48) if yo corresponds to

J(S(J))

OOX)C

OO)OC OOX)( oovx

. O• )C)C r.

O_nvv _r.

VS.)

•r,r,

(51) ifyo corresponds to

J(S(J))

O>(X)C

vvv

r , r , "

)( )( X ()( ><

l xxx

^xv" r , ^V )CXX

(54) if ^y0 corresponds to

a D & I & I & I quadruplefold

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•

0 c.

(/)

ro

(/)

uo,o

0 n2

)

O·

• 8

0 8

(1)

cg

0 0

0

11

0

no,1

0 II3

)

FIGURE 10

FIGURE 11

•O -0

ua CX)

08 00

nI,1

CID

(2)

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J(S(J))

·-<) J

ma(!)

A 8

00 00

Illd

• OO ~t 08

(1 08

000

08 ml,a .

2

• O • O ^•

OO A OO

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00 ^<) 00

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m a(b) IIIb rue

O •

^O^•

⁰⁸

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^'Z)^.

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^<)^•

· 08 >r <J8

000

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₀₀₀

().

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I

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2

ml,a

I

00 00

•O O >i .

^r_<)() _.

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FIGURE 12. (a)

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• 0 ~ • 0

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mo,1

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3

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FIGURE 12. (b)

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12.

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FIGURE 13. (a)

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800 •O 80

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