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Coeur de l’invariant de Casson et cobordismes
d’homologie
Kristell Dequidt Picot
To cite this version:
Kristell Dequidt Picot. Coeur de l’invariant de Casson et cobordismes d’homologie. Mathématiques
[math]. Université de Nantes, 2005. Français. �tel-00009786�
HAL Id: tel-00009786
https://tel.archives-ouvertes.fr/tel-00009786
Submitted on 20 Jul 2005
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 w}~ Σ Ó578V†RÓÏ1Ҕyv98Vv9ˆpt[yvx~R8v95578‡8Ï$w‚8x5•~R698„8ƒ´8x5ÏR8 g †Žy|57†0UÏR„ A 5
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>Å1فÚNØÌËuÌ gËÉ ÆRÙ>ÎÝÙÊØ]ÉÝÌÊÇ
vx{zy†Ž†R89†V„ , w‚†R~Rtw‚89†„89†VM
„‡w-,Ug 69ˆ;Ï$tÛw‚†Rˆ;89†„8 Σ &uÓw tÏR69†R8xÏ$é´8x5u~{Ý,-Ï$w‚8x5•~Žy|~$w‚5 :
Mg = π 0 (
w-,
g
+
(Σg )).
* w-57†åÓ5;„‡w‚†"&uÓ78 D2 †$ÓÏå{zy †$ÓÏ1ÒÝyv98 Σ y|w}57†$w&•Ó&, Ó5ptUw}5•~0[y†Ž8 ∗ ∈ ∂D2 A &8
v9ˆpt{‚6xˆ;8x5u~Žy|w}ÏR8²„8n{Ý, w}5u~R6xÏ$w‚8xÓÏG„8 D2 89†$~g Ó578n†$ÓÏ1ÒÝyv98ntbw}5u~R698 Σ v9ˆpt[yvx~R8
g,1
v95578 8(Ï$w‚8x5u~R698G„8 ƒ´8x5ÏR8(ƒ XÓ578Gv9ˆptb>†Žy|5u~R8G„8 UÏR„ A
) + 3 /A;?/0>D3 '2' 2' &8 Å1Ù>Ú¹ØÌ;˕ÌXËÉ ÆRÙ>ÎÝÙÊØ]ÉÝÌÊÇ Mg,1 „8 Σg,1 89†$~V{‚8ƒ>ÏRÓtU8(„89†
vx{zy†Ž†R89†„ , w‚†R~Rtw‚89†(„89†„‡-w , 69ˆ;Ï$tÛw‚†$ˆ;89†(„8 Σ &uÓwåw}57„‡Ów‚†R8x5•~{Ý, w‚„8x5u~$w}~R6²†$ÓÏ
g,1
{‚8 UÏR„ ∂Σ :
g,1
-w , (Σ ; ∂Σ )).
M =π (
g,1
0
<
g,1
G
g,1
w}~ Ó5¯6x{‚6xˆ;8x5•~J„8²w-,
&5©57~R8xώy ϕ †Ž5 8 ~R8x57†Rw‚52 Σg
g,1 ; ∂Σg,1 )
t[y|Ïn{Ý, w‚„8xϕ5u~$w}~R6†$ÓÏX{‚8!„‡w‚† &•Ó78 D(Σ
%
5


~
$
‚
w
x
8
•
5
~²y|w}57†RwÓ5 ;
ˆ Ï$tÛw‚†$ˆ;8÷v¹y|575w &uÓ78
2A
R
†
Ó
‘
Ï
1
Þ
9
8
x
v
$
~
w
Ò
A
Mg,1 → Mg
+935+T)^4C;\79a5),8 a D),>da 7 7 / +T>6;\> /0798 M
 w}~ γ Ó578 v9ÓÏ$b8†$w}ˆpt{‚8
Ҕ8xÏ$ˆ;698(†$ÓÏ Σ 8x~†Rw}~ I × S 1 Ó5鴁w‚†Rw}5[y|ƒ´8(ÏR6xƒ>Ó{}w‚8xÏt[y|ώy|ˆ;6x~$ÏR6JÏ$w‚8x5•~R6J„8 γ
„7y|57† Σ A &8g,1~ 7w‚†$~„8( 8xÛ5 n„‡ÏRw}~R8G{‚8 {‚5ƒ²„8 γ 89†$~{Ý, w‚„8x5u~$w}~R6J8x5„8xÛ7ÏR†„8
g,1
8x~V89†$~V„6#75w*†$ÓÏ I × S 1 v9ˆpˆ;8G„7y|57†{zy #7ƒ>ÓÏR8 G A G t[y|ÏG:
I × S1
g,1
I × S1 → I × S1
(t, eiθ ) 7→ (t, ei(θ−2πt) )
τγ
G AG
γ
7w‚†$~„8V 8xÛ5 „‡ÏRw}~R8 A ,-Ï$w‚8x5•~Žy|~$w‚5„8{zy #7ƒ>ÓÏR889†$~v98x{}{‚8Ó7†$Ó78x{}{‚8 A
#+9>&)917 ' 2' E ) 7+ 3 h/0] &>D)^/08 *Ì!>Å1فÚNØÌ÷˕ÌaËÉ Æ$Ù>ÎÝÙÊØbɔÌÊÇ Mg,1 ÌÈÇÈÎ
Ìx͕ÌÈÍ]Ë>Å$Æ ‡Ø ׁŠ‚ÌÊÇnÎ 0ÉúÇÈÎâÇnË•Ì XÌ •Í }Ì }فÍ3˕ÌÊÇ (3g − 1) RÙÚ Å RÌÊÇ 3ËuÌÊǎÇÈÉúÍbƎÌÊÇJÇÈÚuÅ
}× Ú $Å Ì
GفڕÇJ×ÈØ´ØÌ }ÌxÅ1فÍ7Ç ŽÌÊÇJÎ 0ÉúÇÈÎâǂÌÊÇJƒ´6x576xώy|~R8xÓÏR†„8( w‚v"´: Ï$w‚†RÛ+
G A<
+ 6x576xώy|~R8xÓÏR†„8G w‚v":´Ï$w‚†$Û
./y|{}Û78xÓÏR8xÓ7†R8xˆ;8x5u~ @{Ý, Û78xÓÏR8ayvx~$Ó78x{}{‚8!5 578a†Žy|w}~Gt[y†J„69v9ˆptb>†Ž8xÏJÓ5 „‡w Ò Ò”69ˆ;Ï$tÛw‚†$ˆ;8D&uÓ78x{‚v95&uÓ78(8x5tÏR „‡Ów}~„8 ~ 7w‚†$~R†„8( 8xÛ5 A
578v957†R6 &uÓ78x57v98(575~$Ï$w}é wzy|{‚8(„‡Ó~$Û769Ï"xˆ;8 G A G A <p89†$~(:
#+9>&)917 ' 2' ( > M ÌÊÇÈÎ&؇ׁŠx×>ÉâÎ Ø‡ÙÚ Å g ≥ 3 0/ "8+( /0.("( (7 % ( (7+"*1 % D2-/ 1&+')8( * /0"')C,- > .2/0 1 :
<$- % 8/@ /0+ $& /31 2 I × S &A > 6' &') 1 .$& >0> /0 2 / ( 8 >&% /0& /@D 2-
:
Σ " 8 > )'68 "$/ )(;+ ! / $ * / , (7 3/@2&/ C * > "!#8$&
5 > "!#8$&
A 282 /0 &1& 2 4+ g 5 $+')(7+8D 2- 9782 /0 "/ & Σ+3C 4 & " +
!#/ &B2 8 > $ 2- > "!#8$& A 2 2;:
g,1
2
1
g,1
g,1
<><
];?/A>d3 7
8?a5)
>
&8÷ƒ>ÏRÓtU8!ґ57„7y|ˆ;8x5u~Žy|{
Mg,1
π1 (Σg,1 , ∗)
∗ ∈ ∂Σg,1
„
8
9
8
$
†
a
~
}
{
}
w
Ï
R
3
8
„
¬
8
ώy|5ƒ 2g v957†$w‚„6xÏR57†!{zy©[y†R8 x y
F := π1 (Σg,1 , ∗)
Σg,1
i
i
R
Ï
x
8
t
R
Ï
9
6
R
†
x
8
u
5
R
~
9
6
(
8
[
t
|
y
V
Ï
z
{
y
7
>
ƒ
Ó
R
Ï
(
8
x
v
w
|
y
t
Ï 9† A
1≤i≤g
#
- "
yi
xi
,šyvx~$w‚53„8
G AH
+ 6x576xώy|~R8xÓÏR†„8{zyX[y†Ž8Ó7†$Ó78x{}{‚8(„8
∗
π1 (Σg,1 , ∗)
†$ÓÏ{‚8 ƒ>ÏRÓtU8 Ҕ57„7y|ˆ;8x5•~Žy|{åyp6x~R6„69vxÏ$w}~R8(t[y|ÏVw‚8x{‚†R8x5 :
#+9>&)917 ' 2' e /079-08\793 Mg,1 ÌÊÇÈÎÉzÇ9فá;فÅPØ ‡ÌJׁÚÇxÙ>ڕ$Ç Å$فÚNØÌp˕Ì
xÙ>ÅÊá²Æ ؇ׁ
Å ‚ÌÊÇJׁڇÎÝفá;فÅPØ •ÉúÇÊá²ÌÊÇ *à´×Í]Î ‚!
Ì ŽÙánáXڇÎÝ×>ÎPÌÈÚ Å
Aut π1 (Σg,1 , ∗)
Ä
>
×
Å
× ÇÊÚ ÉâÎP$Ì nÍbفڕǩÍ]Ù´ÎPÌÈÅ1Ù>ÍÇ
Ž Ì
ωg = [x1 , y1 ] . . . [xg , yg ]
Aut0 (π1 (Σg,1 , ∗))
Ç9فÚuÇ >Å1فÚNØÌ ];?/A>d3 7
Mg,1
8?a5)
~R57† α ÔâÏR89†$t A β Õ{zynvx{zy†Ž†R8„ , Û7ˆ; H1 (Σg,1 , Z)
i
Ô
â
R
Ï
9
8
$
†
t
Õ
b
t


Ó
Ï
{‚ÏR† i
A yi
xi
1 ≤ i ≤ gA
89†$~pÓ578[y†Ž8†$èuˆpt{‚89vx~$w &uÓ78„8 H (Σ , Z) A ,šyvx~$w‚5'„8
(αi , βi , 1 ≤ i ≤ g)
†$ÓÏ H (Σ , Z) 89†$~w}57„‡Ów}~R8t[y|Ï v98x{}{‚8†RÓÏ π (Σ ,1∗) A g,105at[y|Ï$~$w‚vxÓ{}w‚8xχ8x{}{‚8
Mg,1
{zy|w‚†Ž†R8 w}5•é>y|Ï$wz1y|5•~R8Gg,1{zyXґÏ$ˆ;8†Rèuˆpt{‚89vx~$w &uÓ78(„ , w}5•~R8xÏR†Ž891vx~$w‚g,15 A uw {Ý,-557~R8
{‚8Jƒ>ÏRÓtU8X†Rèuˆpt{‚89vx~$w &uÓ78!Ô &uÓ78J57Ó7† tbÓÏ$ÏR57†w‚„8x5•~$-w Sp H ⊂ Aut H1 (Σg,1 , Z)
#[8xÏ8 Sp(2g, Z)Õy|{‚ÏR†
(*),> >&8?/A;?/0>D3' 2' ! Éâáp× •Ì¬ËuÌ M ˴ׁÍÇ Aut H (Σ , Z) ÌÊÇÈÎ ‚Ì Å$فÚNØÌ
{‚ƒ>w‚8G„8 {zypv9MÓÏ$g,1
b8
Sp H
&857¹è•y|Óp„8{zy†$ÓϑÞ$89vx~$w‚5
8x~V89†R~V57~R6 T A
g,1
1
g,1
Mg,1 → Sp H
g,1
89†$~0y|ttb8x{}{‚6ÇxÙ>ڕÇ$ Å$فÚNØÌË•Ì Ù ÅRÌ ¶É
B\`!\FUHJORZbcdI0Z H!MPeJ\cd\VKR\VfVOŽI
Z*> >&)/A8,1 7T87; ] #-2/23 ),798 #>d1 >D-0> D/07
Ó578!†$ÓÏ1ÒÝyv98²v955788
v9ˆpt[yvx~R8 Ï$w‚8x5•~R698„8ƒ´8x5ÏR8 g ≥ 0 •†Žy|57†
Ó578v9ˆptb>†Žy|5u~R8
„8 UÏR„ Ô Σ = Σ Õ A ~R57† H †R5tÏR8xˆpw‚8xσ>ÏRÓtb8(„ , Û7ˆ;{‚ƒ>w‚8a:
g,1
H := H1 (Σ; Z) A
) + 3 /A;?/0>D3 ' E 0' 5RÙ $Ù>Å1ˁÉúÇÊá²ÌË Ù>ápÙ Ù >ɔÌÇÈÚuÅ 89†R~ Ó5(~$Ï$w}t{‚8x~ − +
 M 89†$~ 3Ýé>y|Ï$w‚6x~R6(v9ˆpt[yvx~R8GÏ$w‚8x5u~R698(ˆJÓ5w‚8G„8„8xΣ$Ó t{‚5ƒ´8xˆ;8x5•~R†(M, i , i )
†Žy|~$w‚†1ÒÝy|w‚†Žy|5u~V{‚89†tÏRtÏ$w‚6x~R69††$Ów}éy|5u~R89†(:
i± : Σ ,→ M
w}~
bÏR„/Ô Σ = ΣΣÕåÓ
g
<IH
ÔâwúÕ÷&8 UÏR„!„8 M 89†R~w‚„8x5u~$w-#[6Gy|ÓUÏR„!„8 I × Σ t[y|Ï Ó5„‡w-,U69ˆ;Ï$tÛw‚†$ˆ;8
Ï$w‚8x5•~R6
Ôâw}wúÕ i+ v9Ï$ΦÏR89M†$tU:∂(I
57
„ X×{zyXΣ)ÏR89→
†$~$Ï$∂M
w‚vx~$w‚5/„8 Φ {1} × Σ Ôâw}w}wúÕ i− v9Ï$ÏR89†$tU57
„ X{zynÏR89†$~$Ï$w‚vx~$w‚53„8 ΦM −{0} × Σ Ôâw}éÕ!{‚89†(t{‚5ƒ´8xˆ;8x5•~R† i± : Σ ,→ M w}57„‡ÓMw‚†R8x5u~J„89†(w‚†Žˆ;Ï$tÛw‚†$ˆ;89†G8x5¯Û7 ˆ;{‚ƒ>w‚8 A
Ñåy|ÏV{zyp†RÓw}~R8 557~R8xώy Σ± := ˆ i± A
&89†v9bÏR„‡w‚†Rˆ;89†„ , Û7ˆ;{‚ƒ>w‚8G†R5•~v957†$w‚„6xÏR69† J„‡-w ,U69ˆ;Ï$tÛw‚†Rˆ;89†Vv957†R8x?Ï é>y|5•~²{Ý,-Ï$w‚8x5•~Žy|~$w‚5t"Ï 9† A %5 57~R8 C(Σ) {Ý,-8x57†R8xˆG{‚83„89†;vx{zy†R†R89†p„ ,-6 &uÓw}éy|{‚8x57v98
„89†v9bÏR„‡w‚†$ˆ;89†„ , Û7ˆ;{‚ƒ>w‚8G†$ÓÏ Σ A
uw (M, i± ) 8x~ (N, i± ) †Ž5•~„8x$Ó v9UÏR„‡w‚†$ˆ;89†„ , Û7ˆ;{‚ƒ>w‚8 b5tU8xÓ~y|{‚ÏR†
N t[y|Ï |8xˆptw}{‚8xˆ;8x5•
„6 #75w}Ï{‚8xÓÏVMtÏRu„‡Ów}~ M.N
~ ¬„8 M 8x~ N :
[
M.N := (M
+
N, i−
M , iN ).
− −1
i+
M (iN )
i−
M
M
Σ
1Σ
i+
M
Σ
Σ+
M
Σ−
M
G AL
i+
N
N
Σ
Σ+
N
Σ−
N
&w€tÏRu„‡Ów}~†RÓÏ{‚89†Vv9bÏR„‡w‚†Rˆ;89†„ , Û7ˆ;{‚ƒ>w‚8 A
8VtÏR „‡Ów}~v95‡Ò xÏR8!

:= (I × Σ, id , id )
0
i−
N
1
Ó578V†$~$Ï$Ó7vx~$ÓÏR8„8ˆ;57݄8 A ,-6x{‚6xˆ;8x5•~ Ó5w}~R689†$~
89†$~V{Ý,šy|tt{}w‚v¹y|~$w‚5 id × {} A
C(Σ)
Ô
Õ
id = 0, 1
Ó7†tUÓ鴁57†„6#75w}Ï8XtÏR69†R8x5•~{zyX57~$w‚5¬„8(vxè {}w}57„‡ÏR8(„ , Û7ˆ;{‚ƒ>w‚8²:
) + 3 /A;?/0>D3 ' E E 5 ÉúÍ]Ë>Å$ÌaË Ùá;Ù }Ù ÉÝÌnÇÈÚuÅ 89†$~Ó5¯v9UÏR„‡w‚†$ˆ;8p„ , Û7 ˆ;{‚ƒ>w‚8 (M, i± ) ~R8x{ &•Ó78 (i− )−1 ◦ i+ : H (Σ, Z) →ΣH (Σ, Z) †Rw}~V{Ý, w‚„8x5u~$w}~R6 A
1
1
∗
∗
,-8x57†R8xˆG{‚8G„89†Vvx{zy†R†R89†„ ,-6 &uÓw}éy|{‚8x57v98„89†vxè {}w}57„‡ÏR89†V„ , Û7ˆ;{‚ƒ>w‚8(ґÏ$ˆ;8Ó5
†ŽÓ7?† ݈;57 ݄8(„8 C(Σ) 57~R6 T C(Σ) A
' E ~RÓ~€6x{‚6xˆ;8x5•~ „‡ÓJƒ>ÏRÓtb8 „8„‡-w ,U69~Rtw‚89† 57Ó7†÷tbÓ鴁57†y†Ž†Ruvxw‚8xÏ÷Ó5 v9UÏR„‡w‚ψ†$ˆ;8܄ , Û7ˆ;{‚ƒ>w‚8 †$ÓÏ Σ „6 #7M(Σ)
5wt[y|Ï
89vxw0„6 #75w}~Ó5/ˆ²y|ttw}5ƒvxèu{}w}57„‡ÏRC
8 &uÓ78
Cψ = (I × Σ, i− = id, i+ = ψ) A
57Ó7†y|ttU8x{}{‚8xÏR57†(á;×ÊØ´Ø]ÉâÍ ¶ÉâÍ]ˁÅRÌXË•Ì ψ A
€ynv95•é´8x5u~$w‚5¬Ó~$w}{}w‚†R698 tUÓÏ{‚89†ˆ²y|ttw}5ƒ²vxè {}w}57„‡ÏR89†5&,-89†$~t[y† {zynv95ué´8x$5 ~$w‚53Ó7†RÓ78x{}{‚8Gˆ²y|w‚†8x{}{‚8Xy²{Ý,šy9é>y|5•~Žy|ƒ´8n„8JÏR89†$~R8xÏtÏRuvŽÛ78J„8J{zy;„6 #75w}~$w‚5©„8
' -
84 (7.2 2-/ 1+(/0&
M(Σ)
&A
"/ Σ = Σ Mg
<
L
g
Mg,1
"/ Σ = Σ :
g,1
id
ψ
I×Σ
G A?
.Üy|ttw}5ƒ²vxè {}w}57„‡ÏR8
Cψ
v9bÏR„‡w‚†Rˆ;8(„ , Û7ˆ;{‚ƒ>w‚8G8x~„8†RÓw}éuÏR8{zynv95ué´8x5•~$w‚5/Ó~$w}{}w‚†R698t[y|Ï./Ï$w}~Žy
„7y|57
† -./ G 8x~t[y|Ï+y|ÏRÓ‡ÒÝy|{}w‚„‡w‚†8x~V&8xé w}578(„7y|57† + -&8 G 8x~-&8¹< A
8xˆ²y|"Ï &uÓ7570
† &•Ó78n†$w †R5•~(„8x$Ó Ü„‡-w ,U69ˆ;Ï$tÛw‚†Rˆ;89†G„8
y|{‚ÏR†
{‚8÷ÏR89v9{}{‚8xˆ;8x5•~÷„89†pˆ²ϕy|tψtw}5ƒ vxè {}w}57„‡ÏR89† C 8x~ C 89†$~;„‡-w , 69M(Σ)
ˆ;Ï$tÛ783y|Ó
ϕ
ψ
ˆ²y|ttw}5ƒ²vxè {}w}57„‡ÏR8 C :
ϕ◦ψ
Cϕ◦ψ = Cϕ .Cψ .
w}57†Rw {zy÷{‚w„8;ƒ>ÏRÓtb8²Ó7†$Ó78x{}{‚8²†$ÓÏ M 89†$~Gv9ˆpt[y|~$w}{‚8÷y9é´89v;{zy{‚w„8
g,1
ˆ;57݄8(„8 C(Σ) A
uw ψ 89†$~(Ó5 6x{‚6xˆ;8x5u~X„‡Ó †RÓ7†?݃>ÏRÓtU8²„8;*ÏR8x{}{}w T (Σ) y|{‚ÏR† C 89†$~(Ó5
ψ
vxèu{}w}57„‡ÏR8G„ , Û7ˆ;{‚ƒ>w‚8 A
),>da h7S4B8\8\>F]9/A+ 46a ]> h>&) /08?17T8 #>d1 >D-0>D/07 Ó7†tbÓ鴁57†v957†$-w „6xÏR8xÏn{zy3ÏR8x{zy|~$w‚5 „ ,-6&uÓw}éy|{‚8x57v98†$ÓÏ C(Σ) „6#75w‚8÷t[y|Ïn{zy357~$w‚5 „8!bÏR„‡w‚†Rˆ;8
„ , Û7ˆ;{‚ƒ>w‚8a:
) + 3 /A;?/0>D3 ' E e w‚8x5u~ M 8x~ N „8xÓ$v9bÏR„‡w‚†Rˆ;89†„ , Û7ˆ;{‚ƒ>w‚8 A 57†$w‚„6
ÏR57† Ó578 L Ýé>y|Ï$w‚6x~R6 W ~R8x{}{‚C
8 &uÓ78X{‚8XbÏR„©„8
89†$~~R8x5•Ó¯t[y|Ï{‚8XÏR89v9{}{‚8xˆ;8x5•~
&•Ó78 Σ± †Rw}~Vw‚„8x5u~$-w #[6py9é´89v −Σ± A
„8 M 8x~ −N {‚8G{‚5ƒa„8{‚8xÓÏbÏR„3„8(†RÏ$~R8 W
{‚ÏR† M 8x~ N †Ž5•~ „‡w}~R
† RفÅ1˕ׁÍ[Îâdž$w]8x~ †R8xÓ{‚8xˆ;8x5•~†$Mwb{‚89†w}57vx{}Ó7†$w‚57† M ,→ W N8x~
w}57„‡Ów‚†Ž8x5•~ „89†w‚†Rˆ;Ï$tÛw‚†Rˆ;89†8x5¬Û7ˆ;{‚ƒ>w‚8 A %5/„‡w}ώy &•Ó78 W 89†$~Ó5
N ,→ W
RفÅ1Ë>ÉzÇÈá;Ì 8x5u~$ÏR8
8x~ N A
M
M
−
−Σ−
M = ΣN
G AJ
+
Σ+
M = −ΣN
W
@ ÏR„‡w‚†$ˆ;8
−N
W
8x5•~$ÏR8
M
8x~
N
Ñ^y|Ï{zyJ†$Ów}~R8‡5÷é´8xÏ$ώyG{‚89†UÏR„‡w‚†$ˆ;89†„ , Û7ˆ;{‚ƒ>w‚8 y¹é´89v {‚89†{}w‚8xÓ$÷„ ,šy|~$~Žy vŽÛ78xˆ;8x5•~„8 M y9é´89v −N 6xt[y|w‚†R†$w‚†v9ˆpˆ;8v>,-89†$~{‚8 v¹y†W
„7y|57†{zy%#7ƒ>ÓÏR8(vxw-P„89†R†RÓ7† A
9 / >@> &') ( (7 > 1 *: C 2 % >0> 1&4+* > & , )
" > /@&A * 8
(&""/0+"( / > /0" > )' 2- 972/0"') 5 > ( > $&:
<>?
&8¬tÏR „‡Ów}~„89†÷v9bÏR„‡w‚†$ˆ;89†÷„ , Û7ˆ;{‚ƒ>w‚8ÜtÏR69†R8xÏ$é´83{‚89†÷vx{zy†R†R89†!„ ,-6&•Ów}é>y {‚8x57v98 A 8t{}Ó7†tbÓÏa~RÓ~av9UÏR„‡w‚†$ˆ;8 M {‚8tÏR „‡Ów}~ I × M tb8xÓ~ /x~$ÏR8é Ó
v9ˆpˆ;8GÓ5UÏR„‡w‚†$ˆ;8(„ , Û7ˆ;{‚ƒ>w‚8X8x5•~$ÏR8 M.(−M ) 8x~ I × Σ  −M 89†$~{‚8(v9 UÏR„‡w‚†$ˆ;8 M ˆJÓ5w„8X{Ý,-Ï$w‚8x5•~Žy|~$w‚5 ttU>†R698Ô Σ± = Σ∓ Õ A w}57†$wå{Ý,-8x57†R8xˆJ{‚8
„89†vx{zy†R†R89†„8 UÏR„‡w‚†$ˆ;8(„89†Vv9UÏR„‡w‚†$ˆ;89†V„ , Û7ˆ;−M
{‚ƒ>w‚8GˆGMÓ5w€„8 {zyn{‚w tÏR „‡Ów}~
89†R~Ó5ƒ>ÏRÓtb0
8 &uÓ7857Ó7†57~R8xÏR57† H(Σ) A
%5 57~R8xώy T H(Σ) {‚8a†RÓ7† ݃>ÏRÓtb8 ÷„8 H(Σ) 8x5ƒ´8x57„‡ÏR6÷t[y|Ïn{‚89†nvxèu{}w}57„‡ÏR89†
„ , Û7ˆ;{‚ƒ>w‚8 A
89†$~÷Ó578/†$ÓÏ1Ҕyv98܆Žy|57†!UÏR„ Σ 57Ó7†a57~R8xÏR57†÷tUÓÏ
>6;\4C;?/0>D358 uw
Σ
g
y|ÏR6xƒ´8xÏJ:
8x~[†$w
Cg := C(Σ)
Hg := H(Σ)
T Hg := T H(Σ)
tb>†R† 9„8 Ó578(v9ˆptb>†Êy|5•~R8G„8
Σ
Cg,1 := C(Σ)
Hg,1 := H(Σ)
T Hg,1 := T H(Σ) A
UÏR„w A 8 A
Σ = Σg,1
‡57Ó7†57~R8xÏR57†
7914B),`ba57 8?a#) -07T8 ]>>&) /08?17T8* #>d1 >d-A>D/07 7
Ó7†G鴁Ó{‚57†
y †Ž†Ruvxw‚8xÏ M Ó5Üv9bÏR„‡w‚†Rˆ;8J„ , Û7ˆ;{‚ƒ>w‚8 M †$ÓÏ Σ A Ñ*ΣÓg,1
Ïv98x{zy$bv957†$w‚„6xÏR57†
{‚8(t{‚5ƒ´8xˆ;8x5•~ Σ ,→ Σ tÏR69v969„8xˆpˆ;8x5u~„6#75w A &8(gv9UÏR„‡w‚†$ˆ;8 M 89†R~„6#75w
v9ˆpˆ;8G{‚8ÏR89v9{}{‚8xg,1
ˆ;8x5•~ „8 g M 8x~ I × D2 ~R8x5•Ó/8x5¬w‚„8x5•~$w-#]y|5u~ Φ (I × ∂Σ )
8x~ I × ∂D2 é wzy Φ−1 A €89†t{‚5ƒ´8xˆ;8x5u~R† i± „‡Ó3v9UÏR„‡w‚†$ˆ;8 M †R5uM~„6#75w‚†g,1
t[y|Ï
{‚8 ÏR89v9{}{‚8xˆ;8x5u~„89†VMt{‚5ƒ´8xˆ;8x5•~R† i± y9é´89v{Ý, w‚„8x5u~$w}~R6J†RÓÏ{‚8„‡w‚"† &•Ó78 D2 A
8x~$~R8(v957†$~$Ï$Ó7vx~$w‚5¬w}57„‡Ów}~Ó578Gy|tt{}w‚v¹y|~$w‚53„8 ˆ;57 ݄89†
Cg,1 → Cg
&uÓw t[y†R†R8 5[y|~$ÓÏR8x{}{‚8xˆ;8x5u~ y|Ó &•Ó7~$w‚8x5u~8x5Ó5Û7ˆ;ˆ;Ï$tÛw‚†$ˆ;8X„8 ƒ>ÏRÓtb89† Hg,1 → Hg .
ORK1W[FUT W]OŽ\VQ HXI\eJQXZ[\VQ a\VDJF)KRI0Z fVFb\VD aI0Z
Hg,1
Mg,1
I^W
+y|ÏRÓ‡Ò”y|{}w‚„‡w‚†8x~ &8xé w}578´„7y|57† + >5•~„6#75w7Ó578#7{}~$ώy|~$w‚5atbÓÏ0{‚8 ƒ>ÏRÓtU8
&•ÓwtÏR{‚5ƒ´8;{zy #7{}~$ώy|~$w‚5 „8´Û57†R5 „‡Ó¯ƒ>ÏRÓtU8;„8²„‡w-, 69~Rtw‚89† M A
' ( $+972-/0"') 2 % * +') > +4+/039725 $ > /0 2-32 % * ') > +4/0 &A1&4 > &')& $ > /0 2- 2 % * +') > +4+/0 > (- 2 / 2 2 , $ > /@ 2-& 2 % * ') > +4/@ &A 3$ > /0 2 2 % * +') > +4+/0+) :
> + +/ )(8/ > ?2 &$& >0> 5 3972-/0"') &" 2 ,.$&+9782/0"')&
2 % * ') > +4/@ M N (7 +9 / D97I 8×I2/0×D
"') W &" M NW:
Hg,1
g,1
2
<IJ
y|57†Vv98x~$~R8†Ž89vx~$w‚557Ó7†y|{}{‚57†ÏŽy|ttU8x{‚8xφŽyn„6#75w}~$w‚5¬8x~V8x5é´8xÏ$ÏR57†!&•Ó78x{ &uÓ789†
tÏRtÏ$w‚6x~R69† A
57†$w‚„6xÏR57†¬{‚8¯ƒ>ÏRÓtb8ÜҔ57„7y|ˆ;8x5•~Žy|{
y|w}57†$w &•Ó78©†Žy †R6
Ï$w‚8!v98x5•~$ώy|{‚8¬„89†Rv98x57„7y|5•~R8 {F } A 8xˆ²y|FÏ"&uÓ7:=57†Cπ1&•(Σ
Ó78 g,1H, ∗)= F/F 8x~JtU>†R57†
2
†{Ý,-8‡~R8x57†$w‚5¬v98x5u~$ώy|{‚8
A Ó7†y¹é´57†y|k{‚ÏRk≥1
L = F /F
k
k
k+1
/ Lk
0
/ F/Fk+1
/ F/Fk
8x{}{‚8Pvxw&w}57„‡Ów}~{zyn†$Ów}~R8G8yvx~R8p:
0

/ H ⊗ Lk
/ Aut(F/Fk+1 )
/ 1.
/ Aut(F/Fk )
/1
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"ˆ
G
H ⊗ Lk '
(H, Lk )
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Aut(F/Fk+1 ) → Aut(F/Fk )
6 75w‚†R†R57†
ϕ ∈ Ker Aut(F/Fk+1 ) → Aut(F/Fk ) A
k+1 → F/Fk+1
t[y|Ï α(x) = ϕ(x).x−1 :*v>,-89†$~GÓ5 Û7ˆ;ˆ;Ï$tÛw‚†$ˆ;8÷8x5•é´α¹è•:y|F/F
5u~ F/F
†$ÓÏ L A
k+1
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y†R†R vxw‚8 ϕ ᾱ A
~R57k † ᾱ {Ý, Û7ˆ;ˆ;Ï$tÛw‚†$ˆ;8v9Ï$ÏR89†$tU57„7y|5•~„8 H „7y|57† L A %5k+1
k
#
&
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0
k
) + 3 /A;?/0>D3 ' 2' ~R57† Aut0(F/Fk ) {‚8†ŽÓ7†?݃>ÏRÓtb8„8
„‡ÏR6(t[y|Ï{‚89†y|Ó~Rˆ;Ï$tÛw‚†$ˆ;89† h „8 F/F é´6xÏ$w-#]y|5•~
Aut(F/Fk )
k
[h̃(x1 ), h̃(y1 )] . . . [h̃(xg ), h̃(yg )] = [x1 , y1 ] . . . [xg , yg ]
98 †$~Ó5ÏR8x{ xé´8xˆ;8x5u~„8 h „7y|57† Aut F A
8xˆ²y|Ï"&uÓ757† &uÓ78 Aut (F/F ) 89†$~V{‚8 ƒ>ÏRÓtU8(†$è ˆpt{‚89vx~$w &•Ó78

ˆ;u„
8x5ƒ´8x5$
Fk
h̃
0
2
Sp H A
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„‡Ów}~R8„8V{zyJ†$Ów}~R88yvx~R8tÏR69v969„8x5•~R8 A €857Nè•y|0Ó!„8v98xk+2
~$~R8†RÓϑÞ189vx~$w‚05!89†$~k+1
{‚8†RÓ7†?
ƒ>ÏRÓtU8 D (H) „8 H ⊗L „6 #75w „8{zyˆ²y|5w xÏR8 †$Ów}é>y|5•~R8: L = F /F
tU8xÓ~ /x~$ÏR8kw‚„8x5u~$-w #[6 n{zyXt[k+1
y|Ï$~$w‚8(„8G„8xƒ>ÏR6 k + 1 „8 {Ý,šy|{}ƒ xÏR8G„8(k+1&w‚8 {}w}Ïk+1
R8(†$ÓÏ k+2
HA
%5¬tb>†R8y|{‚ÏR†
Dk (H) = Ker(H ⊗ Lk+1
Ó7†y¹é´57†V{zyn†$Ów}~R8(87yvx~R8
[.,.]
/ Lk+2 ).
0 → Dk (H) → Aut0 (F/Fk+2 ) → Aut0 (F/Fk+1 ) → 1.
( (7 > 1 % >0> &AB21C /0( 8
F1 = F
<
P
Fk+1 = [F, Fk ]
(7 k≥1
:
$ $
T]£´¡# ¥ h¢^# 5V Ÿ)¦P¥ €Ÿ#V¥ ¥ ¡ 6 £I¡Ý¥[¢
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Mg,1
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X
†N, w‚„8x5•~$w-#[8xÏy|Ӄ>ÏRÓtU8 Aut F Ï;{zy©tÏR|Þ$89vx~$w‚5 F → F/F tUÓÏ g,1k ≥ 1
w}57„‡Ów}~!Ó578Üy|tt{}w‚v¹y|~$w‚5'5[y|~$0ÓÏR8x{}{‚8 Aut F → Aut (F/F )k+1
A Ó7†atUÓ鴁57†
0
0
k+1
y|{‚ÏR†„6 #75w}Ï
) + 3 /A;?/0>D3 ' E
Mg,1 (k) = Ker Mg,1 → Aut0 (F/Fk+1 )
Ó7†V~R8x5757†y|{‚ÏR†V{‚8„‡wzy|ƒ>ώy|ˆpˆ;8Jv9ˆpˆGÓ~Žy|~$w ҆$Ów}é>y|5•~G:
1
A
l1
l l
l
l l
ul l
Mg,1 (k + 1)
1 OOO
OOO
OOO
OOO
OO'
oo
o
o
wo o
Mg,1 (k)
OOO
OOO
OOO
OOO
'
τk
1
/ Dk (H)
Mg,1 R
RRR
RRR
RRR
R
/ Aut0 (F/Fk+2 )
RRR
(
/ Aut0 (F/Fk+1 )
/1
€
y #7{}~$ώy|~$w‚5y|w}57†$w„6#75w‚8
89†$~
g,1 (1) ⊂ Mg,1
y|ttU8x{‚6980 ΑÅ1×>ΑɔفÍÜË•Ì Ù •ÍÇ9فÍ38x· ~ · ·{‚8⊂ˆ;M
Ï$tg,1Ûw‚(k)
†$ˆ;8 ⊂w}57„‡· ·Ó·w}~ ⊂τ M
k : Mg,1 (k) → Dk (H)
فá;فá;فÅPØ •ÉúÇÊá²ÌJ˕ÌuÙ •Í7ÇxÙÍ A
k &8ƒ>ÏRÓtU8(„8„‡w-,U69~Rtw‚89† M y|ƒ>w}~
†$ÓÏ M (k) t[y|Ïv95ÞӃ•y|w‚†Žg,15tÓw‚†"&uÓ78v>,-89†$~VÓ5†RÓ7†?݃>ÏRÓtU8(„‡w‚†$~$w}5ƒ>Ó76
8x~†$ÓÏ Hg,1⊗ L ´†ŽÓ7?† ݃>ÏRÓtb8„‡w‚†$~$w}5ƒ>Ó76„8 Aut(F/F ) 8x5²v9ˆptU>†Žy|5u~
k+2
{Ý,šyvx~$w‚5¬„8(v9k+1
5Þ1Ӄ•y|w‚†R5¬8x~{‚8ˆ;Ï$tÛw‚†Rˆ;8 M → Aut(F/F
A
k+2 )
Ñåy|Ïv957†$~$Ï$Ó7vx~$w‚5 τ 89†R~V6 &•Ów}é>y|Ï$wzy|5•~R8GtbÓÏVv989†yvx~$g,1
w‚57† A
k
8xÓ{‚† {‚89†„8x$Ó !tÏR8xˆpw‚8xÏR† Û7ˆ;ˆ;Ï$tÛw‚†$ˆ;89†V„8 ´Û57†R55u~6x~R66x~$Ó7„‡w‚69†:u{‚8
tÏR8xˆpw‚8xÏt[y|Ï ´Û57†R5¬8x~{‚8„8x$Ó w xˆ;8Gt[y|ÏV.3Ï$w}~Žy A
(*),> 5) /0+;\+T8 a ),7.1/07T)#>d1 >D1>&)# /08?1 7! 7F> 3#8\>d3
Ñ^y|τ6#75w}~$w‚5
&•Ów*y|ƒ>w}~~$Ï$w}éuwzy|{‚8xˆ;8x5u~†$ÓÏ F/F = H :‡w}{
0
89†$~{‚8(†RÓ7?† ݃>ÏRÓtU8(„8
2
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w Tg,1 A
g,1
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†RÓÏ [x] ∧ [y] ∈ ∧22 H  x, y ∈ F 8x~ [x], [y] †R5•~{‚8xÓÏvx{zy†Ž†R89†V„7y|57† H2A %52tb8xÓ3~
y|{‚ÏR† w‚„8x5u~$-w #[8xÏ D (H) y9é´89v ∧3 H é Ó!v9ˆpˆ;8G†RÓ7†?݃>ÏRÓtU8„8 H ⊗ ∧2 H é wzyJ{‚8
Mg,1 (1)
1
<IK
ˆ;Ï$tÛw‚†Rˆ;8
a ∧ b ∧ c → a ⊗ (b ∧ c) + b ⊗ (c ∧ a) + c ⊗ (a ∧ b) A
,-6x~$Ó7„83„‡ÓtÏR8xˆpw‚8xÏaÛ7ˆ;ˆ;Ï$tÛw‚†$ˆ;8 τ : T → ∧3 H t[y|Ï ´Û57†R5 y
tU8xÏ$ˆpw‚†V„ ,-8x5¬„6x~R8xÏ$ˆpw}578xÏ{Ý, w}ˆ²y|ƒ´8tÓw‚†„89†Vƒ´6x1576xώy|~Rg,1
8xÓÏR†„‡Ó57Nè•y|Ó :
#+9>&)917 ' F>#' F>dE ÄÙÚ Å g ≥ 2
τ1 ÌÊÇxÎ0ÇÈÚuÅ Ì 9Î‘É æÊá τ1 = D1 (H) ' ∧3 H τ1 ÅRÌÊÇÝØ72Ì 9ÎPÌ ‚ÌÊÇJ× 9ΑɔفÍÇGË•Ì Mg,1 ÇÊÚ Å Tg,1 ØׁŠRÙÍ Ú ´×ÉúÇxÙÍ !ÌxÎ ÇÈÚuÅ ∧3 H
Ë´Ù>Í7ÍbÆRÌ Ø‡×Å
Ù
ÌÊÇxÎ 2 ׁڇÎÝفá;فÅ$
Ø •ÉúÇÊá²ÌJË•Ì Hϕ.(a
ÉúÍ]Ë>Úu∧ɑÎbb؇∧ׁÅc)ϕ= ϕ∗ (a) ∧ ϕ∗ (b) ∧ ϕ∗ (c) ϕ∗
ÌÊÇÈÎ }ÌpÇxفÚuÇ Å$فڹØ7ÌÌÈÍ •ÌÈÍbˁÅ$Æ؇×>Å }ÌÊÇ;Î 0ÉúÇÈÎâÇpË•Ì XÌ •Í
Ker τ1 = Mg,1 (2)
}!
Ì }ÙÍ ÷˕ÌÊ!
Ç RÙ>ÚuÅ RÌÊÇJÇÊÉâáØ }ÌÊÇ ¹ÌÈÅÈá;ƎÌÊÇ(Ç9Æ؇ׁÅ$ׁÍ[ÎPÌÊÇ y|57†V{zyn{}w}~$~R6xώy|~$ÓÏR8 M (2) 89†$~ƒ´6x576xώy|{‚8xˆ;8x5•~57~R6 K A
g,1
g,1
y|ttU8x{‚57†0&•Ó&, Ó578nv9ÓÏ$U8X†Rw}ˆpt{‚8GҔ8xÏ$ˆ;698 γ †$ÓÏ Σ 89†$~„‡w}~R8²Ç¹ÆP؇ׁÅ$ׁÍ[ÎP̲†$w
yG„8x$Ó ²v9ˆptb>†Žy|5u~R89†v95578 89† A &8 uÌÈÍ7ÅRÌË•Ì g,1}× RÙÚ ÅŽÌ γ 89†R~{‚8Vƒ´8x5ÏR8
Σg,1 \ {γ}
„8 {zy;†RÓ7?† P†$ÓÏ1ÒÝyv98 5&,šy9è•y|5u8
~ &uÓ&, Ó578†R8xÓ{‚8(v9ˆptU>†Žy|5•~R8J„8 bÏR„ A
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k+1 .
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8 &•Ó78 π ◦ s = id 8x~ s(1) = 1 A %5tU8xÓ~{}Ów
w 0A 8 A Ó578(Ay|tt{}w‚v¹y|E~$w‚5 s G: G → E1 ~R8x{}{‚0
y†Ž†Ruvxw‚8xÏÓ578Ҕ57vx~$w‚5 f : G × G → A „6 #75w‚8Gv9ˆpˆ;G8 ¤{‚8G„6ÈÒÝy|Ó~„ , Û7ˆ;ˆ;?Ï tÛw‚†Rˆ;8(„8 s!8x5tU>†Žy|5u~ tbÓÏ~RÓ7†6x{‚6xˆ;8x5•~R† g 8x~ h „8 G s(g)s(h) = i(f (g, h))s(gh).
% 5aˆ;5•~$ÏR8 |y {‚ÏR†'&uÓ78 f 89†$~Ó5!< Pv9 vxè vx{‚857Ï$ˆ²y|{}w‚†R6 A uw s0 89†R~Ó578y|Ó~$ÏR8†R89vx~$w‚5
„8²v98x~$~R8;8 ~Rx8 57†Rw‚5 v98x5•~$ώy|{‚8!y|{‚ÏR†({‚8;v9 vxè vx{‚8ay†R†R vxw‚6 s0 „‡w-, xÏR8;„8 f t[y|Ï(Ó5
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#+9>&)917 ' 0' Z*4B8\8\>D3 AZ*4 æ VÌRàÉzÇxÎPÌ÷Ú Í ÉâÍ7ç¤×ÅÈɑ×>Í[ÎGÌÈÍ[ΑÉÝÌÈÅ λ ˕ÌÈÇ ÇPØ ÈÅRÌÊÇXË Ù>ápÙ Ù >ɔÌXفÅÈɔÌxÍ[ÎPÆRÌÊÇGÎPÌ ‚ÌÊ
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× ÙÅÊÇ λ(M ) = 0 π1 (M ) = 1
ÉâÉúÉ ÌxÎ λ(M #M ) = λ(M ) + λ(M ) λ(−M ) = −λ(M )
1
HIH
2
1
2
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Ú Í©Í]ÙNÌÈÚË˴ׁÍÇ(Ú ÍbÌ PÇÝØ ÈÅ$Ì;Ë ‡Ùá;Ù Ù É”ÌnÙ>ÅÊÉÝÌÈÍ[ÎPÆRÌ M ÌxÎÇxفɑÎ
PÇÝØ ÈÅ$ÌnË Ù>ápÙ Ù>ɔÌXÙ xÎPÌxÍ7ÚÌpÌxÍ¯Ì Ì29ΑÚׁÍ[ÎÇÊÚ Å M Ú ÍbÌ
1 •ÉâÅÊÚ Å É”ÌXÇÊÚ Å K × Ù>ŎÇ
K
× Mn (K)
n
λ(Mn (K)) = λ(M ) +
Ù
n 00
∆ (1)
2 K
ÌÊÇÈÎ ‚ ̲؇٠Íá²Ì
(t) = a + a (t + t−1 ) + a2 (t2 + t−2 ) + . . .
Ë ∆
}ÌRà´KׁÍ]˕ÌxÅ(Í]ف0ÅÈá;× ¶ÉúÇ91Æ(ç|ÆÈÅÈÉ ×Í[Î
ß Í;؇×>ÅÈÎ‘É xÚ¶ÉÝÌÈÅ
∆K (1) = 1
P
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$AY ,­ 5"bŸu¡WF*£ ¥³c ¢¢^#
¥ £ &ŸT#V¥ ¥ ¡ 6 £I¡Ý¥[¢
€Ó~G„ ,šy|UÏR„ =# 57†Ó5¯„‡w-,U69ˆ;Ï$tÛw‚†Rˆ;8 „8p{zy÷†$ÓÏ1ÒÝyv98
&•Ów # 8n{‚8
„‡w‚† &•Ó78 D2 tUw}5•~6tUw}5•~8x~&•Ówb89†$~„6 #759w uƒ>Ï v98ıgy|Óa~$Û769Ï"xˆ;8„8Σgw‚8x{‚†R8x5•t[y|Ï:
ıg : π1 (Σg,1 ) → π1 (Σg,1 )
xi 7→ yi−1
57†$w‚„6xÏR57† {‚8nv9Ï$t7†8x5¯y|57†R89†
yi 7→ yi xi yi−1
†$~Žy|57„7y|ÏR„Ü„8JbÏR„
„8X†RÏ$~R8
† H =: vÈΣÒ#7ƒ>ÓÏR8 G AQP A
&uÓ78{‚89†v9ÓÏ$U89† ỹ i = 1, A}A}A , g UÏR„H8x5•~„89† „‡w‚†"&uÓ789† d „7y|57∂H
{‚ÏR† {‚8VÏR89v9{}{‚8xˆ;8x5•~„8 H 8x~ H {‚8V{‚5ƒG„8{‚8xÓÏUÏR„péuwzy ı 89†$~„‡w-, 69ˆ;Ï$tÛ78
{zy;†$tÛx ÏR8 S :
X
g
i
g
i
g
g
g
g
g
3
S 3 = Hg
[
ıg
−Hg .
D 2 ⊂ Σg
ỹi
di
G AQP
D3
Ï$t7†8x5¬y|57†R89†
Hg A
Ñ*ÓÏX~RÓ~J6x{‚6xˆ;8x5u~ „‡Ó ƒ>ÏRÓtb8a„8²„‡w-, 69~Rtw‚89†
&{‚8Ç9ÉúÍ]ËuÌÈá²ÌÈÍ[Π˕Ì
JÌRÌ ´×•×Å1ËpׁǎÇxÙ xÉ”Æ ϕ 89†$ϕ~å{zy 3Ýé>y|Ï$w‚6x~R6 S 3 v9ˆpt[yvx~R8 v95M
578g,1
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ϕ
HL
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{‚8{‚5ƒ„8{‚8xÓÏ^UÏR„
é zw y ı .ϕ  ϕ 89†$~{Ý, w}ˆ²y|ƒ´8(„8 ϕ t[y|ÏV{zyXtÏR|Þ$89vx~$w‚53v¹y|575Hw &ugÓ78 M → M :
g
g,1
[
Sϕ3 = Hg
ıg .ϕ
g
−Hg .
%5 ÏR8xˆ²y|Ï"&uÓ783y|{‚ÏR† &•Ó780†$w ϕ 89†$~pÓ5 6x{‚6xˆ;8x5u~a„‡Ó >ƒ ÏRÓtU8„8*ÏR8x{}{}w
Ý é>y|Ï$w‚6x~R6 S 3 89†$~Ó578 H P†$tÛxÏR8„ , Û7ˆ;{‚ƒ>w‚8 8x5•~$w xÏR8 A Pu5 é´8xÏR†R8xˆ;8x5•~ ~RÓ~R8
3
tU8xÓ~ /x~$ÏR8(ϕ~R8x5uÓ78G„8(v98x~$~R8ˆ²y|5w xÏR8 A
H
å{zy
Tg,1
"
Z
y†R†R58x~./Ï$w}~Žyp5•~ˆ;5•~$ÏR6(w}57„6xtU8x57„7y|ˆpˆ;8x5•~ &uÓ78p:
#+9>&)917 ' E <>&) /A;\4 Z*4B8\8\>d3 ÙÚ ÎPÌ 3 ÝÇPØ ÈÅRÌË Ùá;Ù }فÉÝÌ÷ÌÈÍ[Î‘É ÈÅ$Ì
Ø7ÌÈÚ Î xΑÅRÌ/Ù xÎPÌÈÍ7Ú7Ìp؇ׁÅÚ ÍÇ xÉâÍ]˕ÌÈá²ÌÈÍ[ÎXË•Ì GÌRÌ •×´×Å$Ë ÇÊÚ ÅÚuÍ Æ ‚ÆÈá²ÌÈÍ[ÎJËuÌ K
g,1
ØفÚuÅGÚuÍ ŽÌÈÅxÎÝׁÉâÍ g 89vxw tb8xÏ$ˆ;8x~Xy|{‚ÏR†(„8p„6 #75w}ÏGÓ578;y|tt{}w‚v¹y|~$w‚5 λ∗ : K → Z w}57„‡Ów}~R8pt[y|Ï
g,1
{Ý, w}5uéy|Ï$wzy|5u~„8J y†Ž†R5¯:
λ∗ : Kg,1 → Z
ϕ 7→ λ(Sϕ3 ).
(*),> >&8?/A;?/0>D3 ' <>&) /A;\4 < >De 57†$w‚„6xÏR57†V{Ý, Û7ˆ;ˆ;Ï$tÛw‚†$ˆ;8
(
θ0
ÌÊÇÈÎÚ Í Ù>ápÙ>ápÙ>ÅÝØ u ÉzÇÈá;Ì „6#75w€t[y|Ï
:T →Z
λ∗
θ0 (xi ∧ xj ↔ yi ∧ yj ) = 1,
„ , Ó53y|Ó~$ÏR8ƒ´6x576xώy|~R8xÓÏ ) = 0.
θ0 (
Ó7†y¹é´57†y|{‚ÏR†{zyXtÏRtb>†$w}~$w‚53†$Ów}éy|5u~R8tÏRÓé´698(t[y|Ï{‚8v¹y|{‚vxÓ{:
(*),> >&8?/A;?/0>D3 ' e < >D)^/A;\4 < >De فÉâÎ τγ Ú Í!Î 0ÉúÇÈÎ^Ë•Ì JÌ uÍ JË>Å1فɑÎP̂Ì
}ف
Í Ë Ú ÍbÌ RÙÚ Å ŽÌÇÊÉâáØ ‚Ì ¹ÌÈÅÈá²ÆRÌÇ9ÆPØׁÅ1×>Í[ÎPÌ γ Ë•Ì •ÌÈÍ[Å$Ì h Ì9Î7Ç9فÉâÎ u , v , . . . , u , v
Ú ÍbÌ $×|Ç9Ì(Ç |áØ ‚2Ì 9Î‘É xÚ7ÌØ‡ÙÚ Å }ÌØ]Å$ÌxáXÉÝÌÈÅ Å$فÚNØÌXË Ùá;Ù }Ù ÉÝÌG˕
Ì }1×nÇxÙ>1 ڕ$Ç ÝÇÊÚ Å x×h ÊÌ h
ËuÌ ŽÙá؇Ù|Çx×>Í[ÎPÌnË•Ì $فÅ$Ë γ Ù>ŎÇ
λ∗ (τγ ) = −θ0 ((u1 ∧ v1 + · · · + uh ∧ vh )⊗2 ).
y|ttU8x{‚57†C&•Ó78÷{Ý, w}ˆ²y|ƒ´8„ , Ó5 ~R8x{~ 7w‚†$~;„8 8xÛ5 τ t[y|Ïp{‚8„8xÓ$‡w xˆ;8÷Û7 ˆ;ˆ;Ï$tÛw‚†Rˆ;8p„8 •Û57†R5©89†$~{Ý, w}ˆ²y|ƒ´8p„8 (u ∧ v + ·γ· · + u ∧ v )⊗2 t[y|Ï{zy
tÏR|Þ$89vx~$w‚5av¹y|575w &uÓ78 T → T A , w‚„69889†$~y|{‚ÏR† 1 „8V„16#75w}Ï{Ý, Û7ˆ;hˆ;hÏ$tÛw‚†Rˆ;8 λ∗
8x5Ó~$w}{}w‚†Êy|5•~{‚8„8x$Ó ‡w xˆ;8(Û7ˆ;ˆ;Ï$tÛw‚†Rˆ;8G„D
8 ´Û57†Ž5 τ : K → T 8x~
.Üy|{}Û78xÓÏR8xÓ7†R8xˆ;8x5u~ ]{Ý, Û7ˆ;ˆ;Ï$tÛw‚†Rˆ;8 θ 578(2 tb8xÓg,1
~t[y†R†R8xÏy|
Ó &uÓ7 θ0 : T → Z A
~$w‚8x5u~!v¹y|ϲw}{89†$~;5755•Ó{†$Óϲ{‚857Nè•y|Ó'„8¬{zy©tÏR|Þ$89vx~$0w‚5 T → T A &8¬Ó~²é>y
/x~$ÏR8„ , w}5u~$ÏRu„‡Ów}ÏR8Ó5 57Óé´8x{Û7ˆ;ˆ;Ï$tÛw‚†Rˆ;8 d : T → Z &•Ówé>y¬tb8xÏ$ˆ;8x~$~$ÏR8
„8nt[y†Ž†R8xÏGy|Ó &uÓ7~$w‚8x5•~ A Et[y|Ï$~$w}ÏG„8 8x~ 5¯éy!~R8x5w}ÏJÓ5©Û7ˆ;ˆ;?Ï tÛw‚†Rˆ;8 q : K → Q A ./Ï$w}~Žy¬v957†$~$τÏ$2Ów}θ~X0Ó5 Û7dˆ;ˆ;Ï$tÛw‚†$ˆ;8 d : K → Z
g,1
g,1
H>?
v9ˆpˆ;8 ¤ÏR6¹y|{}w‚†Žy|~$w‚5 ƒ´69ˆ;6x~$Ï$w &uÓ78 „8 d A Ó7†R†Rw9&{‚89†JÛ7ˆ;ˆ;Ï$tÛw‚†$ˆ;89†
鴁5u~†Žy|~$w‚†1ÒÝy|w}ÏR8{Ý,-6xƒ•y|{}w}~R6(†RÓw}éy|5u~R8;:
λ∗ = q +
q
8x~
d
1
d.
24
t {}w‚vxw}~R57†Ó5tU8xÓt{}Ó7†V{zynv957†$~$Ï$Ó7vx~$w‚5/„8v989†VÛ7ˆ;ˆ;Ï$tÛw‚†$ˆ;89† A
$AY 5 5 €Ÿ ¡P¢ '¥ ¥ &Ÿu¡”£I
, Û7ˆ;ˆ;Ï$tÛw‚†$ˆ;8
(
q : Kg,1 → Q
89†$~V„6#75w€t[y|Ï
d:T →Z
d((a ∧ b)⊗2 ) = 0,
d(a ∧ b ↔ c ∧ d) = (a · b)(c · d) − (a · c)(b · d) + (a · d)(b · c),
‡8x~ d †R5u~„89†6x{‚6xˆ;8x5•~R†V„8 H 8x~  · 89†R~{zyGҔÏ$ˆ;8 „ , w}5•~R8xÏR†R89vx~$w‚5¬†RÓÏ
05;t[y|Ï$~$w‚vxÓ{}w‚8xϕ tbÓÏ0~RÓ~R8V†ŽÓ7†?Ý [y†R8†$è ˆpt{‚89vx~$w &•Ó78 u , v , . . . , u , v „8 H HA

a b c
1
1
h
h
1
d((u1 ∧ v1 + · · · + uh ∧ vh )⊗2 ) = h(h − 1).
2
.3Ï$w}~Žyaˆ;5•~$ÏR8py|{‚ÏR0
† &uÓ78
v98&uÓw
1 †¹,šy|55uÓ{‚8X†RÓÏ8xÏ
„6#75w}~Ó5Û7ˆ;ˆ;Ï$tÛw‚†Rˆ;8 q q0: T:=→θ0Q+A 3 d, Û7ˆ;ˆ;Ï$tÛw‚†Rˆ;8 q(T: K→ T→) Q 89†$~
g,1
„6 #75w€t[y|ÏV{zynv9ˆptb>†Rw}~$w‚5 0
τ2
q = q 0 ◦ τ2 : Kg,1
$AY X 5 5 €Ÿ ¡P¢ '¥ ¥ &Ÿu¡”£I
/ T
q0
/ Q.
d : Kg,1 → Z
&8nÓ~89†$~„8Xv957†R~$Ï$Ów}ÏR8nÓ5/Û7ˆ;ˆ;Ï$tÛw‚†Rˆ;8
&uÓw^ÏR6¹y|{}w‚†Ž8
ƒ´69ˆ;6x~$Ï$w &uÓ78xˆ;8x5•~ A 5/„ ,šy|Ó~$ÏR89†~R8xÏ$ˆ;89†b53vRÛ78xÏRvŽÛ78XdÓ:53KÛ7g,1ˆ;→
ˆ;ZÏ$tÛw‚†$ˆ;8 d ~R8xd{
&uÓ78 ]tbÓÏ~RÓ~ ~ 7w‚†$~ „8n8xÛ5 τγ {‚8J{‚5ƒ„ , Ó578nv9ÓÏ$b8X†R6xt[y|ώy|5•~R8 γ „8Jƒ´8x5ÏR8
†Rw}~tÏRtUÏ$~$w‚5578x{ 1 − 1) A
h d(τγ )
Ñ*ÓÏåv98x{z$y N„6 #75w‚†R†Ž57†*„8x$Ó G2 v9h(h
 vxè vx{‚89† „8 Z 2 (M , Z) :N{‚8v9 vxè vx{‚8„8./8xè´8xÏ
g,1
8x~{‚8Gv9uvxè‡vx{‚8„ , w}5•~R8xÏR†R89vx~$w‚5 A
h7S]>F]b]T-07 7S<7D79)
ˆpˆ;8x57Ð957†t[y|ÏVώy|ttU8x{‚8xÏV{zyn„6#75w}~$w‚5/„‡Óˆ²y|ttw}5ƒ;~RÏR8;:
) + 3 /A;?/0>D3 ' w}~ ϕ Ó56x{‚6xˆ;8x5u~÷„8 Mg A €8ˆ²y|ttw}5ƒ ~RÏR8 y†R†Žuvxw‚6
89†$~p{zy H Ýéy|Ï$w‚6x~R6†Êy|57†nbÏR„ ~R8x5uÓ78„8 I × Σ 8x5 w‚„8x5•~$-w #]y|5u~ (0, x) y9é´89v
Tϕ
g
[tUÓÏ~RÓ~ x „8 Σg A
(1, ϕ(x))
$+ "/@2-11&.&A >&% 8$/0+"/,+ /@ > +:
- % 8$/0+ 2
/ > /0"1& ./0Z$&/; % &A ( 8 > %$+ + &/0+ >@> +:
> %$& ++&/@M
g,1
HIJ
w‚8x5•~ ϕ 8x~ ψ „8xÓ$÷6x{‚6xˆ;8x5u~R†„‡Óƒ>ÏRÓtU8„8(„‡w-, 69~Rtw‚89† M A €y L Ýé>y|Ï$w‚6x~R6
89†$~„69vxÏ$w}~R8Gt[y|ÏV{‚8 Σ #7ÏR6Xy|Ó$P„89†R†$Ó7†„ , Ó5¬„‡w‚†"&uÓ78%p„8xÓ$g ~$ÏRÓ7†„5u~{‚8
Mϕ,ψ
UÏR„89†$~{Ý, Ó5w‚5/„‡w‚†‘Þ1w}5u~R8(g„89†ˆ²y|ttw}5ƒ´†~RÏR89† T T 8x~ −T A
ϕψ
&8(v9 vxè vx{‚8„8(./8xè´8xÏ τ 89†$~y|{‚ÏR†V„6 #75w€t[y|Ï ϕ ψ
†$w}ƒ>5 M
τ (ϕ, ψ) =
∀ϕ, ψ ∈ Mg .
ϕ,ψ
%5JtU8xÓ~*6xƒ•y|{‚8xˆ;8x5u~0{‚8„6 #75w}Ï0v9ˆpˆ;8< Pv9 vxè‡vx{‚8tUÓÏ M éuwzyV{zy†$ÓϑÞ$89vx~$w‚5
g,1
v¹y|575w &uÓ78 M → M .
g,1
g
h7S]>F]b]T-07 /23d;\79),8\79]I;?/0>D3
& 8¬ƒ>ÏRÓtU8 !„83v9Û7ˆ;{‚ƒ>w‚8 H 1 (M ; H (Σ , Z)) 89†$~aw}5$#75wvxè‡vx{}w &uÓ78 A
^w-57†VÓ53Û7ˆ;ˆ;Ï$tÛw‚†$ˆ;8nvxÏRw‚†R6 k : Mg,1 →1 H g,1
&•Ów^†Rw}~Ó53ƒ´6x576
ώy|~R8xÓÏa„8 H 1 (M ; H (Σ , Z)) A y|ttU8xg,1
{‚57†&uÓ&1, Ó(Σ5 g,1Û7,ˆ;Z)ˆ;Ï$tÛw‚†$ˆ;8¬vxÏRw‚†Ž6
g,1
1
g,1
é´6xÏ$-w #[8{Ý,-6xƒ•y|{}w}~R6 k(ϕψ)
 ϕ 89†R~p{Ý,šy|Ó~Rˆ;Ï$tÛw‚†$ˆ;83„8 H
= k(ϕ) + ϕ∗ k(ψ)
∗
w}57„‡Ów}~t[y|Ï ϕ ԑvÈÒ0t[y|ώy|ƒ>ώy|tÛ78 G A L7A <´Õ A
%5tU8xÓ~ay|{‚ÏR†²„6 #75w}Ïa{‚8¬v9 vxè vx{‚8„ , w}5u~R8xÏR†R89vx~$w‚5 c 8x5tb>†Žy|5u~²tUÓÏ;~RÓ~
k
:
ϕ, ψ ∈ M
g,1
ck (ϕ, ψ) = −k(ϕ) · ϕ∗ k(ψ)
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vx{zy†Ž†R89†(„8aÛ78xÏ$5¯ÏR8x{zy|~$w}é´89†Jˆ²y|w‚† Þ1Ó7†$~R8;v98x{}{‚89†(„5u~G57Ó7†Xy|ÓÏR57†(U89†Rw}5©t[y|Ï({zy
†RÓw}~R8 A
y|57†G{‚8;v¹y†Jy|7†Ž{}Ó {zytÏR8xˆpw xÏR8avx{zy†R†R8;„8aÛ78xÏ$5 c (η) 89†$~(5uÓ{}{‚8 v¹y|Ó7†Ž8
„8n{Ý,-8 ‡w‚†$~R8x57v98p„8n{zyaҔÏ$ˆ;8p鴁{}ӈ;8pv9ˆpt{‚8 8 A €y÷t{}Ó7† [y1 †R†Ž8pvx{zy†R†R8p„8²Û78xÏ$5
6xé´8x5u~$Ó78x{}{‚8xˆ;8x5•~ 5755•Ó{}{‚8J89†$~{zy 2 Vvx{zy†R†Ž8„8JÛ78xÏ$5
°a¦V¢
ξ0
c2 (η) ∈ H 4 (B; Z).
% 5ÜtU8xÓ~ {Ý, w}5•~R8xÏ$tÏR6x~R8xÏGv9ˆpˆ;8n{Ý,-7†$~$Ï$Ó7vx~$w‚5©tÏ$w}ˆ²y|w}ÏR8 ;{Ý,-8‡w‚†$~R8x57v98p„ , Ó578p†R89v
~$w‚53„‡Ó #7ÏR6tÏ$w}57vxw}t[y|{ Pη A 6#75w‚†R†R57†?Ý{zy;„7y|57†{‚8(v¹y†ÏR8x{zy|~$w ÒV:
?>=
) + 3 /A;?/0>D3 E #7ÏR6
† &•Ó&, w}{å8 ‡w‚†$~R8GÓ578X†R89vx~$w‚5 „‡Ó
uÓttU>†R570
y|Ó$P„89†Ž†$Ó7†„ , Ó5!†RÓ7†?Pv9ˆpt{‚88 L A %5!y|ttU8x{}{‚8xώy 2 $}×|sǎǹÌ˕SU
̇(n)
ÌxÅÊÍÅRÌ }×>ΑPη
ÉúçÌ
×>Ú ÈÅ$Æ η ÌxÎ }×÷ǹÌ29Î‘É”ÙÍ s {Ý,-7†$~$Ï$Ó7vx~$w‚5©tÏ$w}ˆ²y|w}ÏR8 atÏR{‚5ƒ´8xÏ s B A %5Ü{zy
57~R8xώy c (η, s) :
2
c2 (η, s) = c(η0 , s) ∈ H 4 (B, L; Z).
P 5u~R6xÏR89†R†R57†?Ý57Ó7†^y|Ó7†R†$w;{zy t{}Ó7†€Û[y|Ó~R8 vx{zy†R†R8„8Û78xÏ$5 c (η) ∈ H 2n (B; Z) :
8x{}{‚889†R~{Ý,-7†$~$Ï$Ó7vx~$w‚5tÏ$w}ˆ²y|w}ÏR8%X{Ý,-8‡w‚†$~R8x57v98(„ , Ó578†R89vx~$w‚53„‡nÓ #7ÏR68x5†$tÛxÏR89†
η0 A
) + 3 /A;?/0>D3 E e uÓttb>†Ž57%† &uÓ&, w}{8 w‚†R~R8nÓ578;†R89vx~$w‚5 X „‡Ó S2n−1 #7ÏR6 η0
y|$Ó P„89†Ž†$Ó7† „ , Ó5a†ŽÓ7?† Pv9ˆpt{‚8 ‡8 L A %5ay|ttU8x{}{‚8xώy n $ ×|ÇÊÇ9ÌËu
Ì ÌÈÅÊÍ÷ÅRÌ }×>ΑÉúçÌ
×>Ú ÈÅRÆ ÌxÎ }×nǹ2Ì 9Î‘É”ÙÍ X {Ý,-7†$~$Ï$Ó7vx~$w‚5¬tÏ$w}ˆ²y|w}ÏR%
8 JtÏR{‚5ƒ´8xÏ X B A %5÷{zy
57~R8xώy c η(η, X) :
n
E
cn (η, X) = c(η0 , X) ∈ H 2n (B, L; Z).
& ÏR†"&uÓ78 n = 2 ]5¬tU8xÓ~w‚„8x5•~$w-#[8xÏ{‚89† 7# ÏR69†
{zyXw Þ$89vx~$w‚53v¹y|575w &uÓ78 S 3 ' SU (2) „7y|57†{‚89†6#7ÏR89† A
Pη
8x~
η0
éuwzy
°²Ÿ´¥D'¡7Ÿ´¥ R¦W ¢¢¤¥$ ¥ ° #*£>Ÿ^ €¡ Ÿ´¥[¦]£´¡W€¥
w}~
ÓC
5 #7ÏR6é´89vx~RÏ$w‚8x{[ÏR698x{bÏ$w‚8x5•~R6 „8ώy|5ƒ
8xˆ²y|Ï"&uÓ757†
&uÓ78 {‚8 #7ÏξR:6EtÏ$→w}57vxBw}t[y|{*„‡Ó¬v9ˆpt{‚8 -w #[6G„8 ξ 89†R~Ó5 SU (n) #7nÏR≥6257A ~R6 P(ξC) A
y|57†{‚8Vv¹y† y|7†R{}Ó ´{zy(tÏR8xˆpw xÏR8vx{zy†Ž†R8V„8Ñ^5•~$ϑÞRy|ƒ>w}5
„8 89†$~„6#75w‚8
ƒ>ρv98 X{zyp„8x$Ó w xˆ;8Gvx{zy†R†R8„8JÛ78xÏ$5¬„8†Ž5¬v9ˆpt{‚8 -w #[6²p: 1 (ξ) ξ
p1 (ξ) = −c2 (ξC ) ∈ H 4 (B; Z).
%53„6#75w}~„8 ˆ/xˆ;8{zyntÏR8xˆpw xÏR8Jvx{zy†R†R8„8Ñ^5•~$ϑÞRy|ƒ>w}53ÏR8x{zy|~$w}é´8;:
) + 3 /A;?/0>D3 E uÓttU>†R57† &uÓ&, w}{8 w‚†$~R8÷Ó578÷†R89vx~$w‚5 s „‡Ó #7ÏR6÷tÏ$w}57vxw}t[y|{
y|$Ó P„89†R†RÓ7†„ , Ó5)†RÓ7?† Pv9ˆpt{‚8 8 L A %5)„6 #75w}~{zy Ø]ÅRÌÈánÉ xÅ$Ì }×|ÇŽÇ¹Ì ËuÌ
P(ξC )
~ X{zyn†R89vx~$w‚5 s t[y|Ï
ÄVفÍ[Î‘Å × >ÉúÍ p (ξ, s) ÅRÌ }×>ΑÉúçÌny|Ó #7ÏR6 ξ 8x!
1
p1 (ξ, s) = −c2 (ξC , s) ∈ H 4 (B, L; Z).
vxwV8x57v9ÏR8 057Ó7†;578vRÛ78xÏRvRÛ78xÏR57†at[y† Ü
„6#75w}Ï;{Ý,-8x57†Ž8xˆG{‚8¬„89†²vx{zy†R†R89†p„8
Ñ^5•~$ϑÞRy|ƒ>w}5/ÏR8x{zy|~$w}é´89† A
X$
$O `XF 0Z X
Fb\HXDJO$W \c d
c I0Z HJI
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eJO1WbQJI
I^Wm_^KRTVZ[Z]I0Z
³aŸ€¢¢¬¥£RBV Ÿ hV¡Ý£´¢Ÿ´¥[¦]£´¡1¢
€é>y|5•~ „8(ƒ´6x576xώy|{}w‚†R8xÏ {zy;tÏRtb>†$w}~$w‚5¯< A = A G 6x5757v9698J„7y|57†{Ý, w}5u~$ÏRu„‡Ó7vx~$w‚5¯y|Ó
v¹y†ÏR8x{zy|~$w Ò uÏR8xé´N贁57† {zy57~$w‚5„8vxÏR>†R†tÏR „‡Ów}~ 8x~„8vxÓtatÏR „‡Ów}~ÏR8x{zy|~$w Ò*„7y|57†
{‚8v¹y†„89† CW Pv9ˆpt{‚8 89†GÔ -./y ut Õ A
?
G
Z*)^>&8?8 5)^>% a /A; )^79-04C;?/
„8xÓ$ CW Pv9ˆpt{‚889† #75w‚† A ~R57† (C (X), ∂ ) ÔâÏR89†$t A
8(„8(vRÛ[y¤ê‚578G†$w}5ƒ>Ó{}w xÏR8Jy†R†Žuvxw‚6D X Ôâ∗ÏR89†$t A YXÕ A
X,šy|tÏ"9†{‚8 ~$Û769Ï"xˆ;8„ ,-0w}{‚8x5ub8xÏ$ƒ ä]w}{}U8xÏÔ -.Üy ~$ۈ H A G Ձw}{]8 w‚†$~R8 Ó5786&uÓw-
é>y|{‚8x57v98(„ , Û7ˆ;~Rtw‚8 C (X) ⊗ C (Y ) → C (X × Y ) A
∗
∗
∗
w‚8x5•~ A ⊂ X 8x~ B ⊂ Y „8x$Ó †RÓ7?† Pv9ˆpt{‚8 89† A y|ttU8x{‚57† &uÓ78
A 05 Ó~$w}{}w‚†Žy|5•~{‚89†÷tÏRtÏ$w‚6x~R69†„‡Ó tÏR „‡Ów}~~R8x57†RÏ$w‚8x9{ 57Ó7†
C∗ (X, A) = CC (X)
(A)
~R8x5757†
w‚8x5•~
8x~
X
Õ{‚8(v9Yˆpt{‚8
(C∗ (Y ), ∂Y )
∗
∗
C∗ (X, A) ⊗ C∗ (Y, B) =
∼
=
∼
=
C∗ (X) C∗ (Y )
⊗
C∗ (A)
C∗ (B)
C∗ (X) ⊗ C∗ (Y )
C∗ (A) ⊗ C∗ (Y ) ⊕ C∗ (X) ⊗ C∗ (B)
C∗ (X × Y )
.
C∗ (A × Y ) ⊕ C∗ (X × B)
&89†0w}57vx{}Ó7†$w‚57† ×B ,→ (A×Y )∪(X ×B) 8x~
„6#75w‚†Ž†R8x5•~VÓ578Jy|tX
t{}w‚v¹y|~$w‚53„8(vŽÛ[y¤ê‚578;:
C∗ (A × Y ) ⊕ C∗ (X × B)
k
A×Y ,→ (A×Y )∪(X ×B)
/ C∗ (A × Y ) ∪ (X × B)
&uÓw w}57„‡Ów}~VÓ5w‚†Žˆ;Ï$tÛw‚†$ˆ;8 8x5Û7ˆ;{‚ƒ>w‚8G8x~8x5v9Û7ˆ;{‚ƒ>w‚8 A
uw u ∈ C (X, A, G ) 8x~ v ∈ C (Y, B, G ) 7 y|{‚ÏR†
p
u⊗v ∈
q
1
"ˆ
C (X)
∗
C∗ (A)
=
"ˆ
⊗
2
C∗ (Y ) ,
G
⊗
G
1
2
C∗ (B) p+q
C∗ (X × Y )
,
G
⊗
G
1
2
C∗ (A × Y ) ⊕ C∗ (X × B) p+q
0 5t[y†Ž†Žy|5•~y|Ӄ>ÏRÓtU89†a„8/v9Û7ˆ;{‚ƒ>w‚8Ü8x~a8x5 y|tt{}w &•Ó[y|5u~
 ~R8x5757†Ó5Û7ˆ;ˆ;Ï$tÛw‚†$ˆ;8Xy|ttb8x{‚69Å1Ù|ÇÊÇ Ø]Å$Ù¹ËÚ ÉâÎÅ$Ì ×>ΑÉ
×
H p (X, A; G1 ) ⊗ H q (Y, B; G2 )
( ),> 5) /0+;\+
(k ∗ )−1
57Ó7†
/ H p+q (X × Y, (A × Y ) ∪ (X × B); G1 ⊗ G2 )
{‚8VvxÏR>†R† tÏR „‡Ów}~ ÏR8x{zy|~$w Ҁ89†$~5[y|~$ÓÏR8x{*:>tUÓÏ~RÓ~R89†y|tt{}w‚v¹y|~$w‚57†
8x~ g : (Y, B) → (Y 0 , B 0) y|Ó$P„89†R†$Ó7†„8 CW Pv9ˆpt{‚889†
f : (X, A) → (X 0 , A0 )
57Ó7†y9鴁57†G: ∀u ∈ H p(X, A, G ) 8x~ ∀v ∈ H q (Y, B, G ) 1
2
(f × g)∗ (u × v) = f ∗ u × g ∗ v.
" &$&/ % &A( > 3$ 8(7 2-&"( $&B(7 > +4/ 1 && 41& 1 > , / > ! 1 > &( / &
2 1C /0""& $& ( > ,-$&/0"/ !;2 X ×Y : 8 $&+"$($& &$/ &A? %A + 1/ C 1 > 8 1 % / > % 4/0
2 CW ) $&')( > ,-& *2-" ) $&+')( > ,-&&:
?><
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Pv9ˆpt{‚88Ü8x~†Rw‚8x5•~ 8x~ „8xÓ$'†RÓ7†?Pv9ˆpt{‚889† A &89†
„8xÓ$ay|ttX{}w‚v¹y|~$w‚57CW
†„8vRÛ[y¤ê‚578 vxw-P„89†R†ŽÓ7†w}57„‡AÓw}~R89†Bt[y|Ï{‚89†w}57vx{}Ó7†$w‚57† v¹y|575w &uÓ789†
„6x~R8xÏ$ˆpw}578x5u~„89†Vw‚†Rˆ;Ï$tÛw‚†$ˆ;89†8x5¬Û7ˆ;{‚ƒ>w‚8(8x~v9Û7ˆ;{‚ƒ>w‚8²:
g:
C∗ (X × X)
C∗ (A × X) ⊕ C∗ (X × B)
→
C∗ (X × X)
C∗ (A × X ∪ X × B)
C∗ (X)
C∗ (A) ⊕ C∗ (B)
→
C∗ (X)
.
C∗ (A ∪ B)
l:
Ó7†y¹é´57†y|{‚ÏR†{‚8„‡wzy|ƒ>ώy|ˆpˆ;8Gv9ˆpˆGÓ~Žy|~$w Ò:
H p (X, A; G1 ) ⊗ H q (X, B; G2 )
×
H p+q (
"ˆ
C∗ (X×X)
C∗ (A×X)⊕C∗ (X×B) ; G1
O
⊗ G2 )
d∗1
/ H p+q
C∗ (X)
C∗ (A)⊕C∗ (B) ; G1
O
g∗

l∗
H p+q (X × X, (A × X ∪ X × B); G1 ⊗ G2 )
d1
8x~
d2
l∗
8 t{}Ó7†tbÓÏ~RÓ~
/ H p+q (X, A ∪ B; G1 ⊗ G2 )
d : X → X × XA
†R5u~a„89†aw‚†Rˆ;Ï$tÛw‚†$ˆ;89†57Ó7†atbÓ鴁57†÷„6#75w}Ï!{‚8 x ÚNØ
H p (X, A; G1 ) ⊗ H q (X, B; G2 )
X
d∗2
†Ž5•~Vw}57„‡Ów}~R†t[y|Ï{Ý,šy|tt{}w‚v¹y|~$w‚53„‡wzy|ƒ´5[y|{‚8
Ñ0Ów‚"† &uÓ78 ∗ 8x~
ØbÅ1Ù¹Ë>ÚuɑÎÅRÌ }×>Î‘É g :
⊗ G2
∪
u ∈ H p (X, A, G1 )
/ H p+q (X, A ∪ B; G1 ⊗ G2 ).
8x~VtUÓÏ~RÓ~
u ∪ v = d∗ (u × v).
v ∈ H q (X, B, G2 )
³ ¦W ¢¢|¥[¢SRCUŸFR £I7Ÿ•¡¢¤£>¡UV ¥[¢÷Ÿ´¥]¦Wb£´¡!€¥[¢÷¥7£ Ÿ[¢ Ÿ hV¡”£
!
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8x~ ξ 0 : E 0 → B 0 „8xÓ$+#7ÏR69†Xé´89vx~RÏ$w‚8x{‚†nÏR698x{‚†JÏ$w‚8x5u~R69†
„8Gώy|5ƒ n 8xξ~ :mEÏR89→†$tU89Bvx~$w}é´8xˆ;
8x5•~ A w‚8x5•~ 8x~ „8xÓ$¬†R89vx~$w‚57†„8G{‚8xÓÏR† #7ÏR69†
8x5܆$tÛ xÏR89†y†R†R vxw‚69† ξ 8x~ ξ 0 y|Ó$P„89†R†$Ó7† „X8X†RÓ7Y†?Pv9ˆpt{‚889† L ⊂ B 8x~ L0 ⊂ B 0
0
0
ÏR89†Rtb89vx~$w}é´8xˆ;8x5u~ A
y|57†V{‚8(v¹y†y|7†R{}Ó 57Ó7†y9鴁57†V{zyptÏRtb>†$w}~$w‚53†$Ów}éy|5u~R8;:
?IH
(*),> >&8?/A;?/0>D3E e2' < 5)^> , €× ׁǎÇ9ÌnË ßڂÌÈÅX× ÈÇxÙ ÚÌ;Ë Ú Í È ÅRÆ
ØbÅ1Ù¹Ë>ÚuɑΠÌÈÇÈÎ } Ì x Å$Ù|ÇŽÇ ØbÅ1Ù¹Ë>ÚuɑÎËuÌÊÇ }×|ǎǹÌÊÇJË ß Ú‚ÌÈÅJ×ÊÇxÙ Ú7ÌÊÇX˕ÌÊÇ ÈÅ$ÆÈÇ
e(ξ × ξ 0 ) = e(ξ) × e(ξ 0 ) ∈ H n+m (B × B 0 ; Z).
y|57† Ó 5ÜtÏR8xˆpw‚8xÏ~R8xˆpt7†U{‚8XÓ~89†$~„8nv957†$~$Ï$Ów}ÏR8nÓ578p†R89vx~$w‚5 X × Y „‡Ó
#7ÏR6 (ξ × ξ0)0 y|Ó$P„89†R†$Ó7†„8 L × B 0 ∪ B × L0 w}57„‡Ów}~R8t[y|Ï {‚89††R89vx~$w‚57†
S n+m−1
x
8
~
*
8x~Jw‚8x5 „6 #75w‚8 ¬Û7ˆ;~Rtw‚8!ÏR8x{zy|~$w}é´8 t"Ï 9† A 057†$Ów}~R8*57Ó7†
X
8 ‡tÏ$w}ˆ;Y8xÏR57†{zypvx{zy†R†R8„ ,-0Ó{‚8xÏÏR8x{zy|~$w}é´8 pv98x~$~R8G†R89vxL~$w‚×53L8x50 Ҕ57vx~$w‚5¬„89†vx{zy†R†R89†
8x~
tUÓÏV~R8x5w}ÏVÓ578 é´8xÏR†$w‚5ÏR8x{zy|~$w}é´8G„8 {zyXtÏRtb>†$w}~$w‚5¬tÏR69v96 e(ξ, X) e(ξ 0 , Y )
„8x5u~R8 A
† [{zya†R89vx~$w‚5 tU8xÓ~
wzy;{Ý, w}5Þ189vx~$w‚5©v¹y|575w &uÓ78
n „7y|57†{‚89† #7ÏR89
/x~$ÏR8÷é Ó78÷v9ˆpˆ;8÷Ó578†R89vx~$w‚5 S„n−1
8 ξ ,→
y|$Ó P„R89†R†RÓ7†p„8 L A €y #7ÏR8 Rn „8 X
6x~Žy|5u~
ξ
v95u~$ώyvx~$w}{‚8 X †R8tÏR{‚5ƒ´8(8x5÷Ó578 †R89vx~$w‚5ƒ>{‚[y|{‚8 X „8 ξ A 8xˆ²y|"Ï &•Ó757† &uÓ78
„8x$Ó ÷~R8x{}{‚89†8 ‡~R8x57†$w‚57†V†Ž5•~VÛ7ˆ;~RtU89†VÏR8x{zy|~$w}é´8xˆ;8x5u~ R L A
uÓttb>†R[email protected]† &uÓ78 X †Žw}~åv957†R~$Ï$Ów}~R8 >"† &•Ó78x{‚8x~$~R8y|t"Ï 9†0"† &uÓ78x{‚8x~$~R8 >v98x{}{}Ó{‚8y|t"Ï 9†
v98x{}{}Ó{‚8 „8{zyˆ²y|5w xÏR8 †$RÓw}éy|5u~R8:¤†Rw σ 89†$~^Ó578v98x{}{}Ó{‚8„8 B \L y|{‚ÏR† ξ| 89†$~*~$Ï$w}é wzy|{
σ
8x~ (X )| „6 #75w}~VÓ578Jy|tt{}w‚v¹y|~$w‚5
R σ
σ
XR
: σ → Rn .
%5 „8xˆ²y|57„8!y|{‚ÏR† &•Ó78 σ †Rw}~ ÷é>y|{‚8xÓÏR†X„7y|57† S n−1 „ 9†%&uÓ78;v>,-89†$~(tU>†?
†Rw}{‚8 A 05Üt[y|Ï$~$w‚vxÓ{}w‚8xÏ „8nv98x~$X
~R8XRˆ²y|5w xÏR8 X ÏR89†R~$ÏR8xw}5•~R8 L ∪ B n−1 tb8xÓ~ /x~$ÏR8
é Ó78v9ˆpˆ;8Ó578(†Ž89vx~$w‚5t[y|Ï$~$w‚8x{}{‚8G„‡Ó S n−1 R#7ÏR6 ξ &uÓw tÏR{‚5ƒ´8 X A
0
„8 ξ 0 u& Ów tÏR %5vRÛ7w‚†$w}~„8 ˆ²y|5w xÏR8Jy|5[y|{‚ƒ>Ó78(Ó578†R89vx~$w‚5ƒ>{‚[y|{‚8(„8
R
{‚5ƒ´8 Y Ôâé Ó78v9ˆpˆ;8 †R89vx~$w‚5!t[y|Ï$~$w‚8x{}{‚8„8 ξ 0 éuwzyG{Ý, w}5Þ$89vx~$w‚5 SYm−1
„7y|57†
,→ Rm
{‚89B† #7ÏR89†ÊÕ A
YR
57†$w‚„6xÏR57†V{‚8 #7ÏR6(tÏRu„‡Ów}~ ξ × ξ 0 A &89††Ž89vx~$w‚57† X : B → E 8x~
w}57„‡Ów‚†R8x5•~5[y|~$ÓÏR8x{}{‚8xˆ;8x5•~Ó578(†R89vx~$w‚5ƒ>{‚[yR|{‚8
: B0 → E0
XR × Y R : B × B 0 → E × E 0 .
uw σ 89†$~0Ó578v98x{}{}Ó{‚8V„8 B 8x~ †$w τ 89†$~0Ó578v98x{}{}Ó{‚8V„8
ÏR89†R~$ÏR8xw}5•~R8 σ × τ „6#75w}~{Ý,šy|tt{}w‚v¹y|~$w‚5 :
fσ×τ : σ × τ
B0
→ R n × Rm
y|{‚ÏR† {zy †R89vx~$w‚5
σ
XR
(x), YRτ (y) .
(x, y) 7→
XR × Y R
#Ów}~$~R8 VÛ7ˆ;~Rtb8xÏ
N5GÏR8x57Ï$ˆ²y|{}w‚†Ž8{‚ÏR†"&uÓ78v>,-89†R~€tU>†R†$w}{‚8 A w}57†$w9
t[y|Ïåv957†$~$Ï$Ó7vx~$w‚5 X ×YXRw}57×Y
„‡ÓRw}~^Ó578†R89vx~$w‚5 X ×Y „‡Ó S n+m−1 #7ÏR6 (ξ×ξ 0)
0
y|$Ó P„89†Ž†$Ó7†V„8 L × BR0 ∪ BR× L0 A
?
L
#+9>&)917 E e E
€× }×|ǎǹÌË ß Ú‚ÌÈÅÅ$Ì ×>ΑÉâç|Ì e(ξ × ξ0, X × Y ) å ×Êؕ؇ׁÅxÎPÌÈÍ]ׁÍ[Î ç|ÆÈÅÈÉ Ì!2 Æ ´× ɑÎPÆÇÈÚuÉâç¤×Í]ÎPÌ
H n+m (B × B 0 , B × L0 ∪ L × B 0 ; Z)
Ee
e(ξ × ξ 0 , X × Y ) = e(ξ, X) × e(ξ 0 , Y ).
 w‚8x5u~ X 0 8x~ Y 0 „ ,šy|Ó~$ÏR89††R89vx~$w‚57†ƒ>{‚[y|{‚89†„8 ξ 8x~„8 ξ 0
ÏR89†Rtb89vx~$w}é´8xˆ;8x5u~€v957†$~$Ï$Ów}~R89†(v9Rˆpˆ;8;RtÏR69v969„8xˆpˆ;8x5•~ A 0{}{‚89†w}57„‡Ów‚†R8x5u~(Ó578;†R89v
~$w‚5 X 0 × Y 0 „8 (ξ × ξ 0) y|$Ó P„89†R†$Ó7†(„8 B × L0 ∪ L × B 0 A {‚ÏR†{‚89†G†R89vx~$w‚57†
8x~ X × Y †R5u~VÛ70ˆ;~Rtb89†ÏR8x{zy|~$w}é´8xˆ;8x5•0
~ L × L0 8x~
X0 × Y 0
e(ξ × ξ 0 , X 0 × Y 0 ) = e(ξ × ξ 0 , X × Y ).
( ),[email protected]
Ñåy|Ïv957†$~$Ï$Ó7vx~$w‚5utUÓÏ~RÓ~R8 pPv98x{}{}Ó{‚8 σ „8 B 8x~~RÓ~R8 q Pv98x{}{}Ó{‚8
R
~
x
8
}
{
‚
{
9
8
†
&uÓ78 p + q ≤ n + m − 1 ]{Ý,šy|tt{}w‚v¹y|~$w‚5 fσ×τ 89†$~0pé>y|{‚8xÓÏR† „7y|57†
τ
B0
Ô
u
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w
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R
~
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S n+m−1
†Ž89vx~$w‚5 S
„‡Ó S n+m−1 #7ÏR6 (ξ ×ξ 0 ) &uÓwutÏR{‚5ƒ´8 X ×YR (BR×B 0)n+m−1 A
X×Y
ˆpˆ;8(„7y|57†V{‚8 t[y|ώy|ƒ>ώy|tÛ78X< A $< 750 tU8xÓ~{}Ów*y†R†R vxw‚8xÏÓ5v9uvxè‡vx{‚8
„6 #75w&t[y|Ï(:
c(ξ × ξ 0 , S
)
„8
X×Y

c(ξ × ξ 0 , SX×Y )(σ × τ ) = [f∂(σ×τ ) ] ∈ πn+m−1 (S n+m−1 ) ' Z
89†$~;Ó578 P v98x{}{}Ó{‚8„8
89†$~pÓ578 Pv98x{}{}Ó{‚8„8
B 0 \ L0
σ
p
B\L τ
q
p+q = n+m
89†$~{zy(ÏR89†R~$Ï$w‚vx~$w‚5!„8{Ý,šy|tt{}w‚v¹y|~$w‚5 f
 f
σ×τ
∂(σ×τ )
y¹é´89v
y|ÓabÏR„!„8{zy n + m
v98x{}{}Ó{‚8
8x~J [f
89†$~J{zyvx{zy†R†R8a„ , Û7ˆ;~Rtw‚8÷„8;{Ý,šy|tt{}w‚v¹y|~$w‚5 f
éuÓ78
]
∂(σ×τ
)
∂(σ×τ
)
v9ˆpˆ;8Ó578Gy|tt{}w‚v¹y|~$w‚5/„8 {zyp†$tÛxÏR8 S n+m−1 „7y|57†8x{}{‚8݈/xˆ;8 A
Ñ^y|Ï(„6 #75w}~$w‚5U{zy!vx{zy†R†R8X„8nv9Û7ˆ;{‚ƒ>w‚8p„8
„ ,- Ó{‚8xÏÏR8x{zy|~$w}é´8 e(ξ × ξ 0, X × Y ) A
c(ξ × ξ 0 , SX×Y )
by|{‚ÏR† bt[y|Ïv957†$~$Ï$Ó7vx~$w‚5 fσ×τ 89†R~0²é>y|{‚8xÓÏR†„7y|57†
uw
n
vx{zy†Ž†R8p„ , 6=
Û7ˆ;
~Rtw‚8 [f
89†R~5•Ó{}{‚8 A
]
89†$~{zy!vx{zy†R†R8
S n+m−1
8x~{zy
∂(σ×τ )
uÓttb>†R57†Vˆ²y|w}5•~R8x5[y|5•~ &uÓ78
8x~57~R57† n ÔâÏR89†Rt A n Õ{Ý,-8x5u~$w‚8xÏv9Ï$ÏR89†?
σ
tU57„7y|5•~ {zy vx{zy†R†R80„ , Û7ˆ;~Rtw‚8 p[X=σn] ∈ π (S n−1
ÔâÏR89†Rt A [Yτ τ ] ∈ π (S m−1 ) Õ
)
n−1
R
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ÔâÏR89†Rt A π (S m−1
Õy¹m−1
é´89v Z :
)
)
n−1
< e(ξ, X), σ > = nσ
8x~
m−1
< e(ξ 0 , Y ), τ > = nτ .
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A
0 5¯8,U8x[f~ ∂(σ×τ
†Rw n ) ] 89=†$~ n5•σÓ.n{9€τy|{‚ÏR† X σ y!é>y|{‚8xÓÏR†„7y|57† S n−1 8x~ f yaé>y|{‚8xÓÏR†
σ×τ
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τ
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σ
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σ : σ → Rn ÔâÏR89†$t A Y τ : τ → Rm Õ89†R~575©5•Ó{}{‚8;8x5¯„8xÛ7ÏR†(„8 |n | ÔâÏR89†$t A
XR
σ
R
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Õ †$w}5ƒ>Ó{zy|Ï$w}~R69†„‡w‚†$~$w}57vx~R89†vRÛ[yvxÓ578X„ , w}57„‡w‚v98
ÔâÏR89†$t A
n
n Õ A Ñåy|Ï
|n | tUw}5•~R†τ = |n |
„6#75w}~$w‚5 f 89†$~575¬†$w}5ƒ>Ó{}w‚8xÏ8x5¬„8xÛ7ÏR†„8 σ|n= |.|n
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P{ 578÷57Ó7σ×τ
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8x5XÓ~$w}{}w‚†Žy|5u~0{‚8t{‚5ƒ´8xˆ;8x5u~v¹y|575w &•Ó78 „8 S n+m−1 „7y|57σ† Rnτ ×Rm |{Ý,šy|tt{}w‚v¹y|~$w‚5
89†$~Û7ˆ;~Rtb8 X{Ý,šy|tt{}w‚v¹y|~$w‚5
f
|nτ |
σ
σ
τ
τ
∂(σ×τ )
S n+m−1 → S n+m−1
(x1 , . . . , xn , y1 , . . . , ym ) 7→ (σ x1 , . . . , xn , τ y1 , . . . , ym )
¹v y|Ï {Ý,-Ï$w‚8x5u~Žy|~$w‚5„7y|57† {zy0#7ÏR8 Rn × Rm „8 ξ × ξ 0 89†$~w}57„‡Ów}~R8t[y|Ï {zyX†$~$Ï$Ó7vx~$ÓÏR8
tÏR „‡Ów}~ A Ñåy|πv957†R6&uÓ78x5•~¤8x5(w‚„8x5•~$w-#]y|5u~*v¹y|575w &uÓ78xˆ;8x5•~ π
n+m−1 ) y9é´89v
n+m−1 (S
Z 5¬~$w‚8x5u~
8x~V8x5¬ƒ´6x576xώy|{
[f∂(σ×τ ) ] = σ τ ,
[f∂(σ×τ ) ] = nσ .nτ A
w}57†$w
e(ξ × ξ 0 , X × Y ) = e(ξ, X) × e(ξ 0 , Y ) A
2
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t{‚889†(:
Z*>D),>d-2-A46/0)^7E e e Ù>ɔÌÈÍ]Î ξ : E → B ÌxÎ ξ0 : E0 → B0 ˕ÌÈÚ>à ÈÅRÆÊÇpç|Ì29ÎÝفÅÈÉÝÌ Ç
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Ø }ÌRà•ÌÊÇ3˕ÌÅ1×Í n ÌxÎ m GÌxÎGËuÌ Å$فÚNØÌ SU (n) ÌxÎ SU (m) Å$ÌÊÇPØÌ29ΑÉâç|ÌÈá²ÌÈÍ]ΠفɔÌxÍ[Î X ÌxÎ Y ËuÌÊÇ(Ç9Ì 9ΑɑÙ>ÍÇJ˕ÌÊÇ ÈÅ$ÆÈÇnÌÈÍÜÇÝØ ÈÅRÌÊÇX×|ÇÊÇxÙ xÉÝÆÊÇ ξ0 ÌxÎ ξ0 ×>Ú 1˕ÌÊǎÇÈڕÇ
ËuÌÇxÙ>ڕ$Ç $RفáØ ‚Ì$à•ÌÈÇ L ⊂ B Ì9Î L0 ⊂ B 0 Å$ÌÈÇÝØ2Ì ¹Î‘Éúç|Ìxá;ÌxÍ[Î Í ØÌÈÚ‡Î€× }فÅÊÇ0 RÙ>ÍÇÈΑÅÈÚuÉâÅRÌ
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Å ‚ÌVØ]Å$Ù ŽÆ$˕ÆØ]ÅR2Æ ŽÆ$˕ÌxÍ[ÎVÚ ÍbÌGÇ9Ì 9ΑɑÙ>Í X × Y ËÚ S 2n+2m−1 xÅ$Æ (ξ × ξ 0) ×Ú ËuÌÊǎÇÈڕÇË•Ì L × B 0 ∪ B × L0 ÈÉÝÌÈÍÜË•Æ Í7ÉÝÌ ‡Ùá;Ù>ÎÝÙÊØ]ÉÝÌÅRÌ ×´Î‘Éúç|Ì L × L0 0GفÚuÇ
×>ç¤ÙÍÇG× }فÅÊ
Ç 2 Æ ´× ɑÎPÆGÇÊÚ Éúç¤×>Í[ ÎPÌ
cn+m (ξ×ξ 0 , X×Y) = cn (ξ, X)×cm (ξ 0 , Y) ∈ H 2n+2m (B×B 0 , L×B 0 ∪B×L0 ; Z).
X ³ ¦W ¢¢|¥ RCbŸT"R £I7Ÿu¡P¢¤£´¡UV ¥¯Ÿ•¥[¦Wb£´¡!€¥ ¥£3¢^# '¥ ¥ ¡”£¥ !
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8x~ 0 0 B „8xÓ$ #7ÏR69†né´89vx~RÏ$w‚8x{‚†pÏR698x{‚†nÏ$w‚8x5•~R69†
„8÷ώy|5ƒ n ξ8x~ : mE ÏR→89†$tbB89vx~$w}é´8xξˆ;8x: 5uE~ay|→
Ó$P„89†R†$Ó7†²„‡Ó ˆ/xˆ;8 CW Pv9ˆpt{‚8‡8 #75w B A
y|ttb8x{‚57B† &uÓ78{zyX†Rˆpˆ;8 „8 g'Ûw}~$578xè÷89†$~ {‚8 #7ÏR6é´89vx~RÏ$w‚8x{&w}57„‡Ów}~ t[y|Ï{‚8!#7ÏR6
tÏR „‡Ów}~ ξ × ξ 0 8x~ {Ý,šy|tt{}w‚v¹y|~$w‚5¬„‡wzy|ƒ´5[y|{‚8 d : B → B × B A Ó7†Vy9鴁57†Vy|{‚ÏR† {‚8
„‡wzy|ƒ>ώy|ˆpˆ;8Jv9ˆpˆGÓ~Žy|~$w ҆$Ów}é>y|5•~J:
E(ξ ⊕ ξ 0 )
/ E × E0
ξ⊕ξ 0
ξ×ξ 0
B
d
/B×B
 E(ξ ⊕ ξ 0) 89†$~V„6#75w&t[y|Ï {(x, x0 ) ∈ E × E 0 / ξ(x) = ξ 0(x0 )} A 8„‡wzy|ƒ>ώy|ˆpˆ;8
}w 57„‡Ów}~å5[y|~$ÓÏR8x{}{‚8xˆ;8x5u~0Ó578y|tt{}w‚v¹y|~$w‚5;„8 #7ÏR69†å8x5X†$tÛxÏR89† (ξ⊕ξ 0) → (ξ×ξ 0) A
0
?IJ
0
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v9ˆpt{‚889† XL ⊂ YB 8x~ L0 ⊂ B ÏR89†$tU89vx~$w}é´8xˆ;8x5•~ A %5¬v9ξ570 †$w‚„ xξÏR08Xy|{‚ÏR†Ó578G†R89vx~$w‚5
„‡Ó #7ÏR6 (ξ × ξ 0) „6#75w‚8(v9ˆpˆ;8GtÏR69v969„8xˆpˆ;8x5•~†$ÓÏ L × B ∪ B × L0 A
X ×Y
{}{‚8aw}57„‡Ów}~nÓ578÷†R89vx~$w‚5 0X ⊕ Y := d∗ (X × Y ) „‡Ó<#7ÏR6!8x5 †$tÛxÏR8 (ξ ⊕ ξ 0)
0
y|$Ó P„89†Ž†$Ó7†V„8 L ∪ L0 A
#+9>&)917[E e €× ׁǎÇ9ÌVË ß Ú ‚ÌÈÅ e(ξ⊕ξ0, X⊕Y ) çÆÈÅÊÉ Ì Æ ´× ÉâÎPÆÇÊÚ Éúç¤×>Í[ÎPÌ
Ù
e(ξ ⊕ ξ 0 , X ⊕ Y ) = e(ξ, X) ∪ e(ξ 0 , Y ) ∈ H n+m (B, L ∪ L0 ; Z)
e(ξ, X) ∪ e(ξ 0 , Y )
ÌxÎ e(ξ 0 , Y )
e(ξ, X)
ÌÊÇxÎ } Ì x ÚNØ Ø]Å$Ù¹ËÚ ÉâÎGÅ$Ì ×>Î‘É Ë•ÌÊÇ
ׁǎÇ9ÌÈÇË ß Ú}ÌxÅ!ÅRÌ }×>ΑÉâç|ÌÊÇ
8xˆ²y|Ï"&uÓ757† &uÓ78 X ⊕ Y 89†$~Gw‚8x5 „6#75w‚8 ÷
Û7ˆ;~Rtw‚8aÏR8x{zy|~$w}é´8
tÏ"9† A
( ),[email protected] Ñ^y|Ïґ57vx~RÏ$wzy|{}w}~R6
L ∩ L0
e(ξ ⊕ ξ 0 , X ⊕ Y ) = e(ξ ⊕ ξ 0 , d∗ (X × Y ) )
= d∗ (e(ξ × ξ 0 , X × Y ))
= d∗ (e(ξ, X) × e(ξ, Y ))
= e(ξ, X) ∪ e(ξ 0 , Y ).
* y²„8xÏ$5w xÏR8X6xƒ•y|{}w}~R6XtÏR¹é w‚8x5•~„‡w}ÏR89vx~R8xˆ;8x5•~„8J{zy;„6#75w}~$w‚5©„‡Ó/vxÓt3tÏRu„‡Ów}~ÏR8
{zy|~$w Ò A
2
Ó7†V~R8x5757†„857Óé´8¹y|Ó3Ó578tÏRtÏ$w‚6x~R6Jy|5[y|{‚ƒ>Ó78GtbÓÏ{‚89† #7ÏR69†Vé´89vx~R Ï$w‚8x{‚†v9ˆpt{‚8‡89†J:
Z*>D),>d-2-A46/0)^7 E e فÉÝÌÈÍ[Î ξ : E → B ÌxÎ ξ0 : E0 → B ˕Ìxځà xÅ$ÆÊÇ;çÌ29ÎÝفÅÈÉ”Ì Ç
ŽÙá
Ø }ÌRà•ÌÊÇÜËuÌ3Å1×Í ÌxÎ ÌxÎpË•Ì Å$فÚNØÌ
ÌxÎ (m) Å$ÌÊÇPØ2Ì 9ΑÉâç|ÌÈá²ÌÈÍ]ΠفɔÌxÍ[Î X ÌxÎ Y ˕ÌÊÇXnÇ92Ì 9ΑɔفmÍÇ;˕ÌÊÇ xÅ$ÆÊDzÌÈÍ SU
ÇÝØ (n)
xÅ$Ìa×|ǎÇ9ÙSU
xÉÝÆÊÇ
ÌxÎ ξ 0 ×>Ú 1˕ÌÊǎÇÈڕÇ
ξ0
ËuÌaÇxفÚuÇ RفáØ ‚Ì$à•ÌÊÇ L Ì9Î L0 Ë•Ì B Å$ÌÈÇÝØ2Ì ¹Î‘Éúç|Ìxá;ÌxÍ[Î Í/ØÌÈڇ!
Î RفÍ7ÇÈΑÅÈ0ÚuÉâÅ$Ìn؇ׁŠ‚Ì
ØbÅ1Ù ÊÆ$˕ÆØ]ÅR2Æ ŽÆ$˕ÌxÍ[ÎVÚ ÍbÌJǹ2Ì 9Î‘É”ÙÍ X ⊕ Y Ë>Ú S 2n+2m−1 xÅ$Æ (ξ ⊕ ξ 0) ×>Ú 1˕ÌÊǎÇÈڕÇ
0
ËuÌ L ∪ L0 cn+m (ξ ⊕ ξ 0 , X ⊕ Y) = cn (ξ, X) ∪ cm (ξ 0 , Y) ∈ H 2n+2m (B, L ∪ L0 ; Z).
$O `XF _^\c nKRI(O
KRTW]O$SVI0Z
I^W'_^KRTVZ]Z[I0Z_^TFUTV_€W åFbOŽZW]O DXI0Z FbI
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2n
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ξ C : EC → B
Ú Í ÈÅRÆ çÌ29ÎÝفÅÈÉ”Ì Å$ÆRÌ
B
Ç9ÙÍ ξ :ÈEÅRÆ →
ŽÙáØ}ÌRàÉ Æ }فÅÊÇ
c2n (ξC ) = (−1)n e(ξ) ∪ e(ξ)
ÈÌ ÇÈÎ }× 2n ׁǎÇ9Ì˕̇ÌÈÅÈÍa× ÊÇ9Ù ÚÌË•Ì ξ ÌxÎ*Ù e(ξ) ÌÊÇÈÎ × ×ÇŽÇ9Ì
C
Ë ßڂÌÈÅJ× ÊÇ9Ù ÚÌnËÚ ÈÅRÆ ξ &8XÓ~„8Xv98x~$~R8n†R89vx~$w‚5©89†$~„8Xώy|ttÏR vRÛ78xÏv98x~$~R8XtÏRtb>†$w}~R5©„‡Ó/~$Û769Ï"xˆ;8
< A L7A L tÏR69v969„8x5•~!y #75„ ,-8x5 ~$w}ÏR8xÏ;Ó578é´8xÏR†$w‚5ÏR8x{zy|~$w}é´8 A Ñ*ÓÏ!v98x{zy$057Ó7†ay¹é´57†
U89†Rw}5„8tÏR69vxw‚†Ž8xÏv90
8 &•Ó78 {Ý,-538x5•~R8x57„t[y|Ï |†R89vx~$w‚5¬v9ˆpt{‚8 ‡-w #[698 A
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57†$w‚„6xÏR57†XÓ5+#7ÏR6;ÏR698x{Ï$w‚8x5•~R6 ξ : E → B „8²ÏŽy|5ƒ n y|Ó$P„89†Ž†$Ó7†J„ , Ó5
Pv9ˆpt{‚88#75w y|w}57†Rw &•Ó78a†R5 #7ÏR6av9ˆpt{‚8 w-#[6÷y†R†R vxw‚6 ξC : EC → B
CW
„6#75w€tÏR69v969„8xˆpˆ;8x5•B~ A
w}~ Ó578p†R89vx~$w‚5¯„8 y|$Ó P„89†R†$Ó7†„ , Ó5¯†RÓ7?† Pv9ˆpt{‚8 8 „8 B :Uw}{å†N,šy|ƒ>w}~
„57v܄ , Ós578©y|tt{}w‚v¹y|~$w‚5 sξ : L → E &•Ów é´6xÏ$-w #[8 ξ.s = id LA wzy¯
{Ý, w}5Þ$89vx~$w‚5
L
v¹y|575w &uÓ78 n Cn „7y|57†{‚891† #7ÏR89† s w}57„‡Ów}~Ó578†R89vx~$w‚5a5[y|~$ÓÏR8x{}{‚8 „8 ξ y|$Ó „89†Ž†$Ó7†„8 LRy|t,→
tU8x{‚698pÇ9Ì 9ΑɑÙ>Í RفáØ ‚Ì$àÉ ÆRÌaË•Ì s 8x~V57~R698 s A 8xˆ²y|"Ï &uÓ757C8
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C
89†$~{Ý,šy|tt{}w‚v¹y|~$w‚5~R8x5•Ó78 t[y|Ï v9ˆptb>†Rw}~$w‚5÷„8 s y9é´89v{Ý, w}57vx{}Ó7†$w‚5v¹y|575w &uÓ78
sC
Ó7†y9鴁57†{‚8(„‡wzy|ƒ>ώy|ˆpˆ;8Gv9ˆpˆJÓ~Žy|~$w ҆$Ów}é>y|5•~G:
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89†$~GÓ5 †Rv¹y|{zy|w}ÏR8a575 5•Ó{„8 A ~R57†Xy|{‚ÏR†
{zy¬†R89vx~$w‚5 „8
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„89†Ž†$λÓ7†„8 L y|w}57†$w]w}57„‡Ów}~R8„8 s A ,šy|CÓ~$ÏR8†R89vx~$w‚5atÏR6Èґ6xλs
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89†R~
(1 + i)sC A
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y|Ó$P„89†Ž†$Ó7†J„8 A wzy{Ý, w}5Þ$89vx~$w‚5 v¹y|575w &•Ó78 n−1
n „7y|57†X{‚89†D#7ÏR89†€8x{}{‚8
tU8xÓ!~ /x~$ÏR8Jé Ó78JLv9ˆpˆ;8X†R89vx~$w‚5܄8 ξ y|$Ó P„89†R†RÓ7S†„8 L,→A %R53tU8xÓ~ y|{‚ÏR†„6#75w}Ï {‚89†
†Ž89vx~$w‚57† t[y|Ï$~$w‚8x{}{‚89†
8x~
„8 A #Ów}~$~R8a{‚89† ÏR8x57Ï$ˆ²y|{}w‚†R8xÏG8x5©{‚89†
Û7ˆ;~Rt[y|5u2~ ;]5¬tUX8xÓC~{‚89†(1鴁+w}Ïi)X
v9ˆpCˆ;8X„ξ89C††R89vx~$w‚57†t[y|Ï$~$w‚8x{}{‚89† „‡Ó S 2n−1 #7ÏR6
y|$Ó P„89†R†RÓ7†„8 L :•5a{‚89† 57~R8xώyJ8x57v9ÏR8 X 8x~ (1 + i)X ÏR89†Rtb89vx~$w}é´8xˆ;8x5u~ A
(ξC )0
C
C
57v98xÏ$5[y|5•~ X 8x{}{‚8 89†$~ 8x5 Ҕy|w}~ ~R8x5uÓ78mt[y|Ï {Ý, w}57vx{}Ó7†$w‚5 v¹y|575w &uÓ78
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S n−1 ,→ S 2n−1
ËuÌ X A
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#7ÏR6 (ξC)0 y|$Ó P„89†R†RÓ7†„8 L †R5•~Û7ˆ;~Rtb89† A 0{}{‚89†Cy|ÓÏR5u~„57vGˆ C/xˆ;8
S 2n−1
7†R~$Ï$Ó7vx~$w‚5¬tÏ$w}ˆ²y|w}ÏR8w A 8 A ˆ /xˆ;8 n Pvx{zy†R†R8„8JÛ78xÏ$5ÏR8x{zy|~$w}é´8²:
cn (ξC , XC ) = cn (ξC , (1 + i)XC ) ∈ H 2n (B, L; Z).
,-‡Þ18x~„‡Ó÷t[y|ώy|ƒ>ώy|tÛ78(< A ?X89†$~„57v „8v9ˆptÏR8x57„‡ÏR8(v98x~$~R8 n Pvx{zy†R†R8„8
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E E %5atb8xÓ~ƒ´6x576xώy|{}w‚†R8xÏ {zy57~$w‚5÷„8|†R89vx~$w‚5÷v9ˆpt{‚8 w-#[6983:
†Rw α 89†$~Ó578(†Ž89vx~$w‚5t[y|Ï$~$w‚8x{}{‚8G„8 P(ξ) y|Ó$P„89†R†$Ó7†„8 L ‡{zyaÇ9Ì9ΑɑÙ>Í ŽÙáØ}ÌRàÉ ÆRÌ
*†R8xώy{zy¬†R89vx~$w‚5 „8 P(ξ)C y|$Ó P„89†R†RÓ7†J„8 L w}57„‡Ów}~R8!„8 α é wzy
ËuÌ &57~R698
{Ý, w}5Þ$α89vx~$w‚5¬v¹y|α57C5w &uÓ78 SO(n) ,→ SU (n)
„7y|57†V{‚896† #7ÏR89† A
AY
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6 ¦¡ ¥£S ¢¤£Ÿ9VhR £´¡W# Ÿu¡ ¡ÝŸ´¥
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Ó5#7ÏR6é´89vx~RÏ$w‚8x{[v9ˆpt{‚8‡8„8 ώy|5ƒ n 8x~„8ƒ>ÏRÓtU8 U (n) A %5ptU8xÓ~
η:E→B
y|{‚ÏR† „6 #75w}ÏÓ5 #7ÏR6Xé´89vx~RÏ$w‚8x{^ÏR698x{ η 8x5܁Ó{}wzy|5•~ {zy²†R~$Ï$Ó7vx~$ÓÏR8Xv9ˆpt{‚88
R
„7y|57†V{zy #7ÏR8(8x~V8x2n
5tU>†Žy|5•~
x2i−1 = Re(zi ),
€8 7# ÏR6
x2i = Im(zi )
i = 1, . . . , n.
89†$~y|ttU8x{‚6pÅRÆ$× É ÆXË•Ì η A
&89†@#7ÏR69† 8x5a†$tÛ xÏR89†y†R†R vxw‚69† η 8x~ η †R5u~v¹y|575w &uÓ78xˆ;8x5•~w‚†Rˆ;Ï$tÛ789†
é wzy!{Ý, w‚„8x5•~$w}~R6a„7y|57†{zy #7ÏR8 A 57v †$w s 89†R~RÓ578;†R89vx~$w‚5 „8 η y|Ó$P„89†Ž†$Ó7†(„ , Ó5
†ŽÓ7†?Pv9ˆpt{‚8 ‡8 L „8 B &8x{}{‚8;tb8xÓ~ /x~$ÏR8p5[y|~$ÓÏR8x{}{‚8xˆ;8x5•~né Ó78;v90ˆpˆ;8a†Ž89vx~$w‚5 „8
y|$Ó P„89†Ž†$Ó7†„8 L 8x~
(η )
ηR
R 0
cn (η, s) = e(ηR , s) ∈ H 2n (B, L, Z)
+6$& "/@2 *% > 8 > /0 2 "$& > / 1 + i (-'6/0 (-1! 1*34+82- > 3 /@
1 /;AB')/0 > 2 > /0"" >&% /0 / /0+2 21 2- 9 > &')&!)2 8 > & C 9&:
(1 + i)X
'
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Ó7†tUÓ鴁57† y|{‚ÏR†„6#75w}φR5Üv9ˆpt{‚8 w-#[6 ξ y|w}57†$w &uÓ78G{‚8JÏR6¹y|{}w-#[6 (ξ )
nA
C
C R
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ξ⊕ξ
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€y;tÏR8xÓé´8X„8Gv98J{‚8xˆpˆ;8JtÏR¹é w‚8x5•~ tÏ$w}57vxw}t[y|{‚8xˆ;8x5•~(„‡Ó/vRÛ[y|5ƒ´8xˆ;8x5•~(„8G[y†R8
„8 R2n 8x5•é´¹è•y|5u~p{zy[y†Ž8²v¹y|575w &uÓ78 (e , . . . , e ) †$ÓÏ (e , e , . . . , e , e ) A
†RÓ7† Pv9ˆptn{‚8 ‡2n8 L
w}57†$w]Ó578†R89vx~$w‚5 s „‡Ó S 2n−1 #7ÏR6 ((ξ )1 ) y|$Ó 2nP„89†R†$Ó7†„ , 1Ó5!n+1
C R 0
„8 B „6 #75w}~XÓ578²†Ž89vx~$w‚5 s0 „‡+
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0
[y†Ž8 A
Ó7†tUÓ鴁57!† XtÏR69†Ž8x5•~„6xˆ;5u~$ÏR8xÏ{‚8 ~$Û769"Ï xˆ;8J†$Ów}éy|5u~G:
#+9>&)917 E e فÉâÎ ξ ÚuÍ ÈÅ$Æç|2Ì 9ÎÝÙ>ÅÊÉÝÌ ÅRÆRÌ VفÅÈɔÌÈÍ]ÎPÆ˕Ì÷Å$×Í 2k Ì9ÎÇxفɑÎ
ÚuÍbÌ/Ç92Ì 9Î‘É”ÙÍ Ë>Ú xÅ$Æ ÌÈÍ ÇÝØ ÈÅ$ÌÈÇ ξ ×Ú ËuÌÊǎÇÈÚ•Ç¯Ë ÚuÍ Ç9فڕ$Ç $RÙ>áØ }Ì$àuÌ L ËuÌ
X
€× 2k }×|ǎǹÌË•Ì ‡ÌÈÅÈÍ Å$Ì }×>ΑÉâç|0Ì3×Ú S 4k−1 ÈÅRÆ (ξC)0 ÌxÎ }×/ǹ2Ì 9ΑɔفÍ
B
ŽÙá
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C
c2k (ξC , XC ) = (−1)k e(ξ, X) ∪ e(ξ, X) ∈ H 4k (B, L; Z)
( ),[email protected] Ó7†y9鴁57†¬éuÓ u& Ó78Ü{zy †R89vx~$w‚5 }w 57„‡Ów‚†Žy|w‚8x5u~¬„8xÓ$ †Ž89vx~$w‚57†
tÏR6ÈҔ6xÏR6989† X 8x~ (1 + i)X „8 (ξ ) y|Ó$P„89†R†RÓ7†X„8 L A 8nt{}Ó7†bv989†G„8xÏ$5w xÏR89†
C
C 0
†Ž5•~VÛ7ˆ;~RCtU89†„57v
c2k (ξC , XC ) = c2k (ξC , (1 + i)XC ).
€y;†R89vx~$w‚5
w}57„‡Ów}~5[y|~$ÓÏR8x{}{‚8xˆ;8x5•~ {zyp†R89vx~$w‚5
„‡Ó #7ÏR6G8x5
†RtÛxÏR89†y†R†R vxw‚6B(1+
{zy i)X
†RˆpCˆ;8„8 gÛw}~$578xè ξ ⊕ξ y|Ó$P„89†R†$Ó7†„8 XL A ⊕wzXy{‚8 ~$Û769Ï"xˆ;8
< A L7A ?;8x~{‚8{‚8xˆpˆ;8G< A ? A H$57Ó7†~R8x5757†
c2k (ξC , (1 + i)XC ) = e((ξC )R , (1 + i)XC )
= (−1)
2k(2k−1)
2
k
Ñ^y|Ïv957†R6&•Ó78x5u~
e(ξ ⊕ ξ, X ⊕ X)
= (−1) e(ξ, X) ∪ e(ξ, X).
c2k (ξC , XC ) = (−1)k e(ξ, X) ∪ e(ξ, X) A
E w‚8x5•~ X x8 ~ Y „8xÓ$ †R89vx~$w‚57†Jt[y|Ï$~$w‚8x{}{‚89†n„8 ξ0 y|Ó$P„89†R†$Ó7†
„ , Ó5 †RÓ7†?Pv9ˆpt{‚88 L „8 B A {‚ÏR†!{zy †Ž89vx~$w‚5 X ⊕ Y „‡Ó #7ÏR638x5 †$tÛxÏR89†
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2
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R2k × R2k → C2k
(x, y) 7→ x + iy.
{‚ÏR†{Ý,-7†$~$Ï$Ó7vx~$w‚53tÏ$w}ˆ²y|w}ÏR8%ntÏR{‚5ƒ´8xÏ
X + iY
X~RÓ~R8{zyX[y†R8
c2k (ξC , X + iY ) = (−1)k e(ξ, X) ∪ e(ξ, Y ).
JG
B
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|y 57† - ~ | ~$w}è•y|Ûp„6#75w}~0Ó5788‡~R8x57†$w‚5pv98x5•~$ώy|{‚8„‡ÓXƒ>ÏRÓtU8„8 „‡w-, 69~Rtw‚89†
ƒ>ρv98Gy|Ó$¬< ”Ò‘ÏŽy|ˆpw}5ƒ´†„89†ˆ²y|ttw}5ƒ²~RÏR89†
M
g
cg → M g → 1
0→Z→M
(?)
 c 89†$~({Ý,-8x57†R8xˆJ{‚8a„89†Jv9Ót{‚89†
y¹é´89v 6x{‚6xˆ;8x5u~X„8
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y|ttUα8xϕ{‚)57†D&uÓ78nϕ{zy÷57~$w‚5 „8;< M
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6 &uÓw}é>y|{‚8x5•~R%8 n{zyX57~$w‚5/„8 SU Ýt[ϕy|ώy|{}{‚6x{}w‚†Žy|~$w‚5 A
Ñ0Ów‚"† &uÓ78{‚89†ˆ²y|ttw}5ƒ;~RÏR89††R5•~„89† H Ýéy|Ï$w‚6x~R69†Ï$w‚8x5•~R6989††Žy|57† bÏR„ 7{‚8
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„6 #75w}~0y|{‚ÏR†^Ó578†Ž89vx~$w‚5X8x57†R8xˆJ{}w‚†$~R8tÏR6Èґ6xÏR698 „8 (?) y|ttU8x{‚698ǹ2Ì 9ΑɔفÍaË Î‘É >× ;
8x~w}{€ˆ;5•~$ÏR%
8 &uÓ78 {‚8v9 vxè vx{‚8Gy†Ž†Ruvxw‚689†R~ 3τ  τ 89†R~{‚8(v9uvxè‡vx{‚8„8(./8xè´8xÏ A
wzy;{zy²†$ÓϑÞ$89vx~$w‚53v¹y|575w &uÓ78 M → M [5܁~$w‚8x5•~ Ó578J8 ~R8x57†$w‚5Üv98x$5 g,1
g
~$ώy|{‚8G„8 M
g,1
cg,1 → Mg,1 → 1
0→Z→M
(?0 ).
Ó7†y9鴁57† 8x57v9ÏR8XÓ578XÏR6x~$ώyvx~$w‚5/tÏR6ÈҔ6xÏR698n„557698Xt[y|Ï{‚8 σÝw}5•é>y|Ï$wzy|5•~(8x~Ó578
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cg
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ϕ, αϕ ), (ψ, αψ )
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n =< (f ) (σ), µ > .
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i
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g,1
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[email protected]]DM=8[αSTP M]NAr D MLU[βL :]X8P KDMLUP]; 19 MI <919d=1DFE<;>P 7b9 KO~([αDK;>=8],NU[βX KO])α ML Hβ (ΣV ; Z) 9
; <OUP]9 MNF; <Oa= [email protected] Z? KDML M=8NFOaN MI8N KO –L[@ –L ic= <;>=<O <;>L MNF; <O~9
[email protected] MH [email protected] 8P]žMN V W S j 5<X8N (TΣ ) KOUL"[email protected]]O[@BX8S OV E<;>P]OUP]OFOF; <O O DK;>HJH
O KDMLUP]; šI8N yx MN <;>=<O*IŸ;>=8?u; <[email protected]];>NFOvP]9 cLUP 5 MNS [email protected]>D LF;>L[@BS89 (TΣ ) @A? KD
v;>[email protected] Z? KDML M=8NFO~OU=8N Σ I€;>=8NUN[@š98; <DMLUN Z?c= DK;>HJH f=
Σ ×S V
@BI8I8STP][email protected]]; `9 Σ 9<@ <O S V
i
∗
i
∗
i=1
i
i i
1
i
g,1
1
N
g,1 0
3 g,1 0
g,1
8 ' O*kh
P
% O
1
g,1
K/JPJ
1
g,1
i
EhCk )[h
O 8
nX
γ N
/2'if )[hCk 8 /2'[f
hCk
N
%Tf
8 '/O
#f hCk
3
šI M=8L*O |9 [email protected] <9 MN D wic= ?>@B=8L
IŸ;>=8Nt9 KODK;>=8NUX [email protected]]DM=8STP]žMN KOL MSTS KO
ic= [email protected];>=8NUX :;>NUP –L ∂Σ ;>=6S KO~DK;>n=8NUX KOeO [email protected][@ –L KO V
a/`@[^#h"` m RPhjeN` TR
X
γ g,1
Σg,1
HgXhR\OgL IKaPR OfeUI*hTmI*LUm Z
ET E
G γ ‚}ˆy„ ’  c† ‡ ‚J ‡ ™‚yƒG„“ŒF‚
;7>= ’ ‡ˆ
∂Σg,1 nX
γ = 2g − 1.
G ‚}ˆy„ ƒŸ‚  ‚eˆAŒ“‹d’ ’>ƒG„“‚Z”–‚ , ‚Mƒ ‚ ‰ ƒ<z‚Z”–‚  ™‚yƒG„™’R„z>ƒm‘ƒ ” „“‚
‹d’ ‡ γ ‚ ‚ † ”–‚ ’{†cˆK‡ † ˆ ˆ †c‡ ‡ ’ ‚ † ƒ€‚ R‡ ‰\‹8hBˆM’>†ƒ „“‚ ”–‚  ‡ ” ‡ ) ’  ‡ ˆ †
nX
γ = 2h − 1.
235476(3
S <;>=<O:OU= {LY9 MLU=<9dP MN S [email protected]>O
W [email protected]>OY9 KO:DK;>=8NUX KO
O [email protected][@ –L KO\O MN[@J;>X8L –=`DK;>HJH YDK;>NF;>[email protected] YPTHJH γ[email protected] ∂ΣV V
;>PTL I × S = (?u;>P]OUP [email protected] m;>NUP –L [email protected][@BH MLUN m9 [email protected];>=8NUX l;>NUP cL γ =
9<@ <O Σ L MS"i–=
∂Σ
8 γ ={ }×S
; 5 <DMP]9 @K? KDD MSTS P <9d=8PTL [email protected] ;>NUP j
8 S ;>NUP –L[@BLUP]; [email protected] <; 8P]i–= 9 I ×S DK6
L[@BLUP]; 19 Σ Dyxt5<QR=8N : V C V
g,1
-
g,1
g
1
2
1 γ
3
1
1
γ
3
g γ
D2
Iγ × S 1
0)132 4
X
VC
8
W"@[email protected]>OFO 9 9 =8S MN"N [email protected]? e(TD , X| ) H KOF=8N S ;>X<OULUNU=<DMLUP]; I8NF;>S]; 8Q MN
MLa? MNUP 5 DyxN [email protected]= E V o V H
Σ
:
g
2
3
∂D 2
< e(TD 2 , X|∂D2 ), µD2 >= 2 − 2g.
l9 ^@B=8LUN KO L MNUH KOAr X [email protected] 2 − 2g LF;>=8NFOS aS]; 8QY9 ∂D [email protected][@BI8I€;>[email protected]={D[[email protected]
9 {? KDML M=8NFO ic=8PqO ML <[email protected]=29dP]O[i–= V Z;>LF; <O A = #L MSqD[[email protected](9 J? KDML M=8NFOšOU=8N
Dyxv5<QR=8N : V Adr<N [email protected]–=<; <Oei–= PTSsO fI8NF;>S]; 8Q D V
∂D
9
2
$
-
2
2
γ
D2
A
0 132)4
:
V A 8 [email protected] Y? KDML M=8NFO A OF=8N
ETC
∂D 2
W YDK;>HJI8L M=8N KOULaS <;>H X8N š9 :LF;>=8NFO\ic= fx™@BPTL S S]; 8Ql9 γ = ∂Σ
[email protected]\N[@BI8I€;>NUL I n V S I M=8L MLUN :[email protected]]DM=8S nDK;>HJH :[email protected];>XHJH
D MST=8P j DMP KOUL MQ–@BS
8 9d= <;>H X8N n9 :LF;>=8NFO~ic= [email protected] A [email protected]~N[@BI8I€;>NUL
I
r
1
M
~
L
d
9
=
<;>H X8N n9 fLF;>=8NFO\ic= [email protected] X [email protected]~N[@BI8IŸ;>NUL A V
8
bN [email protected][email protected] –Leic= γ = −∂D r <;>=<O\;>X8L <; <O
9
-
X γ
∂Σg,1
γ
g,1
2
nX
∂Σg,1
= 1 − (2 − 2g)
= 2g − 1.

‚}ˆy„ † Ÿƒ ‚Y‹d’> ‡ ‚  ‡ ”u’>ƒ „“‚6”–‚ , y‚ ƒ ‡ ‚
7
; >=]7
(γ, γ 0 )
Y
Y
1
nγ + nγ 0 = 2h.
’ ‡ˆ
2
h
† ‡‚
,
F
γ
α
γ0
> *
* h
7 MHJI8S 9 [email protected] YX€;>NF9<@ cL γ r γ [email protected]>O~OUL[@ <9<@BNF9 V
235476(3 ;>PTL
= :[email protected] YXŸ;>NF9<@ –L n9
9 :Q 8N S [email protected]>O[O KO
9 E<;>H{;>S];>QRP :9 KO|DK;>(γ,
=8NUX γKO)? MNUP 5 –L [γ] = −[γ ] V 9 ΣSTS KO|O [email protected] [email protected] OF=8N [email protected]>D Σ
b9 M=d7bOF;>=<O j OU=8N [email protected]>D KO Σ ML Σ V
; <OUP]9 MNF; <O|= fDK;>=8NUX ZOUPTHJI8S ex MNUH 9
i–=8PŸO [email protected] [email protected]=8N [email protected]>D
= lOU=8N [email protected]>D Σ ML = [email protected] {9 [email protected] –L[@BS]; αP ML:ΣL MSTS {ic= S JX€;>NF9_9 P OF;>PTL
e;>LF; <O P S Z9dP]OFic= e;>X8L c=69 P 0N KDK;>[email protected] cL~9 KO|9dP]OFi–= KO d ML d
α∪γ ∪γ V
DK;>HJH 9<@ <[email protected]<QR=8N : V H V
;>PTL Y = bD[[email protected] :? KDML M=8NFO <; 1OFP 8QR=8STP MN\9 Σ V W ;>X<OFLUNU=<DMLUP]; I8NF; j
S]; 8Q MN Y | P KOULa= [email protected]>OFO :9 9 =8S MN~N [email protected]? e(TP , Y | ) ic=8P ? MNUP 5
V
0)132 4
h,1
0
α
0 0
g−h,3
0
h,2
g,1
g,1
h,2
1
0 0
: : 8 9
g,1
0
< e(TP 0 , Y |α ), µP 0 >= 2 − 2h.
2
α
g ^@BI8NFžKO\[email protected]šLUE K;>NUP 9 :DK; <D cLUN[@BLUP]; `9 ;>X<OULUNU=<DMLUP]; sr <;>=<[email protected]?u; <O~S MQ–@BSTPTL
3
< e(TP 0 , Y |α ), µP 0 >= − < e(Td1 , Y |γ ), µd1 > − < e(Td2 , Y |γ 0 ), µd2 > .
e;>LF; <O l :=< e(Td , Y | ), µ
;>X8L <; <Oe98; <D:S MQ–@BSTPTL
Y
γ
3
1
γ
d1
ML
>
lγY0 :=< e(Td2 , Y |γ 0 ), µd2 >
lγY + lγY0 = 2h − 2.
ET
A
V Z;>=<O
P
γ
α
d1
γ0
d2
P0
0)182)4
V H 8 o @BPTN n9 :[email protected] –L[@BS];
:
P
MLa?>@BNUP ML
P0
A
d1
γ
V o 8 [email protected] :? KDML M=8NFO A OU=8N γ
@BS]DM=8S]; <O I8N KO cL xz; <DMLUP]; 9
DK; <OUP]9 MNF; <ObS _D[[email protected] 9
? KDML M=8NFO I <; 1OUP 8QR=8STP MN~nS fS]; 8Q09 γ 9 y5 8P"DKl;>HJH :I8N KD K9 MHJH cL V
;>PTL A S [email protected] 9 a? KDML M=8NFOOU=8N γ 9 y5 8P 9<@ <[email protected]<QR=8N : V o PTSGO ~I8NF;>S]; 8Q
W KO –LUP MNFO ML OF; –LwS <;>H X8N Z9 ~LF;>=8NFOwi–= [email protected] S aS]; 8Qš9 [email protected]
N[@BdI8IŸV ;>NUL A ML Il N KOUI nKDMLUPT? MH –L V N A x™@BPTLa= bLF;>[email protected][[email protected]Ÿ;>NUL I γ@BP <OUP
0)132)4
γ
-
1
Y γ Y γ
:
Y
γ
Y
γ
nYγ = 1 + lγY .
γ 0 nYγ0 = 1 + lγY0
nYγ + nYγ0
γ
g fHMH Rr ; b;>X8LUP –LeI€;>=8N
γ
ML
= lγY + lγY0 + 2
= 2h.
2
K/JPJ
)[h
/21
Eh 'ifLh %(8 /
h N 8 Eh 'if k
Q
)ihCk *k k )[h " h -/Q k'if /O hCk 8 -5%(1
k
N
fg @ <OeS š[email protected] šI8N KD K9 –LAr <;>=<O:@K?u; <OZ? =Cic= YS YQRNF;>=8I š9 n9dPnm K;>LF;>I8P KO
@BQRPTL:OU=8NZS [email protected]>D [email protected] KDML ML:ic= D MLUL @>DMLUP]; [email protected] ML:DK;
M
[email protected]@ –L V [email protected] MS]; <O\ic= YIŸ;>=8N\LF;>=8L(ΣMS MH ) –L ϕ 9 M ML~I€;>=8NeLF;>=8L\IŸ;>P –L xj
9 Σ r
3 g,1
g,1
g,1
g,1
(ϕ∗ X)x = Tϕ−1 (x) ϕ(Xϕ−1 (x) ).
ETX:
>; PTL γ = ~DK;>=8NUX aOUPTHJI8S |x MNUH a;>NUP –L eOU=8N Σ ML τ OF; JL aP]OULq9 ag ME
9dNF;>PTL V [email protected] MN[@BPTLeO}@K?u;>PTN\O PTS KOULaI€;RO[OUPTX8S :9 y7 I8NUPTH MNeS YDK;>HJI8L M=8N n
xz; <DMLUP]; 19 n I€;>=8N\LF;>=8L YDK;>=8NUX :OFPTHJI8S fx MNUH ;>NUP –L β 9 Σ V
";>=8L\9 ^@BXŸ;>NF9 r<N [email protected]=<; <O~S Zx™@[email protected] –L
= @;7>= ;7~W :L aP]OUL
KOUL~9 y5 [email protected]~S P]9 cLUPTL b9 ME<;>NFOe9 = b?u;>P]OUP j
[email protected] [email protected][@BH MLUN ML~;>NUP –L I τ× S 9 γ ML~OF=8N I × S [email protected]
$
X
β
g,1
γ
# (τγ )∗ X
β
g,1
(
γ
1
γ
3 '
1
γ
τ : Iγ × S 1 → S 1
(t, eiθ ) → ei(θ−2πt)
W [email protected]`9 f? KDML M=8NFO
-
(τγ )∗ X
KOFLaE<;>H{;>LF;>I
Xxγ = Xτγ−1 (x)
;<
Xγ
Xγ
KOUL\9 y5 [email protected]
∀x ∈ Σg,1 .
~ †c‡ ™„  † „“‚  c† ‡ ‚nˆy‘‰e‹ ‚ ‚ ‡ ‰lŒF‚ ⠈ †c‡ Σ )
Rœ = ±1
= n + (β · γ) n
n
‚K„>œ · ‚}ˆy„ ŠƒG„“‚ ‡ ˆK‚ „>ƒ_’ , Œ y‡  I † ‚{”u’>ƒ8ˆ H (Σ ; Z) 1
235476(3 ;>PTL
= _?u;>P]OUP [email protected] {N MQR=8STP MN ;>NUP –L {9 γ 9 y5 8PqDK;>HJH
I ×S
I8N KD K9 MHJH cL V
P"S KO\DK;>=8NUX KO β ML γ nO P –L MNFO KDML [email protected]>OAr8S D[[email protected]`9 ? KDML M=8NFO (τ ) X
KOFLaE<;>H{;>LF;>I X S :S]; 8Ql9 β V =<OFOFP n = n V
StN KOUL MLU=<9dP MN:S š[email protected]>Of9 = #DMNF;>P]O MH cL:I€;ROFPTLUP xw9 KO:DK;>=8NUX KO β ML γ MLfS
[email protected]>O\9 = 1DMNF;>P]O MH –L MQ–@BLUP x V
;7>= )0
g,1
(τγ )∗ X
β
X
β
X
γ
1
γ
1
$ g,1
(τγ )∗ X
β
Iγ
γ
β
Iβ × S 1
:
VG
X
β
0 132)4
γ ∗
Iβ
Iγ × S 1
8 NF;>P]O MH –LeI€;ROUPTLUP x9
ET
H
β
ML
γ
=8I8I€;ROF; <Owi–= |S KOqDK;>=8NUX KO ML O P cL MNFO KDML –LDK;>HJH \9<@ <[email protected]<QR=8N G
[email protected] MS]; <O\i–= <;>=<[email protected]?u; <O\LUNUPT?cβ[email protected]]O YSγ S j 5<X8N (TΣ ) QRN >D @B=1D[[email protected]: 9V V
? KDML M=8NFO O rdic=8P KOUL|= N MI8N KO –L[@ cLe9 <;>LUN :;>NUPTQRP :I8N yx MN :9 KDML (Σ )
S [email protected] 9 1? KDML M=8NFO X OU=8N Σ I M=8L MLUN C9 y5 8P\DK;>HJH b= `@BI8I8STP][email protected]];
`I M=8LfOU=8I8IŸ;RO MNfi–= Rr S S]; 8Q69 I × S r D MLUL [email protected]][email protected]]; KOUL
Σ →S V
98; [email protected]
-
$ g,1
1
#
1
g,1
g,1 0
1 γ
g,1
(Iγ × S 1 ) → S 1
X
(t, eiθ ) 7→ einγ θ .
g ^@BI8NFžKO\[email protected]šN [email protected]–= : V E V : r (τ ) X [email protected]];>NFO~E<;>H{;>LF;>I :S YS]; 8Ql9
β ∩ (I × S )
γ
X
(t, eiθ ) 7→ einγ (θ+2πt)
; <OUP]9 MNF; <[email protected]];>NFO\S KO\@BI8I8STP][email protected]]; <O
β ∩ (Iγ × S 1 ) → S 1
γ ∗
1 fβX : β → S 1
x 7→ ∠(Iβ )x , Xx
ML
(τγ )∗ X
: β → S1
fβ
x 7→ ∠(Iβ )x , ((τγ )∗ X)x
bN [email protected]–= [email protected]];>NFO~i–= Rr<I€;>=8NaLF;>=8L
(τγ )∗ X
fβ
x = (t, eiθ ) ∈ β ∩ (Iγ × S 1 )
r
X
(x) = einγ (θ+2πt)
X
= fβX (x) + einγ (2πt) .
P <OUP™r€IŸ;>=8N:[email protected]>ic= šDMNF;>P]O MH –LYIŸ;ROUPTLUP xw9 β @K? KD γ r ; •N[@U;>=8L n LF;>=8NFO
@B=0DK;>HJI8L M=8N n Vuo @BNS aHMH \I8NF;cD K9 Rrc; {I M=8LwH{; –LUN MN|i–= \OUP<S ~DMNF;>P]O MH cL
KOFL MQ–@BLUP x @BS];>NFO\; bN[@ ;>=8L −n LF;>=8NFO V
o @BNZDK; <O Ki–= cLAr n = n + (β · γ) n .
X
β
(τγ )∗ X
β
X
γ
X
β
X
γ
X
γ
2
/ $ !$ !$ + "!
! !+ "! "! +
kX
"!-(
W MLU=<9 n9d=•DK;>HJI8L M=8Nf9 y5 8P kS @{O KDMLUP]; CI8N KD K9 –L [email protected] <;>=<OeO MNU?cPTN AD @BS j
DM=8S MN y7 8I STP]DMPTL MH cL\S ;>X<OULUNU=<DMLUP]; k €I ;>=8N~= 6I8NF; 9d=8PTL\9 :L aP]OULFO~9 Yg ME V
3
X
ET
o
# [email protected] MS]; <O~i–=
KOFLa= bE<;>H{;>H{;>NUI8E8P]OUH DMNF;>P]O
kX
kX (ϕ.ψ) = kX (ϕ) + ϕ∗ .kX (ψ)
∀ψ, ϕ ∈ Mg,1 .
=<OFOUP™r*PTS <;>=<O0OU= {L09 bO[@K?u;>[email protected]]DM=8S MN y7dI8STP]DMPTL MH –L k IŸ;>=8N{= L aP]OULl9
g ME V
3 ;7>=0~
„™ „w„›kˆM„q”–‚"'n‚ ƒ ” ‡ >„“‚ τ ‚ Rƒ , ” † ƒ€‚  † ‡ ‚
ˆy‘‰e ‹ ‚ ‚ ‡ ‰lŒF‚ 㠈 † ‡ Σ†c‡ ) †
>œ = ±1.
k (τ ) = n [γ],
1;>X8LUP –LePTHJH [email protected] MH –LZS :DK;>NF;>[email protected] [email protected] –L

X3 ;7 Ÿ>z‚MƒG„
‚}ˆ(ˆ}Š‰e‹ ‚yˆ ‚ ‡ ‰lŒF‚}ˆ#ˆ †c‡ Σ 1
γ , . . . , γ n †c‡ R„™>ƒ8ˆ τ ‚•„›‘ˆy„ ”–‚ 'š‚ ƒ ” ‡ >„“‚ ‚ >ƒ , ”–‚ γ ) ‚M„ = ±1 ‹d †c‡ „™ † „
 ˆ
i = 1, . . . , n 1 ‡
2
X
# %'&
γ
g,1
γ
X
X
γ
1
n
g,1
i
i
kX (τγ11
. . . τγnn )
=
1 nX
γ1 [γ1 ]
n
X
+
i
i−1
1
i nX
γi (τγ1 . . . τγi−1 )∗ [γi ].
S <;>=<OaN KOFL I8NF;>=8? MN\S fLUE K;>NFžMH : V C V E V
i=2
K/JJ
fh h gf h
K,JJCJ
vP 78; <OY= 0DK;>=8NUX 0OUPTHJI8S Jx MNUH l;>NUP cL γ 9<@ <O Σ V <;>L <DK;>N
= 2?u;>P]OUP [email protected] l;>NUP cL 09 ML O[; #LaP]OULš9 {g ME 9dNF;>PTL 09 y5 8P]O
I ×S
DK;>HJH 9<@ <O~S [email protected][@BQRN[@BI8E YI8N KD K9 γ cL V τ
; <OUP]9 MNF; <[email protected] A j [email protected] ML I × Σ @BP <OUPi–= S 5<X8N ? KDMLF;>NUP MSN MS
h [email protected]=<; <Ošic= mD MST=8P j DMP KOFL LUNUPT? [email protected]]O[@BX8S MLšD[E<;>P]OUP]OFOF; <Oš[email protected]
ξ := I × TΣ V
O KDMLUP]; _I8N yx MN JDK; <OUL[@ cL JS S]; 8Qb9 I ML:P <9d=8PTL [email protected]:S [email protected](9 ? KDML M=8NFO
e;>=<OwIŸ;>=8?u; <O|@BP <OFP P]9 cLUP 5 MN~OF; [email protected]>D eLF;>L[@BS
g eHMH Rr
OV
<;>=<OvIŸ;>=8?u; <O"P]9 cLUP 5 MNvS [email protected]>D wLF;>L[@BSc9 wOF; S j 5<X8N |@I>×OFOF;ΣDMP ξ× RI V×Σ ×S V
e;>LF; <O X [email protected] KDMLUP]; [email protected] MSTS e9 ξ 9 y5 8P [email protected]= j 9 KOFOU=<Ow9 ∂(I × Σ ) [email protected]
X : ∂(I × Σ ) → S
OUP t = 1
(τ ) X
OUP <;
( t , x ) 7→
X
[email protected] MS]; <O\ic=
KOUL~P –L MNUI8N ML šQ K;>H MLUNUP]ic= MH –LfDK;>HJH [email protected]@>OFO
9 E<;>H{;>S];>QRP Rr"?c= 9<[email protected] <O (τH) (Σ ; Z) rs9 = JDK;>=8NUX {9 ;>X<OULUNU=<DMLUP]; I8NF;>S]; 8Q MN
Z;>LF; <O I = [ , ] ⊂ I V
X
I ×Σ V
W :L aP]OFL\9 :g ME τ O fI8NF;>S]; 8Q b= b9dnP m K;>H{;>NUI8E8P]OUH ψ 9
P –L
9 y5 [email protected] id × τ OF=8N [ , 1] × Σ r ML\S P]9 cLUPTL
I × Σ \ (I × I × S )
b9 ME<;>NFO V
D
1
γ
g,1
3 g,1
γ
2
0
g,1
1
g,1
0
γ
1
# γ
g,1
2
g,1
1
g,1
γ ∗
x
x
γ
X
2
g,1
-
# γ
g,1
2
γ
1
γ
1
g,1
1 2
3 3
(
I
ET
G
γ
2
3
γ
g,1
3 Iγ × S 1
ψγ
I2
0)182)4
V
:
idI2 × τγ
id
8 ψγ
mIŸ;RO Xe [email protected]šO KDMLUP]; 69d= S j 5<X8N (ξ) P <9d=8PTL [email protected] X ML|DK; <OFL[@ –L fS ZS]; 8Q
9 W 9dPnm K;>H{;>NUI8E8P]OUH @BQRPTLZOF=8N I€;>=8NZ9 y5 8PTNe= nO KDMLUP]; C9
9 ME<I;>V NFO|9 \S P –L MNUP M=8Na9d=0LF;>N ψ I × I × SXe ic= <;>=<O <;>L MNF; <O (ψ ) Xe V h (ξ)[email protected] j
ic=<; <O~ic= MSTS :I8NF;>S]; 8Q X V
4 3 3 uW I8NF;>X8S]žMH q9 ;>X<OFLUNU=<DMLUP]; nN M? P –L DFE MNFD[E MN I8NF;>S]; 8Q MN
= [email protected] 9 ~? KDML M=8NFO <; 0OUP 8QR=8STP MN|9 y5 8P OF=8N S ~XŸ;>NF9l9d=0LF;>N ∂(I × I × S )
@B= LF;>N I × I × S V [email protected]; <DfI8NF;>S]; 8Q MN X 69 ME<;>NFO~9 fDK;>I8P KOa9 fS BH
9 :D YLF;>N :P V V 19 ME<;>NFO\9 :DK;>I8P KOe9 :[email protected] DK;>=8NUX γ V
e;>LF; <O ζ S 5<X8N 1? KDMLF;>NUP MSZ;>X8L c= [email protected]{N KOFLUNUP]DMLUP]; 9 ξ I × I × S
ML [email protected] KOULUNUP]DMLUP]; 9
h [email protected]=<; <Onic= {D KO 5<X8N KOnOF; cL
[email protected]ζe=8N MSTS MH cLeLUNUPT? [email protected]]O ζKO V ∂(I × I × S ) V
3 3 \ X3 ( [email protected] <; <Of9 I8ST=<O\I8NFžKOeS :5<X8N
COUI8E<žMN ζe V ;>PTL
S <O MH X8S {9 KO:O KDMLUP]; <O 9 ζe DK; <OFP]9 MN KO E<;>H{;>LF;>I8P JI8NFžKO D KOULf=
S(ζ̃ )
[email protected]>D @ @B= j 9 KOFOU=<O\9
H (∂(I × I × S ); Z) ' Z[α] ⊕ Z[β]
; < ML OF; cL"S KOvDK;>=8NUX KOvOUPTHJI8S KO x MNUH KOv;>NUP cL KO
ML ∂(I ×I )×{∗}
{∗}×S
N KOFIαKDMLUPTβ? MH cL V
; <OUP]9 MNF; <OYS {D[[email protected] J? KDML M=8NFO I <; 2OUP 8QR=8STP MN OU=8N ∂(I × I × S )
P <[email protected]\[email protected] KDMLUP]; CI8N yx MN :9 I PTSs9 y5 8PTL~= :O KDMLUP]; bDK; <OFL[@ –L Y9 ζe V
h [email protected]–=<; <O\i–= I I M=8L~O fI8NF;>S]; 8Q [email protected]=6LF;>N I × I × S V
o ;>=8NeLF;>=8L YO KDMLUP]; Y 9 ζe r8; 6I M=8Le9 y5 8PTN~S KO\@BI8I8STP][email protected]]; <O
2
1
γ
γ
0
3 - 1
/
0
γ ∗
γ
)
2
0
1
2
1
γ
2
γ
γ
1
3
1
γ
0 , 1 2
2
gαY : α → S 1
x 7→ ∠(Iγ )x , Yx
gβY : β → S 1
x 7→ ∠(Iγ )x , Yx
ET
γ
0
ML
γ 1
γ
0
2
2
γ
γ
2
1
3 1
γ
γ
2
γ
1
0
1
@BP <OFPsi–= :S :DK;>=8I8S 9 –LUP MNFO
X
(mX
α , mβ )
ML m =< (g ) (σ), µ > .
=<OFOFP™r*S KO –LUP MNFO ML DK;>HJI8L –LlS <;>H X8N 19 bLF;>=8NFOlic= [email protected]
[email protected] N[@BI8IŸ;>NUL I S CS]m; 8Q 9 KO mDK;>=8NUX KO α ML β N KOUI KDMLUPT? MH –L V KO cLUP MYNFO
9 ML MNUHJP cL cLUP]žMN MH –[email protected] [email protected]>OFO :9 E<;>H{;>LF;>I8P š9 = YO KDMLUP]; 19 (ζ) V
W ;>X<OULUNU=<DMLUP]; I8NF;>S]; 8Q MNb= •O KDMLUP]; Y 9 ζ @B= LF;>N •LF;>=8L cLUP MN1O
[email protected]]DM=8S x™@>DMPTS MH –L ML n9 MI <9`ic= 9 nS –LUP MN m V o =8P]OFi–= I O I8NF;>S]; 8Q
[email protected] [email protected]>O[O :9 9 =8S MN\N [email protected]? e(ζ , Y ) ? MNUP 5
ζ
mYα =< (gαY )∗ (σ), µα >
γ
Y
α
β
0
Y
β
3 0
Y
β
Y ∗
β
Y
β
0
γ
0
DP (e(ζ0 , Y )) = mYβ [β] ∈ H1 (I2 × Iγ × S 1 ; Z) ' Z[β].
3 4 35476(3 \4 & 3 ; @>= sh M? <; <O I8N KO cL <;>LUN
I8NF;>X8S]žMH 9 ;>X<OULUNU=<DMLUP]; ML:@B=`D[[email protected]_9 ? KDML M=8NFO <; •OUP 8QR=8STP MN (ψ ) X̃ VGo @BN
N KOFLUNUP]DMLUP]; sr8PTSsP <9d=8PTL~= YO KDMLUP]; X̃ 9 ζe 9 y5 8P [email protected]
γ ∗
0
γ
OUP t = 1
OUP <; V
I8N KO cLAr8PTS <;>=<O~OF= {L\9 :[email protected]]DM=8S MN m V
[email protected] MS]; <Olic= KOUL{S <;>HnX8N C9 bLF;>=8NFO0ic= x™@BPTL [email protected][@BI8I€;>NUL @B=
D[[email protected] \? KDML M=8NFO In S \S]; 8QJ9 γ V h [email protected]–=<; <O|9 ZOU=8PTL Zi–X= \[email protected] DK;>=8NUX α KOUL
S BH n9d=1LF;>N I × I × S MLAr<[email protected]\9 y5 8PTLUP]; sr = DK;>I8P n9 [email protected];>=8NUX γ @BP <OUP
m =n V
o =8P]OFic= I KOFL DK; <OUL[@ –L S |S]; 8QY9 I ×S r>PTS KOUL*E<;>H{;>LF;>I S ^@BI8I8STP][email protected]];
→ S1
(τγ )∗ X x
(t, t0 , x) →
7
Xx
X̃γ : ∂(I2 × Iγ ) × S
-
3
2
X
γ
X γ γ
1
γ
X̃γ
α
1
KOUL*DK; <OUL[@ –L*S S]; 8Q:9
X
Iγ × S 1 → S 1
Iγ X
γ ∗
P <OUP X̃ KOFLaE<;>H{;>LF;>I
Iγ × S 1 → S 1
γ
X
X
einγ (θ+2πt)
X
einγ θ
(t2 , t, eiθ ) 7→
ET
;
OUP t = 1
OUP <; V
2
3
(t, eiθ ) 7→ einγ (θ+2πt) .
∂(I2 × Iγ ) × S 1 → S 1
(
KOUL*@BS];>NFOtE<;>H{;>LF;>I
X
(t, eiθ ) 7→ einγ θ .
g ^@BI8NFžKO\[email protected]šN [email protected]–= : V E V : r (τ ) X [email protected]];>NFO~E<;>H{;>LF;>I
(t, eiθ ) 7→ 1.
Iγ × S 1 → S 1
nI M=8LtOF=8I8I€;RO MNti–=
1
γ
γ
X̃γ
β
r <;>=<O";>X8L <; <O"S MQ–@BSTPTL m = n V
×I )×{∗}
[email protected]@B; Nt<O9 Kiy5–= 8PTLUcLP]A; rGn9; CI βM=8=Le∂(I
I8NF;>S]; 8Q MN X `9 ME<;>NFOe9 n DK;>I8P KOZ9 :[email protected]{DK;>=8NUX
o ML
γ
k (τ ) = n [γ] V
o @BNeS fH MH :I8NF;cD K9 RrG; bH{; –LUN ni–= (τ ) X KOFLaE<;>H{;>LF;>I
∂(I × I ) × S → S
(
OUP t = 1
e
(t , t, e ) 7→
OUP <; V
e
`@B=8N[@[email protected]];>NFO k (τ ) = −n [γ] V
2
γ
3
X
X̃γ
β
X
γ
X
γ
γ
2
−1
∗
γ
1
inX
γ (θ−2πt)
iθ
2
inX
γ θ
−1
γ
X
1
γ
2
X
γ
2
hOf i h k i f *O h k j k
K/JJ
X
γ
γ
f
M
h
N
W :LUE K;>NFžMH : V C V E I M=8L j PTS <;>=<O\N <O MPTQ MN\OU=8NaS :QRNF;>=8I V >
";>=8L bDK;>=8NUX 6O [email protected][@ –L KOUL –=8STS 6E<;>H{;>S];>QR= V =<O[OUP <;>=<O N MLUNF;>=8?u; <O{S
DK;>NF;>[email protected] [email protected] –L MLaS P <DMST=<OUP]; K ⊆ V
X3 ;[email protected] G
‚}ˆM„ † ƒ€‚  †c‡ ‚lˆ}Š‰e‹ ‚ ‚ ‡ ‰lŒF‚JˆKŒ ‹d’ ‡ ’>ƒG„“‚Jˆ † ‡ Σ ‚M„
γ
ˆy τ ‚}ˆM„ ‚f„›‘ˆy„”–‚ 'n‚ ƒ ” ‡ >„“‚ ‚ >ƒ , ”–‚ γ ) ’  ‡ ˆ k (τ ) = 0 = ±1.
[email protected] MS]; <OYi–= V KOUL:= #OF;>=<O j QRNF;>=8I {9 T V W JLUE K;>NFžMH : V C V E <;>=<O
I MNUH ML MQ–@BS MH cLe9 [email protected]]DM=8S MN k IŸ;>=8Na= :[email protected] :XŸ;>NF9<@ –L
X3 ; @ ;7 €>„
ƒ€‚~‹8’> ‡ ‚  ‡ ”u’>ƒ „“‚ ”–‚ , ‚yƒ ‡ ‚ h ”u’Rƒ8ˆ Σ 1— ƒ
(γ, γ ) †
ˆ † ‹–‹dBˆK‚ ‡ ’ I † ‚ γ ‚K„ 㠈M>ƒ „ ‰ † ƒ<™‚}ˆ\”–‚  ‡ ™‚yƒG„™’R„z>ƒlŠƒ ” † „“‚t‹d’ ‡ ’YˆM † ˆ ˆ † ‡ ’ ‚
I ‚ ‚}ˆn”–ŒM„“‚ ‰ Šƒ€‚yƒG„
 ˆ
† ‡
1 ‡
Σ
-
Vg,1
,
g,1
2
3 g,1 g,1
g,1
γ
X
γ
2
g,1
-
g,1
X
0
g,1
0
h,2
kX (τγ τγ−1
0 ) = 2h[γ].
g ^@BI8NFž[email protected]šI8NF;>IŸ;ROUPTLUP]; : V E V Adr <;>=<[email protected]?u; <O~S MQ–@BSTPTL [email protected] –L
235476(3
3
X
nX
γ + nγ 0 = 2h.
[email protected]]i–[email protected] cL"S *LUE K;>NFžMH : V C V E ML :=8LUPTSTP]O[@ cL"[email protected]I8NF;>I8NUP ML q9 E<;>H{;>H{;>NUI8E8P]OFH
DMNF;>P]O Y9 k r <;>=<O~;>X8L <; <O
9
X
X
0
kX (τγ τγ−1
= nX
0 )
γ [γ] − nγ 0 [γ ]
X
= (nX
γ + nγ 0 ) [γ]
= 2h [γ].
2
EET
h [email protected]=<; <O6OUP
ML
OF; –L69 M=d7 [email protected] KO6X€;>NF9<@ cL KOb9 `HMH
Q 8N MLaL MSTS KO\ic= fS K(γ,O~DKγ;>=8)NUX KO (α,γ MαL α) ; –L~HMH [email protected]>OFO :9 E<;>H{;>S];>QRP Rr€@BS];>NFO
2
(
0
0
−1
kX (τγ τγ−1
0 τ α τ α0 ) = 0
ML τ τ τ τ @[email protected] –L V VAo =8P]OFi–= MSTS KOt; –LvH MH Q 8N RrD KO [email protected] KOvO[; –L
DK; =8QR= [email protected]= MS MH cLw9d= QRNF;>=8I \9 9dPnm K;>LF;>I8P KO V P MSTS KOqOF; –L DK; =8QR= KO
[email protected]= MS MH cL\9d=bOF;>=<O QRNF;>=8I f9 ";>N MSTSTP™[email protected]];>NFO
KOFL= bDK;>HJHn=
L[@BL M=8N:9 T ;>N [T ,j T ] ⊂ K r 98; <D D [email protected] τ τ<;>=<τO:@BI8τIŸ;>NUL š[email protected]>OZ9 P dxz;>N jj
[email protected]]; _OU=8NfS y7dP]OUL <D 9 = MS MH –LY9 V i–=8P O MN[@[email protected]>O:9<@ <O K V W"@
ic= KOULUP]; KOUL@BS];>NFO9 ZO[@K?u;>PTN|O PTS KOULqIŸ;ROFOUPTX8S e9 \LUNF;>=8? MN|9 [email protected] KOwX€;>NF9<@ cL KO
9 {H MH {Q 8N lDK;>HJH lI8N KD K9 MHJH –LJic=8P lOF; [email protected]>OšDK; =8QR= KO S = l9
S ^@B=8LUN [email protected]\9 KO MS MH –LFOZ9d= ";>N MSTSTP V
- g,1
3 g,1
−1 −1
γ 0 α
α0
( γ
g,1
$ g,1
3
g,1
g,1
−1 −1
γ γ 0 α α0
g,1
3 / $ !$ !$ + "!
! !+ "! "! +
dX
"!-(
@K?u;>[email protected]]DM=8S MN OU=8N = YI8NF; 9d=8PTL"9 *LaP]OULFOv9 g ME I MNUH ML"9 DK; <OFLUNU=8PTN
y7dI8STP]DMPTL MH –L:[email protected] SUk j [email protected][@BSTS MSTP]O}@BLUP]; _N [email protected]? š9 bB`;>NUPTL[@ ϕm (T ) 9d=`[email protected] 8Q
LF;>N T @>OFOF; DMP [email protected] –LtS KO"P <OFLUNU=<DMLUP]; <O <; <D [email protected]= [email protected] A V [email protected]\ic= KOULUP];
[email protected] –L [email protected] –Lf9 YO[@A?u;>PTN\OUPsD KOUL~OF= lO[@ –L~IŸ;>=8N\[email protected]]DM=8S MN\O[; σj P c?>@[email protected] cL
/
ϕ
X
, # X
ϕ
dX (ϕ) = σ(Tϕ , mX (ϕ)).
W"@JDMS KOUL~S :[email protected]>O\9d=bLaP]OULe9 :g ME
3 ;[email protected]>= €RŠ„
ƒ „›‘ˆy„:”–‚S'š‚ ƒ ” ‡ >„“‚ ‚ Rƒ , ” † ƒ€‚  † ‡ ‚
ˆy‘‰e ‹ ‚ ‚ ‡ ‰lŒF‚ 㠈 † ‡ Σ τ1 †  ‡ ˆ )
>œ = ±1.
d (τ ) = (n − 1)(n + 1)
bN M? [email protected] –L [email protected] y5 8PTLUP]; `9 :S ^@BI8I8STP][email protected]]; d ML~9 KOeDK;cD ; DMS KO~9 2B ; MN
9
ML~9 P cL MNFO KDMLUP];
%'&
γ
g,1
γ
X
X
γ
X
γ
3
X
cX (ϕ, ψ) = 3τ (ϕ, ψ) + dX (ϕ) + dX (ψ) − dX (ϕ.ψ)
∀ϕ, ψ ∈ Mg,1 ,
<;>=<O|;>X8L <; <O|[email protected]N KDM=8NUN <D Z= \xz;>NUH =8S y7dI8STP]DMPTL Z9 d I€;>=8NLF;>=8LwI8NF; 9d=8PTL
9 fL aP]OULFO\9 g ME MLe98; <DRr 6LUE K;>NUP Rr<IŸ;>=8NaLF;>=8L\9dPnm K;>H{;>NUI8E8P]OUH 9 M V
W fN KOUL f9 fD [email protected][@BQRN[@BI8E :?>@nDK; <OUP]OUL MN I8NF;>=8? MNaS fLUE K;>NFžMH : V A V E V8o ;>=8N
D [email protected] <;>=<[email protected]]; <OArt9<@ <On= I8N MHJP MN L MHJI<OAr"587 MNš= (N MI8N KO –L[@ –L{9 [email protected] j
[email protected][@BSTS MSTP]O}@BLUP]; m9 B`;>NUPTL[@ m (T ) r–I8=8P]OqDK; <OULUNU=8PTN \= :>j [email protected] ML W i–=8P<SU
I MN j
H MLULUN[@dr<[email protected][email protected]=8PTL Rr89 [email protected]]DM=8S MN y7 I8STP]DMPTL MH cLeS σj P –[email protected]@ –L V
X
g,1
-
X
τγ
EEE
γ
2X 3
h MI8N <; <O{S KO <;>L[@BLUP]; <O09 [email protected]`I8N M=8? C9d= LUE K; j
M
C
L
K
D
;
<O
U
]
P
9
MN
F
;
<O
<;>=8? [email protected][email protected] O KDMLUP]; ˜[email protected] MSTS X 9d= S j 5<X8N
: VC VE
B
@
=
9
K
F
O
U
O
<
=
e
O
9
9 y5 8P YI8N KD K9 MHJH cL
ξ := (I × TΣ )
∂(I × Σ )
j
X : ∂(I × Σ ) → S
OUP t = 1
(τ ) X
OUP <;
(t, x) 7→
X
g ^@BI8NFžKOZS nLUE K;>NFžMH : V C V E r OFP n KOUL –=8S™r @BS];>NFOf; CI M=8LZI8NF;>S]; 8Q MN X
= O KDMLUP]; bQRS];>[email protected] Y9 ξ V
P KOFL <; c=8S™[email protected]];>NFO [email protected] KDMLUP]; O aI8NF;>S]; 8Q l9 ME<;>NFO9 DK;>I8P KO
9dP]O ;>P cLnKO 9 {[email protected];>=8NUX γ I8S]; 8Q KO 9<@ X<On= JH MH 0DK;>I8P l9 Σ V ne;>LF; <O S OUPTQ J9 n MLYO[;>P –L x , . . . , x |n | I€;>P cLFOY9dP]OULUP <DMLFO:9 S P cL MNUP M=8N 9
W KO |n | DK;>=8NUX KOf9 y5 8P KO:DMP j 9 KOFOF;>=<OfOF; cLZ9 KOfDK;>=8NUX KO
({ } × I ) ⊂ I × I ) V
9 ;>X<OFLUNU=<DMLUP]; 1IŸ;>[email protected] KDMLUP]; X
P cL
L = {x } × S ⊂ (I × I × S )
i = 1, · · · , |n |
lI M=8Lq98; <D~I8NF;>S]; 8Q MN l9 ME<;>NFOw9 = {?u;>P]OUP [email protected] ~N MQR=8STP MN9 ~D KO DK;>=8NUX KO
W YDK;>HJI8S y7 P 5 š9 YD MLUL yX7 L <OUP]; CO :I8NF;>S]; 8Q 1= O KDMLUP]; ŝ 9d= S j 5<X8N V
ic=8P KOFL Kic=8PT?>@[email protected] cL lIŸ;>=8NšS ^@>DMLUP]; 9 S S 0S]; 8Q•9 [email protected]>ic= m?u;>P]OUP [email protected]
(ξ )
N MQR=8STP MNe9 KO\DK;>=8NUX KO L V
L[@BX8PTSTP]OF; <O Q–@B=<DFE [email protected]@f9dPTN KDMLUP]; lI8N yx MN a9 ; {;>X8LUP cL = |LUNUP
? [email protected]]O[@BLUP]; bDK;>HJŝI8S y7 Y9 T(I × Σ ) V 9 [email protected]šI8NF;>S]; 8Q [email protected] –L I 6= [email protected]]O}@BLUP]; j
DK;>HJI8S y7 ~9 T(I × Σ ) DK;>HJH \@[email protected][@BQRN[@BI8E A V C V : ML JP]9 cLUP 5 @ –L {0} × Σ
@A? KD {1}×Σ ? [email protected] τ rpD MLUL 9 MN 8P]žMN P <9d=8PTLv= wLUNUPT? [email protected]]O[@BLUP]; šDK;>HJI8S y7 m (τ )
[email protected] 8Q LF;>N i–=8P KOFLq= {N MI8N KO cL[@ –La9 [email protected] SU j [email protected][@BSTS MSTP]O[@BLUP]; 9 B`; j
NUPTL[@ m (T ) V T
[email protected] MS]; <Osi–= S P c?>@[email protected] cL*@>O[OF;cDMP D MLUL qDK; <OULUNU=<DMLUP]; KOULs9 y5 8P DK;>HJH
OF=8PTL DK; <OUP]9 MNF; <O|= σj:>j [email protected] ML W DK;>[email protected]>DML ZDK; y7 ZDK; –? [email protected] MH –LZ;>NUP j
L q9 X€;>NF9 ∂W = T V W"@[email protected]]O}@BLUP]; nDK;>HJI8S y7 m (τ ) rpOFL[@BX8PTSTP]O Q–@B=<D[E
[email protected]\S [email protected]•9 :? KDML M=8NFO <;>[email protected];>NUL[@ cL\N [email protected]? MH –L W r89 y5 8PTL\= nO KD j
LUP]; m (τ ) 9d= SU (4)j 5<X8N P(TW ) @B= j 9 KOFOF=<O~9d=bXŸ;>NF969 W r<@BP <OFP
OUPTQ W.
σ(T , m (τ )) =< p (TW , m (τ ) ), µ > − 3
W X8=8L [email protected] –L [email protected] cL*9 wDK; <OULUNU=8PTN = L MSTS :>j [email protected] ML qic=8P <;>=<OsI MNUH MLULUN[@
9 :[email protected]]DM=8S MN\[email protected]>DMPTS MH cLeD σj P –[email protected]@ –L V
\4
" (
NFžMH
2
0
g,1 0
γ
g,1
γ
1
1
g,1
γ ∗
x
x
X γ
γ
0
X γ
,
1
2
X γ
γ
γ
1
γ
2
X
γ
|nX
γ |
X
γ
g,1
X
γ
3 γ i
γ
'
X
γ
1
γ
C 0
1
i
(
3
1
3
X
i
X
g,1
g
γ
g
/
g
X
γ
τγ
X
τγ
X
'
X
γ
C
τγ
X
γ
-
τγ
1
γ
EE C
X
γ
,
W
K/JJ
Qjk
Ef [Q h
Q
Q Q[h K
M
Tf
;>PTL = A [email protected] ML f9 ZX€;>NF9 i–[email protected] E<;>H{;>S];>QRP :9 = bDK;>NUI<O [email protected] <O KO
MLfL MSTS JicH= γ X€;>NFj 9 = _9dP]OFic= 9<@ <OΣH V ;>PTL D = #9dP]OFic= šI8S]; 8Q J9<@ <O H
XŸ;>NF9<@ –L γ DK;>HJH :9<@ <[email protected]<QR=8N : V ; V ; <OUP]9 MNF; <O\= 6?u;>P]OUP [email protected] :;>NUP –L [email protected][@ j
H MLUN I × D 9 D ML <;>LF; <O M := H \ (P –L (I ) × D) V
γ
3
g
γ
Wγ
γ
γ
γ
γ
γ
D
Iγ
Iγ × D
V
0)132)4
: ; 8 Iγ × D
=8P]OFic= XŸ;>NF9 := 19dP]O[i–= Y9<@ <O r<S :LaP]OULZ9 Yg ME O YI8NF;>S]; 8Q
= b9do Pnm K;>H{;>NUγI8E8P]OUH τ̃ 9 H 9 y5 8P"HOU=8N M [email protected]~S P]9 –LUPTL Rr MτL\OU=8N I × D [email protected]
γ
γ
# 3 (
γ
γ
γ
τ̃ : Iγ × D → Iγ × D
e;>LF; <O
{1} × Hγ
( t , r eiθ ) → ( t , r ei(θ−2πt) ).
[email protected] :>j R? @BNUP ML n;>X8L c= J9
? kP @ DK;>HJH OU=8PTL
Wγ
τ̃γ
I × Hγ
`P]9 –LUP 5 @ cL
ML
−{0} × Hγ
Wγ = I × Hγ /(0,x)∼(1,τ̃γ (x)) .
@[email protected]=8PTL Rr <;>=<[email protected]=8NF; <O:P –L MN ML [email protected]?u;>PTNYDK;>HJH J;>X8L c= {9
9 Io × I × D 9 [email protected] [email protected] 8P]žMN [email protected] –L
Wγ = I × M/(0,x)∼(1,x)
[
ML
I ×M
γ
;<
{t}×Di ∼{(t,i)}×D
I × Iγ × D/(0,x)∼(1,τ̃ (x)) .
r
V
W LUE K;>NFžMH b9 [email protected]~9 A [email protected] <;>=<OJI MNUH ML{9 [email protected]]DM=8S MN{[email protected]•OUPTQ [email protected]=8N 69
D MLUL :>j ?>@BNUP ML
;[email protected] "’Jˆy , ƒ ’R„ ‚ ”–‚
† ‡ W ‚}ˆy„wƒ †5 ‚ 1
i=0,1
Di := {i} × D ⊂ M i = 0, 1
-
,
γ
h%J PR g e eN^#g f^#g LNO a/^#hXOfeLNg`CLNR ghXRHeNR/QlQ<R KI*LN`<d/eNdRPhbLNR/aP^#Q<QlIKhjeHmRPO
I × D2
QlR Q<K^ hXVm R V aP^#g L RPO7RPOORPhjeN`<RPQ<Q<RPO a/^KLLNRPa/eNR/_RPhje7acW^#`<O`<RPOPZ
I × Σg,1
EE
A
Q I-i #I-LN`ldceNd
235476(3
S ^@BNULUP]DMS nA [email protected] 3
Z;>=<OY=8LUPTSTP]O MNF; <OnS KO <;>L[@BLUP]; <Oš9
3 ;[email protected]@ Y− = I × M/(0,x)∼(1,x) = S 1 × M
Y+ = I × Iγ × D/(0,x)∼(1,τ̃ (x))
P cL (D ∪ D )
X0 = I × (D0 ∪ D1 )/∼ = I × ∂Iγ × D/∼
X− = S 1 × (M \
0
1
X+ = I × Iγ × ∂D/∼
[email protected]]DM=8S [email protected]];>NFO S KO QRNF;>=8I KO"9 E<;>H{;>S];>QRP H (Z; Z) r H (X ; Z) r H (X ; Z)
ML H (X ; Z) V 1;>X8LUP cL
MN
A =
(H (Z; R) → H (X ; R))
Z = I × (∂D0 ∪ ∂D1 )/∼ = I × ∂Iγ × ∂D/∼
1
1
0
1
+
−
1
1
1
0
= < [{x} × ∂D0 ]; [{x} × ∂D1 ] >
MN (H (Z; R) → H (X ; R))
B =
1
1
1
1
+
= < [S × {x0 }] − [S × {x1 }]; [{x} × ∂D0 ] − [{x} × ∂D1 ] >
MN
(H1 (Z; R) → H1 (X− ; R))
< [S 1 × {x0 }] − [S 1 × {x1 }]; [{x} × ∂D0 ], [{x} × ∂D1 ] >
=
< [S 1 × {x0 }] − [S 1 × {x1 }]; [{x} × ∂D0 ] − [{x} × ∂D1 ] >
C =
[email protected] <O~LF;>=<O~S [email protected]>OAr V =
B∩(A+C)
(B∩A)+(B∩C)
OFPTQ
,
OUPTQ
=0
,
Wγ =
OUP γ KOFLa= YDK;>=8NUX O [email protected][@ –L
OUP <;
9 ;
V = 0.
W"@ :>j ?>@BNUP ML W KOUL1DK; y7 #DK;>[email protected]>DML 2;>NUP cL 29 2
XŸ;>NF9`S [email protected] 8Q6LF;>N T V 9 STS <;>=<OfI MNUH ML:98; <D 9 [email protected]]DM=8S MN:S σj P –[email protected]@ –L
=8P]OFi–= |ic= |O[@ZOUPTQ [email protected]=8N KOUL –=8STS RrR; KOUL*N[@BH = {[email protected]]DM=8S
σ(T , m (τ )) VRo
9 ;>X<OFLUNU=<DMLUP];
4 3 3
)
X
τγ
τγ
,
γ
γ
σ(Tτγ , mX (τγ )) =< p1 (TWγ , mX (τγ ) [email protected] –Lq9 |[email protected]>O[O MN D [email protected]]DM=8SG9 ;>X<OULUNU=<DMLUP]; sruP –L MN KOFOF; <O <;>=<O S ;>NUP cL[@BLUP];
9 W [email protected]@>OFO xz; <9<@BH –L[@BS µ KOULS |Q MN[@BL M=8Nw9 Hj (W , ∂W ; Z) ic=8P
KOFL~DK;>[email protected] [email protected]? KDYS ;>NUP –L[@BLUP]; •9 ∂W 98; :[email protected] m (τ ) V
e;>=<O@K?u; <O MQ–@BS MH –L= lDFE<;>P 7 [email protected]=8N MS€9 ;>NUP –L[@BLUP]; 69 P <[email protected] [email protected]
OFLUNU=<DMLU=8N eI8NF; 9d=8PTL|9 I × H MLwS KO|;>NUP cL[@BLUP]; <OaI8N yx MN KO|9 IWML H <;>LF; <O
S YQ MN[@BL M=8Ne9 H (W , ∂W ; Z) @>OFOF; DMP D YD[E<;>P 7 V
µ
W"@\ic= KOULUP]; [email protected]];>NFOt9 DK;>[email protected] MND KO*;>NUP –L[@BLUP]; <O PTS O MN[@\I8ST=<O*DK; –? 8P cL
9 =8LUPTSTP]O MNnS ;>NUP cL[@BLUP]; P]O[OU= l9 [email protected]=<DMLU=8N 0I8NF; 9d=8PTL V 9 y7 I8STP]DMPTL[@ cL µ ML
_DK;>[email protected][@ cL @K? KD S ;>NF9dN 9 KOf? KDML M=8NFO:9 m (τ ) r <;>=<[email protected]]; <OfH{; cLUN MN
S fS MHJH [email protected] –L
γ 3
0
Wγ
3
Wγ
4
γ
γ
γ
4
γ
EE
:
γ
γ
γ
γ
X
γ X
3
γ
3
), µWγ > .
0
W
γ
3
3 ; @@
µWγ = −µ0Wγ .
235476(3 Z;>LF; <O
S eD[[email protected] \? KDML M=8NFOa9 <;>[email protected]ŸO[;>NUL[@ –LaN [email protected]
? MH cL H PTSdP <9d=8PTLN= JOF;>=<O j 5<X8N ν 9 N[@ 8Q E 9 Σ(TH )| V W ;>NUP cL[@BLUP]; j
I8N yx MN :9 H KOULaD MSTS :ic=8P KOFL|DK;>[email protected] [email protected]? [email protected] 9 KDK;>HJIŸ;ROUPTLUP]; 6OF;>HJH
9 PTLUE (TH )| = ν ⊕ TΣ . P <OUP™r S ;>NUP cL[@BLUP]; 9 I × H 98;
[email protected]\[email protected] OULUNU=<DMLU=8N :I8NF; 9d=8PTL~P <[email protected] KDK;>HJIŸ;ROUPTLUP]; bO[;>HJH Y9 E8PTL
γ S
S
γ
Σg
γ
S
g
g
Σg
γ
3
γ
T(I × Hγ )|I×Σg = (TI × Σg ) ⊕ (I × νS ) ⊕ (I × TΣg ).
W [email protected]#9 ? KDML M=8NFOY9 ∂W <;>[email protected];>NUL[@ cL:N [email protected]? MH –L W KOULfS
D[[email protected] ? KDML M=8NFOYP <9d=8PTL:[email protected] MLYDK; <OUL[@ –LYS S]; 8Qb9 D KOULY98; <D =
LUNUPT? [email protected]]O[@BLUP]; J9 I ×ν V h [email protected]–N=<; <O*ic= m (τ ) I8NF;A? P –L9 I = [email protected]]O}@BLUP];
DK;>HJI8S y7 |9 T(I ×Σ ) H = 8P<9 wS ;>NUP –L[@BLUP]; lP <9d=8PTL |[email protected]@eOULUNU=<DMLU=8N |I8NF; 9d=8PTL V
P <OUPS ;>NUP cL[@BLUP]; DK;>[email protected] [email protected]? KD m (τ ) KOUL D MSTS li–=8P KOFL DK;>[email protected]
@A? KD:[email protected];>HJH 9 PTLUE
-
3
g
S
S
γ
X
γ
γ
γ
, X
3
/
(I × νS ) ⊕ (TI × Σg ) ⊕ (I × TΣg ).
o @BNZDK; <O Ki–= cLAr µ = −µ V
2
[email protected] <[email protected]=8PTL \9d= [email protected] Rr <;>=<OwDK; <OFP]98žMN MNF; <OwS ~5<X8N \L[@ 8Q cL TW Hn= 8P
9 S ;>NUP –L[@BLUP]; P <9d=8PTL [email protected]@~OULUNU=<DMLU=8N I8NF; 9d=8PTL V Z;>LF; <O m (τ ) [email protected] KDMLUP];
9d= <;>=8? [email protected]= SU (4)j 5<X8N [email protected]>OFO[;cDMP @B= j 9 KO[OU=<Oe9 ∂W ;>X8L c= 9 m (τ ) KD[[email protected] 8Q [email protected] –L\S ;>NF9dN :9 KOa9 M=d7 I8N MHJP MNFOa? KDML M=8NFOa9d=bD[[email protected] fN MI€žMN KO MSTS
KOFL~98; <DY98; :[email protected]
8 [email protected] 9dPTN KDMLUP]; 1I8N yx MN Y9 I r
r
8 S :[email protected]`9 f? KDML M=8NFO <;>[email protected][;>NUL[@ –L~N [email protected]? MH cL
W
[email protected] O KDMLUP]; 9d= 5<X8N
Z;>=<8 O~;>X8L <; <[email protected]̂S];>NFO~S MQS–@BjSTPTL ((I × TΣ ) ) V
0
Wγ
Wγ
3
3
γ
γ
0
γ
X X
γ
3
X
γ
3
g C 0
dX (τγ ) = − < p1 (TWγ , m0X (τγ ) ), µ0Wγ > .
W P]9 [email protected]];>NFOZ9 <;>=<OZN[@BH MN = •[email protected]]DM=8S*9 YI8N MHJP]žMN [email protected]>OFO n9 ;
LUN [email protected] b9d= SU (4)j 5<X8N [email protected]>O[OF;cDMP :@B= T(I × H ) rdH = 8P"9 fS ;>NUP cL[@BLUP]; 1I8NF; 9do =8PTLAj r
N [email protected]? LF;>=8LvS XŸ;>NF9 Vpo ;>=8NtD [email protected] <;>=<Osx™@B=8LsI8NF;>S]; 8Q MN m (τ ) ∂I ×H V
(
K/JJ
[h
h Eh
fhCkk
J
γ
RQik
Q 3
/Qih
Ef
3
0
X
PO k γ
RQ
γ
O/h
N
%h
h M? <; <O|@B=mDK;>NUI<O [email protected] <O KO H 9 y5 8P I8N KD K9 MHJH cL V e;>=<Ow?u;>=8S]; <O 587 MN
= [email protected]]O[@BLUP]; CDK;>HJI8S y7 Y9 TH
V
;>=8NZD [email protected]<; 1DK; <OUP]98žMN := y7 L <OUP]; [email protected]`9 :? KDML M=8NFO\DK;>HJI8S y7dP 5
o
L[@BX8PTSTP]O Q–@B=<DFE [email protected] :[email protected] Z? KDML M=8NFO <;>[email protected][;>NUL[@ –L r MSTS
X
Σ V
2 THγ
γ
γ
C
g (
EE
H
P <9d=8PTL = mO KDMLUP]; β 9d= SU (3)j 5<X8N P(TH ) @B= j 9 KOFOF=<Oš9 Σ V S [email protected]
@B=<DM= ;>X<OULUNU=<DMLUP]; I8NF;>S]; 8Q MNZD MLUL YO KDMLUP]; H LF;>=8L –LUP MN V
[email protected] <Ol= I8N MHJP MNmL MHJI<OAr <;>=<O0?u;>=8S]; <O{[email protected]•587 MNlOU=8NlS 6?u;>P]OUP [email protected] 1;>NUP cL
";>=8Le9 ^@BX€;>NF9 r ; 6I8NF;>S]; 8Q YS YD[[email protected] f? KDML M=8NFO <;>[email protected]"OF;>NUL[@ –LZOU=8N
I × DV
N
[email protected]? MH –L I ×D DK;>HJH a9<@ <[email protected]<QR=8N : V ET V Z;>LF; <O N D [email protected]
I ×S
Σ
1
γ
γ C
γ
-
g
γ
γ
γ
N
[email protected]`9 :? KDML M=8NFO N
9 |? KDML M=8NFO <; lOUP 8QR=8STP MN OU=8N I × D V S<9 ML MNUHJP \= a9 KDK;>HJI€;ROUPTLUP]; m9d= <5 X8N
L[@ 8Q cL
T(I × D) = ν ⊕ η
; η ML ν OF; –LY9 KO:5<X8N KO:? KDMLF;>NUP MS]O:N MS]OY;>NUP cL KO 9 N[@ 8Q 2 ML 1 N KOUI KDMLUPT? j
H cL V N [email protected]= bi–= 6D MLUL 6O[;>HJH b9 E8PTL 6DK;>NUN KOUIŸ; <9 [email protected]>O [email protected]
9 KDK;>HJIŸ;ROUPTLUP]; `[email protected] <; 8P]i–= Y9 = b5<X8N :I8NF;c9d=8PTL ML~i–=
ML η| = T({t} × D) ∀t ∈ I .
= T(I × S )
η|
W D[[email protected]•9 ? KDML M=8NFO <; •OUP 8QR=8STP MN P <9d=8PTLe= nO KDMLUP]; C9d= 5<X8N
@B= j 9 KO[OU=<Oa9 I × S V Z;>=<O|I€;>=8?u; <O\OU=8I8XI€;RO MN|ic= X KOULaDK; <OUL[@ cL~SS Zj S]; 8Q{9η
E MNFD[E MN I8NF;>S]; 8Q MN X|
b= YO KDMLUP]; `DK; <OUL[@ –L S :S]; 8Q
I V
I ×D
9 I N M? P –L DFE MNFD[E MN I8NF;>S]; 8Q MN X| D V
;[email protected] ;7 "’ ’Bˆ}ˆK‚ ” † ‚ ‡Y‡ ‚ ’R„ŠŽB‚ e(TD, X| ) ŽBŒ ‡  F ‚
5
0)182)4
/ V
: E T 8
γ
γ
γ
{t}×D
1
Iγ ×S
γ
2 1
γ
1
γ
-
Iγ ×S 1
γ
γ
γ
1
0
γ
< e(TD, X|γ ), µD >= nX
γ −1
h [email protected]=<; <O\ic= fS :[email protected] Y? KDML M=8NFO
9 KDMNUPTL\9<@ <[email protected]š5<QR=8N
A
K
U
O
a
L
8
X
P
19
y5
8P
"
U
O
8
=
N
: V EE
W"@[email protected]>OFO 19 9 =8S MNlN [email protected] LUPT? < e(TD, X| ), µ > [email protected]];>NFOlS <;>H X8N C9
LF;>=8NFOli–= mx™@BPTL X S S]; 8Q29 γ [email protected][@BI8IŸ;>NUL A V [email protected] MS]; <O{ic= n KOULJS
<;>HnX8N {9 šLF;>=8NFO:i–= nx™@BPTL X [email protected][@BI8I€;>NUL I S šS]; 8Qb9 γ VŸo =8P]OFic= I [email protected]
LF;>=8N\[email protected]~N[@BI8I€;>NUL A r8; b;>X8LUP –L
−1
235476(3
2
D
γ
γ
X γ
γ
< e(TD, X|γ ), µD >= nX
γ − 1.
/a I-L
4
H (Hγ , ∂Hγ ; Z)
2
RPOfe7eLN` `lIKQ]Z
EE
o
A
D
γ
V EE 8 D[[email protected] f? KDML M=8NFO A
P
rd; mI M=8L|I8NF;>S]; 8Q MN
= :O KDMLUP]; 69 η DK; <OUL[@ cL
S fS]; 8Q0n9 =I 1V <;>L MN[@ s̃ OF; 1DKX;>HJI8S Iy7d×P 5 DV
P
KOUL <; c=8S™r€; `I8NF;>S]; 8Q X I × D •9 ME<;>NFOf9 ?u;>P]OFP [email protected] KO
N MQR=8STP MnNFOe9 −KO~1DK;>=8NUX KO
[
0)132)4
X
γ
0
γ
X
X
γ γ
:
γ
0 Iγ × {yj }
; KOUL:S {OUPTQ l9 n − 1 ML ;#S KO y OF; cLn9 KO:IŸ;>P –LFO C C 9dP]OFLUP <DMLFO 9
P cL (D)
<;>L MN[@ D MLUL y7 L <OUP]; ML <96OF; DK;>HJI8S y7 P 5
= O KDMV LUP]; s̃ 9d= sS j 5<X8N (η ) 9 šV O[;>NUL šic= s̃ OF;>PTL:DK; <OUL[@ –L S I S]×; 8QD69
[email protected]>ic= :?u;>P]OUP [email protected] :N MQR=8STP MNZ9 :D KO~DK;>=8NUX KOe9 ;>X<OULUNU=<DMLUP]; V
I
JOUL[@BX8PTSTP]O[@ –L Q–@B=<D[E |[email protected] [email protected]{9 ? KDML M=8NFO DK;>HJI8S y7 N r[email protected] KDMLUP];
9
P <9d=8PTL\= :[email protected]]O}@BLUP]; 1DK;>HJI8S y7 β 9 T(I × D) V
s̃
[email protected] MS]; <Oqic=
P cL
W"@:O KDMLUP]; 9 |5<X8N
@B= j 9 KO[OU=<O:9 ∂H =M Σ= HI8NF;>\S]; (8Q (I IG)@B×N D)β V P <9d=8PTLf= ⠚O KDMLUP]; #9dP(TH
= SU (3)) j
5<X8N P(TM ) @B= j 9 KOFOU=<O~9 ∂M V S @[email protected]>O~9 ;>X<OULUNU=<DMLUP]; [email protected]šI8NF;>S]; 8Q MN
LF;>=8L cLUP MN . V m587 Z= fL MSTS y7dL <OUP]; β V h KDK;>STS β r [email protected]šO KDMLUP];
M
P <9d=8PTLe= [email protected]]O}@BLUP]; CDK;>HJI8S y7 β X8P 19 y5 8P Y9 TH V
β
j=1,··· ,|nX
γ −1|
0 ,
X
γ # X
3
X
j
C 0
X
γ
γ
C
Iγ ×D
X
γ
γ
γ
g
C
γ
#
Iγ ×D
-
2
M
M
Σ
fh h gf h
Iγ ×D
γ
K,JJ
h M? <; <[email protected][email protected]]DM=8Sv9 ;>X<OULUNU=<DMLUP]; Cic=8P <;>=<O~P cL MN KOFO
2 Hγ
K/JJ
γ C
dX (τγ ) =< p1 TWγ , m0X (τγ ) , µ0Wγ > .
e;>=<O|?u;>=8S]; <O <;>=<OaN[@BH MN = [email protected]]DM=8Ss9 fI8N MHJP]žMN :[email protected]>OFO f9 ; –LUN [email protected]
9 <5 X8N P(T(I ×H ) ) N [email protected]? LF;>=8L"S XŸ;>NF9 V 9 [email protected]]DM=8STP [email protected]|LUNUPTo [email protected]]O}@BLUP];
DK;>HJI8S y7
I M=8L MLUN ~?c= ~DK;>HJH \= ~O KDMLUP]; l9 \D
5<X8N [email protected]=
9 KO[OU=<OJ9 mI ×(τ∂H) V I M=8L{@BS];>[email protected];>S]; 8Q MN −{0} × HSU (4)=j 8LUPTSTP]O[@ cLj
S [email protected][@BQRN[@BI8E 0I8N KD K9 –L V #[email protected] DFE MNFDFE MN I8NF;>S]; 8Q MNnD MLUL l9 MN 8P]žMN #=
0
X
E/a I-L
γ
γ C
γ
RPOfe7eLN` `lI*[email protected]
EE
G
H 4 (M, ∂M )
γ
O KDMLUP];
@B= 9 KOFOU=<OZ9
9 nOF;>NUL ni–=
O YN KDK;>STS nS];>NFO
9 ~S P]9 –Lm̃UP 5GDA(τ@BLUP]); 9 j −{0} × H∂([email protected]? ×KD H{1}) × H ? [email protected] τ̃ m̃; [email protected](τB=8N[)@ @BS];>NFOS MQ–@BSTPTL
X
3 γ
3
γ
< p1 TWγ , m0X (τγ ) j X4
X
γ
γ
3
γ γ
, µ0Wγ >=< p1 T(I × Hγ ), m̃X (τγ ) , µI×Hγ > .
";>=8Le9 ^@BXŸ;>NF9 r8P cL MN Km̃OFOF; (τ<O j )<;>=<O ∂I × M ; 1DK; <OUP]98žMN :S [email protected] YN j
IŸžMN KO~DK;>HJI8S y7 β OF=8N {0} × M OFL[@BX8PTSTP]O Q–@B=<D[E [email protected][email protected] KDMLUP]; CI8N yx MN
9 I Vdo =8P]OFic= τ̃ KOFLwS P]9 cLUPTL :OU=8N M rc; I8NF;>S]; 8Q m (τ ) 9 [email protected] ex™@K;
OF=8N {1} × M V
S <;>=<Otx™@B=8L <DK;>N a9 y5 8PTN m̃ (τ ) OU=8N ∂I ×I ×D V [email protected];>HJH a9 E8PTL
P <[email protected]{9 KDK;>HJI€;ROUPTLUP]; C9d=65<X8N
T(I × D) = ν ⊕ η
)
X
γ
M
3 γ
γ
γ
X
X
γ
γ
1DK; <OUP]9 MN :[email protected] O KDMLUP]; C9 P(T(I × H ) ) 9 y5 8P @B= j 9 KO[OU=<O~9
[email protected]
{0} × I × D
8 S YDK;>HJI8S y7dP 5 9d=1D[[email protected] Y? KDML M=8NFO I P <9d=8PTL\[email protected][email protected] KDMLUP]; 1I8N yx MN
9 r
S8 :[email protected]`9 f? KDML M=8NFO N r
8 [email protected]\O KDMLUP]; s̃ ? = DK;>HJH |O KDMLUP]; J9 (I ×η ) @B= j 9 KOFOF=<Ot9 {0}×I ×D V
;>HJH –L~98;>PTL j ; b9 y5 8PTN|S eD[[email protected] ZN MI€žMN KO|DK;>HJI8S y7 ZOF=8N {1} × I × D
IŸ;>=8Nlic= PTSeDK;65 <DMP]9 •@K? [email protected]#O KDMLUP]; 9 PT(I × H ) 9 y5 8P 1DMP j 9 KOFOU=<OmOU=8N
S];>NFO9d=mN KDK;>STS MH –L?> mN [email protected]–= fic= τ @BQRPTL[email protected] MH cL\OU=8N
{0} × I × D
S [email protected]<Oš9 J? KDML M=8NFO N ML I V S <;>=<OnOU= {L 98; <D{9 09 y5 8PTNY= XŸ; @
O KDMLUP]; 19 P(I × η) @B= j 9 KOFOF=<O~9 {1} × I × D V
P
@BS];>NFO KOUL:S JDK;>HJI8S y7dP 5 l9 = ([email protected] J? KDML M=8NFO <; 2OUP j
QR=8STP MN ns =ic=81PtI8NF;>S]; 8Qs̃ X|
=<OFOUP™rŸ; •I M=8LY9 y5 8PTN:S D[[email protected]
I × DV
9 m? KDML M=8NFO <; OUP 8QR=8STP MN (τ̃ ) s V ; DK;>HJI8S y7 P 5 69 y5 8PTN[@`= O KDMLUP]; 9
i–=8PsDK; c?cP <9dN[@ V
P(I × η)
P
KOFL <; c=8S™r@BS];>NFO KOULsS qDK;>HJI8S y7 P 5 9d=nD[[email protected] ? KDML M=8NFO <;
OFP 8QR=8STPnMN s−1 9 y5 8P"OU=8N I × D s̃ b9 ME<;>NFO\9 :?u;>P]OFP [email protected] KOaN MQR=8STP MNFOe9 KO\DK;>=8NUX KO
T(I × Hγ )|I×Iγ ×D = T(I × Iγ × D) = (TI × Iγ × D) ⊕ (I × ν) ⊕ (I × η).
γ C
γ
C
-
γ
γ
γ
X
γ C
X
γ
X
γ
γ
C
C 0
X
Iγ ×S 1
γ ∗ X
γ
( C
X
γ
- (
X
X
γ
[
0 Iγ × {yj }.
KD ; <OUP]98žMN OU=8N {1}×I ×D S D[[email protected] ? KDML M=8NFO <; YOUP 8QR=8STP MN (τ ) (s
PTS KOFL~9 y5 8P b9 ME<;>NFO\9 KO~?u;>P]OUP [email protected] KO~N MQR=8STP MNFOe9 KO~DK;>=8NUX KO
j=1,··· ,|nX
γ −1|
-
IhX^*eUI*eN`<^#h
_ IKQ<R
m R
γ
[
j=1,··· ,|nX
γ −1|
γ ∗
X) 0 (τγ )(Iγ × {yj })
RPOfem d geN`<Q<`<OdPR\mIKhO(Q<R a I*O X
Z gXOO`][
ORcLUIQ]G R eNRPhO`l^Kh *_ I `Ci
sX RPh Ue I*hje gR\aUWTIK_J"mR#RPa/n
eNRPγg LN6=OhX1^#h"O`lhV#gsQl`<XR/L7LNdPR/[email protected]
Iγ × D
EE
N MI8N KO –L KOZ9<@ <[email protected]š5<QR=8N : V E C V
V E C 8 ;>=8NUX KO~9 ;>X<OFLUNU=<DMLUP]; C9 (τ ) (s )
0I8NF;>S]; 8Q ZOF; DK;>HJI8S y7 P 5 [email protected] \HMH \I8NF; D K9 Zi–= Z9 [email protected]=<9 l=
O KDMLUP]; (τ ) (s̃ ) 9 PT(I × η) @B= j 9 KOFOU=<Oa9 {1} × I × D ic=8P DK; –? P <9dN[@ V
e;>=<O ? <; <On9 0DK; <OULUNU=8PTN l= 0O KDMLUP];
9
@B=
9 KO[OU=<O~9 ∂(I × H ) i–=8P I8NF;>S]; 8Q m (τ ) Mm̃L~ic=8(τP ? )MNUP 5 P(T(I × H ) ) j
0)132)4
:
γ ∗
#
X
X
γ
γ C
γ
dX (τγ ) = − < p1 T(I × Hγ ), m̃X (τγ ) , µI×Hγ > .
) 4 3 d (τ ) 3 5 3 4
3 T X3 63 X 4 γ (I × η)C
)
γ
γ
3
X
C
3
X
γ ∗
)
) 4 ]. 4 3 354
3
)
3
h [email protected]=<; <Olic=
O I8NF;>S]; 8Q [email protected]=8N MSTS MH –L
9 [email protected]
8P]žMN _DK; <OFL[@ –L #9 •S]; m̃8Q (τ9 )I V MLUL y7dL <OUP]; P <9d=8PTLb= _OIKDM×LUP]M; m̂ 9 j
@B= 9 KOFOU=<O9
@BN|DK; <D –LUN[@BLUP]; 69 ;>X<OULUNU=<D j
P(T(I × I × D) ) j
LUP]; sr <;>=<O\@K?u; <O\S MQ–@BSTPTL [email protected] –L ∂(I × I × D) Vco
&
2
γ
X
C
3
γ
X
γ
< p1 T(I × Hγ ), m̃X (τγ ) , µI×Hγ >=< p1 T(I × Iγ × D), m̂X , µI×Iγ ×D >
;
KD ;>NUN KOUIŸ; <9 S ;>NUP –L[@BLUP]; `I8NF;c9d=8PTLe9 I × I × D V
W"@l9dPTN KDMLUP]; •I8N yx MN n9 I MLeS [email protected]_9 n? KDML M=8NFO N P <9d=8P]O –L:9 KOfO KD j
LUP]; <O\9 P(TI × I × D) ML P(I × ν) N KOFI KDMLUPT? MH [email protected] <OUP"ic= = :P KDMLUP];
µI×Iγ ×D
3
γ
- γ
(I × η)0 ,→ P(T(I × Iγ × D))
MLAr<[email protected]@>OFO[@BQ @B=bDK;>HJI8S y7dP 5 Rr
(I × ηC )0 ,→ P(T(I × Iγ × D)C ).
@BNšDK; <OULUNU=<DMLUP]; sr"S [email protected](9 {N MI€žMN KOnDK;>HJI8S y7 m̂ KOUL 9 y5 [email protected]=
o
O KDMLUP]; Se 9 (I × η ) @B= j 9 KOFOU=<O 9 ∂(I × I × D) OUL[@BX8PTSTP]O –Q @B=<DFE [email protected]
X
C 0
γ
g` m d Th`Ce g hXR2ORPa/eN`<^Kh
R/eeNRR Ne RPhXO`<^Kh!RPOfe m ^#hhXdPR2JTI-L"QlI m`CLNRPaceN`l^Kh!JLNd d/LNdPRmR
aP^#hOfeUIKhjeNR\m R
[ RceJTI-LQlI ORPaceN`l^Kh m R
`<IhTm g`CeNR\mR
R/(
e aP^#hOfeUIKhjeNR
P(I × TM )C
βM
QlR Q<^KhXVm R Z P(TI × M )C I
EE
;
[email protected] KDMLUP]; •I8N yx MN š9 I ML\[email protected] D[[email protected]•9 ? KDML M=8NFO
@B=<O[OUP
3
X3 9<@ <OeD MLf;>NF9dN r
NC dX (τγ ) = − < p1 T(I × Iγ × D), m̂X , µI×Iγ ×D >
= < c2 (I × η)C , Se , µI×Iγ ×D > .
)
63
3
dX (τγ )
4 X \4 !3 4 )
3
/. 4 3 3
)
W X8=8L KOUL:9 =8LUPTSTP]O MN:[email protected] JOULUN[@BL MQRP Ji–= 9<@ <[email protected] M=8? J9d=•LUE K;>NFžMH
9d=bDK; D dDMS Y9 P –L MNFO KDMLUP]; [email protected]>OFO[@ [email protected]~S P]OF;>H{;>NUI8E8P]OFH
-
3
DK;>HJH JS šN KDK;>STS MH –LnOFL[@ <9<@BNF9_9
∂(I × Iγ × D)
(I × ηC )0 ' −(I × η ⊕ I × η)0 .
g KDK;>HJIŸ;ROF; <O:S XŸ;>NF9
9 M=d7 FL ;>N KO
∂(I × Iγ × D) = ∂(I × Iγ ) × D ∪ I × Iγ × ∂D .
[email protected] MS]; <O DK;>HJH –L KOULDK; <OULUNU=8PTL [email protected] KDMLUP]; Se OF=8N D KO 9 M=d7šLF;>N KOqOU=8PT?>@ cL
S KO~?>@BS M=8NFO~9 n V
[email protected] <; <OS [email protected]>O9d=0LF;>N I × I × ∂D Se KOULqS ZDK;>HJI8S y7 P 5 Z9 = ZO KDMLUP];
9 (I9 × η)
8 9 y5 8P YOU=8NaLF;>=8L I × I × ∂D S];>NFOFic= n = 0 r
9 KO2DK;>=8NUX KO L = {x } × ∂D r
8 ML29 y5 8P OF=8N I × I × ∂D I8NUPT?
rdS];>NFOFic= n KOUL <; –=8S V
i = 1, . . . , |n |
D 6i–=8P~DK; <D MN bS LF;>N ∂(I × I ) × D Se KOULJS 6DK;>HJI8S y7dP 5 b9 =
9
O KDMLUP]; 19 (I × η)
8 9 y5 8P YOU=8NaLF;>=8L ∂(I × I ) × D S];>NFOFic= n = 1
19 ME<;>NFO~9 Y?u;>P]OUP [email protected] KO~N MQR=8STP MNFOe9 KO |n − 1| DK;>=8NUX KO
8 9 y5 8P
X
γ
0
γ
γ
X
γ
X γ
i
0
i
γ
X
γ
γ
γ
X
γ
X
γ
Kj = 0 (−{0} × Iγ × {yj }) ∪ (I × ∂Iγ × {yj }) ∪ ({1} × τγ (Iγ × {yj }))
I€;>=8N
r8S];>NFOFic= KOUL~9dPnm MN –Le9
W KOwDK;>=8NUjX =KO 1,K. .r .j, =|n 1, −. . .1|, |n − 1| runOF; –LwN MI8N KO –L KO|9<@ 1<OV [email protected]<QR=8N : V E A V
e;>=<O <;>L MNF; <O L S <O MH X8S š9 YD KO~DK;>=8NUX KO\9 ;>X<OULUNU=<DMLUP]; ML S [email protected]{O KDMLUP];
9 (I × η) @BP <OUP9 y5 8P 9 ME<;>NFOš9 = ?u;>P]OUP [email protected] 0N MQR=8STP MN V 9 0S –LUN [email protected]>DKO
bI M=8LaP]9 –LUP 5 MN
DK;>HJH Y9 [email protected]=<9 W [email protected]`9 :? KDML M=8NFO
LV
DK;>HJI8S y7 Se KOUL\;>X8L c=CV DK;>HJLH × Dy7dL <OUP]; `9d=1DK;>HJI8S y7 V P 5 n9 S DK; <OUL[@ –L [email protected]
N[@BI8IŸ;>NUL L S fS]; 8Ql9 V ' L × D V
; <OUP]9 MNF; <O:9 M=d7CDK;>I8P KOf9dP]O U;>P –L KO L ML L 9 nS –LUN [email protected]>DKO L 9<@ <O V ML
<;>LF; <O W ML W 9 KO?u;>P]OUP [email protected] KO|N MQR=8STP MNFO|9 fD KODK;>I8P KOa9<@ <O V V OU=8I8IŸ;RO
ic= lD KOš9 MN 8P MNFO OF; –L 9dP]O U;>P –LFO Vvo @BNšN MLUN[@>DMLUP]; sr S O {I8NF;>S]; 8Q 9 ME<;>NFOš9
-
X
γ
X
γ
j
−
+
3 2
2
(
3 0
X γ
E CT
−
+
3 I × {1} × D
{1} × Iγ × D
{0} × Iγ × D
γ
Iγ
I
0)182)4
I × {0} × D
V E A 8 DK;>=8NUX KOe9 ;>X<OULUNU=<DMLUP];
:
r
Kj j = 1, · · · , |nX
γ − 1|
@A? KD
0 = 1
V
N KOUI V W 6= O KDMLUP]; S N KOUI V S V
W KO:[email protected]<OY9 š? KDML M=8NFO <; _OUP 8QR=8STP MNFO S ML S 9 y5 8P]OFO –L:= O KDMLUP];
9 (I × η ⊕ I × η) @B= j 9 KOFOU=<O 9 ∂(I × I × D) DK; <OUL[@ cL •[email protected]
S ⊕S
N[@BI8IŸ;>NUL L S YS]; 8Q09 V ' L × D V<o @BNZDK; <O Kic= –LAr DK;>HJH 9<@ <O\[email protected] M=8?
9d=6DK; D DMS f9 P cL MNFO KDMLUP]; sr <;>=<O|IŸ;>=8?u; <O|H{; –LUN MN\ic= S ⊕ S KOFLE<;>H{;>LF;>I
[email protected]`O KDMLUP]; 9d= S j 5<X8N (I × η ⊕ I × η) P <9d=8PTL 69 Se [email protected] P]OF;>H{;>NUI8E8P]OFH
=8P]OFic= D ML*P]OF;>H{;>NUI8E8P]OFH aN –? MNFO |S ;>NUP cL[@BLUP];
(I ×η ) ' (I ×η ⊕I ×η) VRo
W− +
-
# +
+
−
−
0
3
−
2
C 0
γ
−
0
+
+
3
3
0
e = − e(I × η ⊕ I × η , S− ⊕ S+ )
c2 (I × ηC , S)
= − e(I × η, S− ) ∪ e(I × η, S+ )
L
V
4 3 3
4 X \4 =o 8P] OFic= [email protected]@>OFO l9 9 =8S MN e(I × η, S ) ?cPTL 9<@ <O = 29 M=d7dP]žMH {QRNF;>=8I {9
DK;>E<;>H{;>S];>QRP N [email protected] xUr S O lI8NF;>S]; 8Q 9 ME<;>NFOJ9 = = 8P]; U 9 OU=8N x™@>D KO
I8S]; 8Q KO\9<@ <O I × I × D ML~9 fX€;>NF9 S cLUN [email protected]>DKO L V e;>=<OaIŸ;>=8?u; <O~OU=8I8IŸ;RO MN
ic= MSTS KO|DK; <OULUPTLU= –La9 KO|OU=8N x™@>D KO|9 MP x MNULwIŸ;>=8N|[email protected]>ic= ZDK;>HJIŸ;RO[@ cL ZDK; y7
9
dX (τγ ) = − < e(I × η, S− ) ∪ e(I × η, S+ ) , µI×Iγ ×D >
)
)
- )
±
γ
±
3 EC E
±
±
9
V
[email protected] S]DM=8S MN S DM=8I I8NF;c9d=8PTL 9 KO [email protected]>OFO KO 9 9 =8S MN N [email protected]? KO
N M? P –L [email protected]]DM=8S MN6S P cL MNFO KDMLUP]; ˜@BSTQ MX8NUP]ic= _9
e(I × η, S ) ∪ e(I × η, S )
8
=
]
P
F
O
c
i
=
M
L
OF; cLb9 KO6= 8P]; <Ob9 #OF=8N [email protected]>D KO69 MP x MNULbI8S]; j
U · U Vao
U
U
Q KOJ9<@ <O I × I × D ' D 98; cL{S KO X€;>NF98OJO[; –L{9 KO <; M=<98O I8S]; 8Q KO{9<@ <O
r8S P cL MNFO KDMLUP]; #@BSTQ MX8NUP]i–=
KOUL\98; [email protected]\S [email protected] j
∂(I × I × D) ' S
D MH cLe9 KO cLUN [email protected]>DKO L · L 9<@ <O ∂(I × I ×UD)·V8Uo @BN\DK; <O Kic= –LAr
L±
−
−
−
+
γ
3
γ
3 (
S
+
4
Li )
i=1,··· ,|nX
γ |
∪(
S
−
+
3 γ
MLl9 0 [email protected]];>NFO L ML
<;>L MN[@
K )V
dX (τγ ) = −L− · L+
1
3 +
[
i=1,··· ,|nX
γ |
L+
−
j
j=1,··· ,|nX
γ −1|
L± =
S];>NFO
KOFLl9dPnm MN cLm9
nX
γ
−
P
+
L±
i ∪
[
j=1,··· ,|nX
γ −1|
OF; –L09 M=d7 DK;>I8P KOl9
Kj± .
|nX
γ |
dX (τγ ) = −
X
i,k=1
+
L−
i · Lk
X
|nX
γ | |nγ −1|
−
X X
i=1
+
−
+
(L−
i · Kj + Kj · Li )
j=1
|nX
γ −1|
X
−
N
Kj− · Kl+ .
j,l=1
+
L−
= 0
i · Lk
L−
i
−
Kj
·
·
Kj+
Kl+
=
Kj−
= 1
∀i, k
0
· L+
i = − ∀i ∀j = 1
∀j, l
P <OUP™rcS];>NFOFic= n KOFL9dPnm MN –La9 0 ML 1 r–S KO cLUP MNFO
HMH OUPTQ @B=<OFOUP = ML
X γ
,
0 nX
γ −1
ML
nX
γ
X
X
2
dX (τγ ) = 2nX
γ (nγ − 1) − (nγ − 1)
Pn
X
γ
=0
r<@BS];>NFO
L−
ML
X
= (nX
γ + 1)(nγ − 1)
L+
OF; –L\9 M=d76DK;>I8P KO\9 = <; M=<9
dX (τγ ) = −K1− · K1+ = −1.
E CC
0
K1
ML
OF; –L|9
Pn
X
γ
=1
r<@BS];>NFO
L−
ML
)
3
4
L1
ML
X
dX (τγ ) = (nX
γ + 1)(nγ − 1).
−1
γ
3 τγ−1
9<@ <O:D š[email protected]>OprGS cLUN [email protected]>DKO
3 Li = −{xi } × S 1
|nX
γ − 1|
DK;>=8NUX KO
j = 1, · · · , |nX
γ |,
Kj = 0 (−{0}×Iγ ×{yj })∪(I×∂Iγ ×{yj })∪({1}×τγ−1 (Iγ ×{yj }))
W KO~DK;>=8NUX KO K OF; cL~N MI8N KO cL KOe9<@ <[email protected]<QR=8N : V E : V
-
ML~9 KO
X
X
γ
8
0
`I M=8 [email protected] šS(τnHM)H nN[@BP]O[; MH –LYIŸ;>=8N
9 ;>X<OFLUNU=<DMLUP]; CO MN[@šS <O MHnX8S 9 KO
8 |n | DK;>=8NUX KO
+
dX (τγ ) = −L−
1 · L1 = 0.
1DK; <DMST=<OUP]; sr
9
OF; –L\9 M=d76DK;>I8P KO\9 = <; M=<9
L+
j
j = 1, · · · , |nX
γ −1|.
I × {1} × D
{1} × Iγ × D
−γ
{0} × Iγ × D
Iγ
I
0)132)4
:
V E : 8 DK;>=8NUX KO9 ;>X<OFLUNU=<DMLUP];
I × {0} × D
r
Kj j = 1, · · · , |nX
γ −1|
EC
A
r>IŸ;>=8N τ ML = 1 V
−1 γ
0
;>HJH :I8N K9 MHJH –LAr <;>=<[email protected]=8NF; <O [email protected]]DM=8S MN~S KO [email protected]>D MH –[email protected] –LFO
+
L−
= 0
i · Lk
∀i, k = 1, · · · |nX
γ |
+
0
X
X
L−
= Kj− · L+
i · Kj
i = ∀i = 1, · · · |nγ | ∀j = 1, · · · |nγ − 1|
Kj− · Kl+ = −1
∀j, l = 1, · · · |nX
γ − 1|
o @BNZDK; <O Ki–= cL
X
X
2
dX (τγ−1 ) = −2nX
γ (nγ − 1) + (nγ − 1)
X
= −(nX
γ + 1)(nγ − 1)
= :; ]7
@BS];>NFO\; bN MLUNF;>=8? YX8P
P γ KOUL\= DK;>=8NUX nOFPTHJI8S x MNUH šO [email protected][@ –L 9 šQ 8N
d (τ ) = 4h(h − 1) V
X
γ
E CX:
2
h
( nn8
2! $+_*),+ &
% ˜ ).*$ %&
- ! % d+ !2˜-(•$ !(*),+ .
) )- % ˜ % d4&) ),*),*
!#$+_*) +  $+ !#- * ! ).*
%&
- ) &- ) ),(- % ˜
% d4&) ) *) *
[email protected] <O•D [email protected] Rr Σ O MN[@ = OU=8N x™@>D (;>NUP cL 9 2Q 8N g O[@ <O`XŸ;>NF9
;>= = •DK;>HJIŸ;RO[@ –L 29 CXŸ;>NF9 Σ = Σ V ;>PTL X = D[[email protected]
Σ = Σ
9 n? KDML M=8NFO:DK;>HJI8S y7 <; `OUP 8QR=8STP MN:OU=8N Σ PTS*I M=8L MLUN š? = nDK;>HJH š= šLUNUP j
? [email protected]]O[@BLUP]; 1DK;>HJI8S y7 Y9 TΣ V
LF;>=8LDK;>XŸ;>NF9dP]OUH f9 E<;>H{;>S];>QRP OU=8N rc; [email protected] @>O[OF;cDMP MN= ZO KDMLUP];
9d= SU (3)j 5<X8N P(TM ) @B= j 9 KOFOU=<OšM9 ∂MΣ P <9d=8PTL [email protected] X V (DK; <OUP]98žMXN eMN[@
@BS];>NFO\S <O MH X8S P (M, X) 9 KO SU j [email protected][@BSTS MSTP]O[@BLUP]; <OZ9 M N [email protected]? Xe PTS
O ^@BQRPTN[@69 Ÿ= [email protected]>D [email protected] [email protected]= 9 KOFOU=<O 9
Z;>=<Of587 MNF; <O
= <;∂M
>=<O ; Z)@BI8I 'MSTS MZNFV; <O σj P c?>@[email protected] cL
= X8P KDMLUP]; JI8N yx MN cLUN P j (M, X) ML HZ ic(M,
N [email protected] x V :9 MN 8P MNZO MN[@šI8N KO MNU? :[email protected][email protected]šN [email protected]]; `9 fX€;>NF9dP]OFH V
<OU=8PTL Rr <;>=<O69 y5 8PTNF; <O = `N [email protected]]; 9 [email protected] <D _OU=8N0S <O MH X8S _9 KO
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S fQRNF;>=8I Y9 Y9dPnm K;>LF;>I8P KO M(Σ) V
h [email protected]=<; <O ic= JS KO y7dL <OUP]; <O D cLUN[@BS KOn9d=([email protected] bOF; –Lš9 y5 8P KO
DK; <OFP]9 MN[@ –LZS [email protected] 8QuOZLF;>N KO ; 1I M=8L~S KO~?u;>PTNZDK;>HJH n9 KO ?>@BNUP ML KOZO[@ <O
XŸ;>NF9šDK; <OULUNU=8PTL KO [email protected]*9 [email protected] 8QZD STP <9dN KO V W P]9 KOUL*@BS];>NFO*j 9 wDK; <OULUNU=8PTN
9 [email protected] 8P]žMN [email protected]=8N MSTS :9 KO j ?>@BNUP ML KOaO}@ <OwXŸ;>NF9 [email protected]~9 KO|DK;>XŸ;>NF9dP]OUH KOa9 E<; j
H{;>S];>QRP V
(
2
(
&JJ
Q
gf *k
h
Q
/Qih Tf
k PQik
*h
gf
7kk
#h
Q
e;>=<OwI€;>=8?u; <[email protected]>OFO[;cDMP MN LF;>=8L|DK;>X€;>NF9dP]OUH f9 E<;>H{;>S];>QRP Z= j ?>@BNUP ML ZO[@ <O
XŸ;>NF969 y5 8P DK;>HJH OU=8PTL
(') ;>PTL i ) = bDK;>XŸ;>NF9dP]OUH :9 E<;>H{;>S];>QRP [email protected]= j 9 KO[OU=<O
9 Σ V W"@ j ?>@BNUP ML T K(M,
[email protected] ?>@BNUP ML eDK;>[email protected]>DML \DK; y7 ~;>NUP –L ZO[@ <O
X€;>NF96;>X8L c= 9 M 6P]9 –LUP 5 @j cL −Σ @K? KD Σ
g
±
M
−
+
TM = M/i− (x)=i+ (x), x∈Σg .
i+ (x)
−
i (x)
V
j [email protected] ML fx MNUH
TM
;>X8L –= 9
M
bP]9 cLUP 5 @ –L
−Σ−
@K? KD Σ V
+
P
KOUL\= •DK;>X€;>NF9dP]OFH n9 E<;>H{;>S];>QRP {@B= 9 KOFOF=<OZ9 r€DK; <OUP]9
FN ; <O(M,Mi OF); _DK;>X€;>NF9dP]OUH 9 E<;>H{;>S];>QRP {@>OFO[;cDMP šOUj =8N Σ V Σ<;>L MN[@ @BS];>NFOj
[email protected] j [email protected] ML O[@ <O|XŸ;>NF9 T V
T
h [email protected]=<; <O~i–= :OFP M KOULa= [email protected] 8QlD STP <9dN @BS];>NFO T KOUL|S [email protected] 8Q
LF;>N [email protected]>OFOF; DMP V
;>=8NšLF;>=8LnDK;>XŸ;>NF9dP]OUH l9 E<;>H{;>S];>QRP M r <;>=<O IŸ;>=8?u; <O 9 y5 8PTNšS <O MHnX8S
o
9 KO SU j [email protected][@BSTS MSTP]O[@BLUP]; <Ow9 T V S KOUL Hn= 8PG9 = ~OULUNU=<DMLU=8N e9 [email protected]>D
P (T )
@ [email protected]= 9 KOFOU=<Oš9
[email protected] MS]; <O ic= JS σj P c?>@[email protected] cLš587 J=
X8P KDMLUP]; bIj 8N yx MN H (T , Z) ' Z V
±
g,1
g
M
2
M
M
su
M
T f
M 3 σ : Psu (TM ) → Z
k
M
SU
/h
3
ikLfh
hCkLh
3
gf *k
hCk >; PTL X = [email protected] š? KDML M=8NFOYDK;>HJI8S y7 <; #OUP 8QR=8STP MNYOU=8N Σ N[@BI8I MS]; <O
ic= PTSqI M=8L MLUN l?c=(DK;>HJH l= [email protected]]O[@BLUP]; DK;>HJI8S y7 09 TΣ MLni–= X KOUL
= 8P]ic= E<;>H{;>LF;>I8P I8NFžKO V
[email protected] <OaD [email protected][@BQRN[@BI8E Rr <;>=<[email protected]]; <[email protected]>OFOF; DMP MN LF;>=8L~DK;>XŸ;>NF9dP]OUH :9 E<;>H{;>S];>QRP
U
O
=8N Σ = qO KDMLUP]; Xe 9 P(TM ) @B= j 9 KOFOF=<O"9 ∂M P <9d=8PTL w9 X V KDMP–[email protected]
M
<;>=<OYI MNUH MLULUN {9 l9 y5 8PTNYS <O MH X8S P (M, X) 9 KO SU j [email protected][@BSTS MSTP]O[@BLUP]; <O 9
N [email protected]? KO Xe PTS<O ^@BQRPTL 9 = [email protected]>D [email protected] \@B= j 9 KOFOF=<O H (M, ∂M ; Z) ' Z V
M
CN M? [email protected] cL:@B=•[email protected]>OZ9 KO
[email protected] ML KO:O[@ <O\XŸ;>NF9 r <;>=<OZIŸ;>=8NUNF; <Oe587 MNe= X8P KD
LUP]; 1I8N yx MN cLUN P (M,3j X) ML Z i–= <;>=<[email protected] MSTS MNF; <O σj P –[email protected]@ –LZN [email protected] x j V
d5 sr <;>=<O ? MNUNF; <OJi–= mS
P –[email protected]@ –L{N [email protected] x KOUL I8N KO MNU? [email protected] E<;>H{;>LF;>I8P
σj
cLUN n9 M=d7bD[[email protected]`9 ? KDML M=8NFOeDK;>HJI8S y7 KO <; `OUP 8QR=8STP MNFOZ9 TΣ ML\[email protected]@ N j
[email protected]]; C9 :X€;>NF9dP]OFH V
;>PTL i ) = •DK;>X€;>NF9dP]OFH nOU=8N Σ V [email protected] MS]; <OZi–= nOF; `X€;>NF9 KOUL\P]9 cLUP 5
@B=6XŸ;>NF9 (M,
9 I × Σ ? [email protected]= b9dnP m K;>H{;>NUI8E8P]OFH <;>L Φ : ∂(I × Σ) → ∂M ML~i–=
S KOfI8S]; 8Q MH cLFO i ML i DK;>NUN KOUIŸ; <9 –[email protected]=d7•N KOULUNUP]DMLUP]; <OY9 Φ {0} × M
ML {1} × M N KOUI KDMLUPT? MH –L V
; <OUP]9 MNF; <O\[email protected]{[email protected]]O}@BLUP]; CDK;>HJI8S y7 9 T(I × Σ) DK; <OUL[@ –L nS :S]; 8Q09
M
m
L
P <9d=8PTL `[email protected]@(9dPTN KDMLUP]; I8N yx MN •9 I ML X 9<@ <O D ML6;>NF9dN V [email protected]#S
I
9dnP m K;>H{;>NUI8E8P]OUH Φ r MSTS P <9d=8PTL:= O KDMLUP]; X 9d= SU (3)j 5<X8N P(TM )
@B= j 9 KO[OU=<O~9 ∂M ?u;>PTN|5<QR=8N V V h [email protected]=<; <O\ic=
r KOFL\98; [email protected] [email protected]•9 :? KDML M=8NFO <;>[email protected] cLUN[@ –LfN [email protected] j
OU=8N
Σ X
LUPT? MH –L M ML i X r
r KOUL{98; [email protected] [email protected] 9 b? KDML M=8NFO <;>[email protected]\OF;>NUL[@ cL
ML{OU=8N
N [email protected]? MΣH –L X M ML i X V
S [email protected]:[email protected]>O9 ;>X<OULUNU=<DMLUP]; I8NF;>S]; 8Q MN X 0= \[email protected]]O}@BLUP]; DK;>HJI8S y7
9 TM V
-
M #
2 3
su
+
3
+
M
−
∗
2
M
M
M
M
−
M
C
su
− -
C
3 ±
M
$
+
∗
M
i+
i−
X
X
(i+ )∗ X
Σ = Σg
(i− ) ∗
X
i+
i−
X
(i+ )∗ X X
Σ = Σg,1
(i− )∗ X
V g KOFDMNUPTI8LUP]; `9 X
)(' ' W <O MH X8S
KOUL S <O MH X8S w9 KO SU j [email protected][@BSTS MSTP]O}@BLUP]; <O
D [email protected] X)@B= j 9 KOFOU=<O"9 H (M,
9 M N [email protected]? KO X D KOUL"= [email protected]>(M,
∂M ; Z) ' Z V
[email protected] <Oš[email protected];>I€;ROFPTLUP]; [email protected] –L Rr <;>=<O @BSTS]; <O 9 y5 8PTNš= 0X8P KDMLUP]; I8N yx MN
cLUN P (M, X) ML Z ML <; <D MNaic= MS]i–= KOwI8NF;>I8NUP ML KO V g:@ <OwS KOqx™@BPTLFOAr <;>=<O
@B=8NF; <O0X KOF;>P 9 6H{; cLUN MN0LF;>=<O0S KO{N KOU=8STL[@BLFOm9 [email protected]#I8NF;>I€;ROUPTLUP]; I€;>=8NlS [email protected]>O
@[email protected] –Le9 :I€;>=8?u;>PTN\S [email protected] :IŸ;>=8NaS [email protected]>O Σ = Σ V
Σ=Σ
% ' €>„
ƒ " ’>‰e‹ ”–‚0ŽB$‚ A„“‚ †c‡ !ˆ F>‰e‹ -]‚ ‚6ƒ >ƒ ˆ}Šƒ / † - ™‚ ‡
X †
ˆ †c‡ Σ 4 0a † ‡ „™ † „ F  ‡ ”>‘ˆ}‰l‚6,” + " >‰{. -T0 />™‚ (M, i ) ’ † ”–‚}ˆ}ˆ † ˆ ”–‚ Σ  -‚ kˆM„“‚
† ƒ€‚  K0‚ K„Rƒ [’>ƒ >ƒ< † ‚
M
su
(
su
3 3
M , #
g
g,1
±
Psu (TM ) → Psu (M, X)
† ‚yˆy„w’ €ƒ ‚ ’ † –” ‚}ˆ[ˆ † ˆ ”–‚-$+ kˆK>‰{ ‡ ‹"–‘ˆ}‰l‚ >”,+ ‚ M‘ˆ}z>ƒ
H 3 (TM ; Z) → H 3 (M, ∂M ; Z).
z’-T‚ σ Šƒ<Ž’ ‡ z’>ƒG„ σ : P
[‚K„z„“‚  K‚0K„Rƒ0”–Œ Gƒ Š„ † ƒ σj P –[email protected]@ –L
N [email protected] x
 X ‚}ˆM„ † ƒ{’ † „ ‡ ‚ " ’>‰e‹m”–‚σŽB‚0K„:“P‚ † ‡ ˆ (M,
F>‰eX)‹-]‚ →‚aƒZ.>ƒšˆ}Šƒ / † - ™‚ ‡ ˆ † ‡ Σ ’.-T ‡ ˆ -$+ " 
‰{R„™y‹ z‚l‚yƒG„ ‡ ‚ X ‚M„ X ‘ƒ€” † „ † ƒ€‚  K‚0K„Rƒ‚yƒG„ ‡ ‚ P (M, X) ‚M„ P (M, X )
† Ÿ‹ ‡ Œ}ˆA‚ ‡ ŽB‚8-T‚ σ Šƒ<Ž’ ‡ z’>ƒG„ ‡ ‚#-T’R„ →Z
su (TM )
r
su
0
0
su
0
su
Psu (M, X)
σr
/Z
'
% 5 Psu (M, X0 )
σr
/Z
;>PTL (M, Σi =) = Σ6DK;>X€;>NF9dP]OUH f9 E<;>H{;>S];>QRP @B= j 9 KO[OU=<O|9 Σ V [email protected] j [email protected] ML
I M=8L MLUN J? = {DK;>HJH m;>X8L –= 09
#N KDK;>[email protected] –L Σ
T
@K? KD −{0} × Σ ML −Σ @A? KD {1} × Σ MV ∪ (I × Σ )
±
g
M
g
−
M
g
g
g
+
M
E<;>H{;>LF;>I8P _I8NFžKOpr|PTS y7 P]OUL `= •= 8P]ic= _O KDMLUP]; 9
@B=
9 KOFOU=<O~9 I × Σ DK; <OFP]9 MNF; <O\[email protected] KDMLUP]; Xe DK; <OUL[@ cL P(T(T
S :S]; 8Qm)9 )I P jj
9d=8PTL |[email protected]@f9dPTN KDMLUP]; lI8N yx MN a9 I ML X VRo ;>=8NqLF;>=8L SU j [email protected][@BSTS MSTP]O[@BLUP];
9 r PTS y7 P]OUL Z= N MI8N KO cL[@ –Leic=8P DK; <DMP]9 [email protected]? KD OU=8N
P <OUP
LF;>=8TL SU j [email protected][@BSTS MSTP]O[@BLUP]; 9 T KOUL cLUP]žMN MH –L{9 XMLe MNUHJP I ×Rr Σ VE<;>H{; j
LF;>I8P N [email protected]? I × Σ I8NFž[email protected]= y7 L <OFP]; 9 Xe T V Z;>LF; <O
e S <O MHnX8S b9 KO SU j [email protected][@BSTS MSTP]O[@BLUP]; <O09 T N [email protected]? KO
P (T , X)
e;>=<O:;>X8L <; <[email protected]];>NFO:= X8P KDMLUP]; P (T ) → P (T , X)e @ {@BXe= j V
9 KOFOU=<Oa9 :S P]O[;>H{;>NUI8E8P]OUH 9 [email protected] H (T ; Z) → H (T , I × Σ ; Z) V
E MNFDFE MN I8NF;>S]; 8Q MN T N M? P –L DFE MNFD[E MN I8NF;>S]; 8Q MN X M
1L[@ cLfic= O KDMLUP]; `9 XeP(TM
KDMPtP <9d=8PTLf= YX8P KDMLUP]; `[email protected] <; 8P]ic=
) V
i–=8P KOULš@ [email protected]= 9 KOFOU=<O 9 JS P]OF;>H{;>NUI8E8P]OUH
P (T , X)
9 y7dDMP]OFP]; eH→(TP , I(M,× ΣX); Z) → H (M, ∂Mj ; Z) V Z;>=<O:;>X8L <; <O @BP <OUP
= 1X8P KDMLUP]; [email protected] <; 8P]i–= P (T ) → P (M, X) @ `@B= j 9 KOFOF=<Om9
S P]OF;>H{;>NUI8E8P]OUH [email protected] <; 8P]ic= H (T ; Z) → H (M, ∂M ; Z) V
W"@_X8P KDMLUP]; I8N yx MN –LUN
ML 98; [email protected] P –[email protected]@ –L
P <9d=8PTL~= :X8P KDMLUP]; bI8N yx MN YPi–= (T<;>=<)[email protected] MSTS MNF; <O σj P –[email protected]@ σ–Lj \N [email protected] x
g su
M
g
(
M
3
M
3
M
M
M
M
su
3
M
su
3
M
g
su M
3
M
3
su
3
M
g
su
g
M
su
su
3
C
M C
3
M
3 M
M
#
σX : Psu (M, X) → Z.
)(' , W"@CX8P KDMLUP];
P c? MNFO 69
I M=8L MLUN 19 KDMNUPTL `OUPTHJI8S MH cL 9 [email protected][email protected] 8P]žMN P`OU=8([email protected] –L) → POF;>PTL (M,
= X)
α
[email protected]]O}@BLUP]; DK;>HJI8S y7 m9 M ic=8PqI8NF;>S]; 8Q X V W 0N KDK;>STS MH –LJ9
MLl9 α| 9 y5 8PTLl= 6LUNUPT? [email protected]]O[@BLUP]; DK;>HJI8S y7 γ 9 TT V
α|
W"@ X8P KDMLUP]; [email protected] <; 8P]i–= P (M, X) → P (T ) c?u;>P :[email protected] [email protected]>OFO :9 α
9<@ <O P (M, X) [email protected]@>OFO :9 E<;>H{;>LF;>I8P n9 γ V
H{ž MNF; <[email protected]=`[email protected]]DM=8S
o9 ;>= =8Nfσ[email protected] P S]–[email protected] [email protected] S –Lnσ?cj PkP @m–[email protected]@[email protected] P –L KfDMNLUMP]; [email protected] Ax @r <;<;>8P=<]O ic=<;>=<PO\N[@B(M,
X) → P (T )
DMP j 9 KOFOU=<O V
;>P –L X ML X 9 [email protected]<O09 ? KDML M=8NFO{DK;>HJI8S y7 KO <; OUP 8QR=8STP MNFO
OU=8N e;>=<Oe?u;>=8S]; <OeH{; –LUN MN:i–= YS E<;>H{;>LF;>I8P cLUN ML P <9d=8PTL
= JΣX8P V KDMLUP]; –LUN lS KO [email protected]>D [email protected] KO P (M, X ) ML XP (M,XX ) ic=8P
I8N KO MNU? {S σj P c?>@[email protected] cLnN [email protected] x V"o ;>=8N D [email protected] <;>=<[email protected]]; <OnDK; <OUP]9 MN MNš[email protected] 4j
?>@BNUP ML `;>NUP –L I × M @BP <OUPfi–= = CE<;>H{;>LF;>I8P h –LUN X ML X
MSTS JP <9d=8PTLš= JE<;>H{;>LF;>I8P h –LUN lS KOnO KDMLUP]; <O (X ) ML (X ) 9
@B= j 9 KOFOU=<O\9 ∂M V
P(TM )
;>PTL = {LUNUPT? [email protected]]O[@BLUP]; DK;>HJI8S y7 09
ic=8PqI8NF;>S]; 8Q
I M=8L αST=8P@>OFOF; DMP MNY= 0O KDMLUP]; (9d= SU (3)jTM
5<X8N P(I × TM ) ([email protected]= )j 9 KV OFOU=<O
9 M := ({1} × M ) ∪ (I × ∂M ) 9 y5 8P
OU=8N {1} × M r
[email protected]
α
OU=8N I × ∂M V
[email protected]~S E<;>H{;>LF;>I8P
h
=8P]OFic= [email protected]šDK;>E<;>H{;>S];>QRP YN [email protected]?
KOUL –=8STS Rr8D MLUL O KD
LUo P]; {I M=8L O aI8NF;>S]; 8Q MN I × M 9 aHHl([email protected] 8P×]žMN M,~= M8P]ic;= Z) E<;>H{;>LF;>I8P ~N [email protected]? j
−Σ−
M
su
M
su
0
1
g
0
-
1
su
1 0
1 M
1 M
1
M
0
su
0 M
C
1
3
M
M
su
3
0
- M
C
1
su
su
-
M
su
su
Σ+
M
M
∗
0
I8NFžKO V Z;>LF; <O α = |L MSTS y7dL <OUP]; ML α O}@\N KOFLUNUP]DMLUP]; {0}×M
9
y5
[email protected]];>NFOq= ZLUNUPT? [email protected]]O[@BLUP]; mDK;>HJI8S y7 f9 ic=8PGI8NF;>S]; 8Q
α
W šX8=8L [email protected]];>NFO:9 n? MNUP 5 MNYi–= σ (M, α ) = Mσ (M, α ) V o ;>=8(XNYD )[email protected]
<;>=<O\@BSTS]; <O <;>=<O~N[@BH [email protected][email protected]]DM=8S"9 = σj P c?>@[email protected] cL ML~DK; <OUP]9 MN [email protected]
@A? KD
j ?>@BNUP ML W"[email protected]{×O KTDMLUP]; ;>X8L 9 –= `9 I × M P <9P]d9 =8PT–LLAUrGP ?c5 [email protected] @JcLS :IN K×DK;>(−Σ
)
STS MH –LAr€=
I ×Σ V
P(I × TM )
O KDMLUP]; γ 9d= SU (3)j 5<αX8N P(I
P
<9
d
8
=
T
P
a
L
=
\
<
E
>
;
{
H
;>LF;>I8P cLUN
× TT ) V γ
S
K
f
O
U
L
U
N
T
P
c
?
k
P
B
@
T
S
]
P
[
O
B
@
U
L
]
P
;
<O
:
K
D
>
;
J
H
8
I
S y7 KO γ ML
M
L
γ := γ|
γ := γ|
9
@B=8NF; cLe98; <D:H MH σj P c?>@[email protected] cL Dyx*N [email protected]= V V V Z;>=<O
γ
@K?u; <OeTT
98; <D:S KO MQ–@BSTPTL KO
M0
0
-
M
+
1
X0
0
X1
C
{1}×T
M −
M
1
1
M C
{0}×TM M
0
0 M
0
M
0
σ(TM , γ0 ) = σX0 (M, α0 )
σ(TM , γ1 ) = σX1 (M, α1 )
GI @[email protected] N [email protected]= V V V KDMP @>DFE<žM? [email protected]:I8N M=8? f9<@ <OS [email protected]>O Σ = Σ V Z;>=<O
<;>L MNF; <[email protected][email protected] OF=8PTL σ S σj P –[email protected]@ –LZN [email protected] x V
e;>=<Oe?u;>=8S]; <O <;>=<OeN[@BH [email protected][email protected]>OeI8N KD K9 –LfQRN >D
@B=6S MHJH YΣOU==8PT?RΣ@ –L
)(')() €>„
ƒ " ’>‰e‹ ”–‚JŽB‚$K„“‚ †c‡ ˆ{ƒ >ƒ_ˆy‘ƒ / † - ™‚ ‡ ˆ † ‡ Σ 4
X †
>ƒ8ˆy”–Œ >ƒ<ˆ
ƒ
F
U ”>kˆy‰l‚6”,+ " >‰{.-T0/>™‚ ’
”–‚}ˆ}ˆ † ˆm”–‚ Σ ‚M„
i ) †
‡ ˆM>ƒ (M,
‡
†
FF ”Rkˆy‰{‚ ”,+ "d>‰{.-  />z‚ ’Bˆ}ˆM KzŒ ’ ”–‚yˆ[ˆ ˆ ”–‚ 4 0~
(M , i )
„™ † „“‚š‚ „“‚yƒ8ˆy>ƒ X‡ ”–‚ X Σ  -v‚ ‘ˆy„“‚ † ƒ€‚ y K$‚ A„† >ƒ F’†Rƒ >ƒ< Σ† ‚ †c‡
r
g
2
g,1
g,1
±
±
g
g,1
g
Psu (M , X) → Psu (M, X)
† ‚}ˆy„’ €ƒ ‚n’ † –” ‚}ˆ}ˆ † ˆš”–‚ -0+ ‘ ˆMR‰J ‡ ‹"ckˆy‰{‚ ”,+ ‚ M‘ˆ}z>ƒ
H 3 (M , ∂M ; Z) → H 3 (M, ∂M ; Z).
n‚|‹- † ˆ ˆ} X ‚}ˆM„ † ƒ€‚š’ † „ ‡ ‚ ‚ „“‚yƒ<ˆ}z>ƒ#”–‚ X Σ ’ -  ‡ ˆ:ƒ  † ˆ ’>Ž>ƒ8ˆ ]- ‚
”>z’ / ‡ ’>‰ ‰{‚ FR‰š‰ † „™’R„ ”–‚y A‚$K„z>ƒ8ˆ8[’>ƒ >ƒ< † ‚}ˆYˆ † ŠŽ’>ƒG„
0
/ P (M , X0 )
su
o
o
o
oo
ooo
o
w oo
Psu (M , X)
NNN
NNN
NNN
NN'
Psu (M, X)
† ‚}ˆy„’ €ƒ ‚n’ † –” ‚}ˆ}ˆ † ˆš”–‚}ˆY‘ˆMR‰J ‡ ‹"ckˆy‰{‚yˆ F’Rƒ >ƒ< † ‚}ˆ
id
H 3 (M , ∂M ; Z)
RRR
RRR
RRR
RRR
R(
W [email protected]>O
-
g
/ H 3 (M , ∂M ; Z)
ll
lll
l
l
lll
l
v ll
H 3 (M, ∂M ; Z)
Σ = Σg,1
O :9 [email protected]];>[email protected]>O
Σ = Σg
[email protected] KOULUNUP]DMLUP]; V
S <;>=<OaN KOFL I8NF;>=8? MN\S fS MHJH :I8N KD K9 cL
% )(' ;>PTL = [email protected] š? KDML M=8NFOYDK;>HJI8S y7 <;
OFP 8QR=8STP MN\5 OU=8N Σ5 ML~OF;>PTL X = y7dL X<OUP]; b9 X Σ V <;>L MN[@
[email protected] O KDMLUP]; 19 P(TM ) @B= j 9 KOFOF=<O~9 ∂M P <9d=8PTL [email protected] X r
X
[email protected]{O KDMLUP]; 19
@B= 9 KOFOF=<O\9 ∂M P <9d=8PTL :[email protected] X
ML
DK; <OFLUNU=8PTL XKO\DK;>HJH :I8N KD K9 MP(TM
HJH cL )V j
E<;>H{;>LF;>I8P eN [email protected]? ∂M I8NFžKOAr–PTS y7dP]OUL ~= \= 8P]i–= y7dL <OUP]; m9 X
; <OUP]9 MNF; <O|S = 8P]ic= y7dL <OUP]; s ic=8P OF;>PTL|DK; <OUL[@ cL eS eS]; 8Q
∂M ∪ (I × D ) V
9 I OU=8N I × D V
";>=8L [email protected][@BSTS MSTP]O[@BLUP]; 9 M N [email protected]? X KOUL cLUP]žMN MH –L9 ML MNUHJP Rr
E<;>H{;>LF;>I8SUP aN j [email protected]? ∂M ∪(I ×D
I8NFž[email protected]= y7 L <OFP]; 9 s M V>o @BN
)
y78DMP]OUP]; sr LF;>=8L SU j [email protected][@BSTS MSTP]O}@BLUP]; C9 M N [email protected]? X KOULa98; <D –LUP]žMN MH cL
9 ML MNUHJP Rr E<;>H{;>LF;>I8P eN [email protected]?
I8NFžKOAr–[email protected]= y7 L <OUP]; m9 X M V
Z;>=<O~;>X8L <; <Oe98; <D:= :X8P KDMLUP]; 1DA∂M
@ <; 8P]i–=
2
g,1
M
2
3 g
M
X
M
M
2
X
2
C
C
M
M
Psu (M , X) → Psu (M, X)
ic=8P KOUL\@ [email protected] = j 9 KO[OU=<O\9  S P]OF;>H{;>NUI8E8P]OFH 9 y78DMP]OUP]; -
3
H 3 (M , ∂M ; Z) → H 3 (M , M ∪ (I × D 2 ); Z) → H 3 (M, ∂M ; Z).
' W"@YX8P KDMLUP]; P –? MNFO ZI M=8LO e9 KDMNUPTN :@BP]O MH cL OF;>PTL =
LUNUPT? [email protected]]O[@BLUP]; lDK;>HJI8S y7 \9 M i–=8PGI8NF;>S]; 8Q X Vuo ;>=8Nw;>X8L [email protected]? [email protected]]O[α@BLUP];
DK;>HJI8S y7 Y9 M @>OFOF; DMP :i–=8P I8NF;>S]; 8Q X rdPTSsOU= {L\9 :I8NF;>S]; 8Q MN α [email protected] s V
;>P –L X ML X 9 M=d7#[email protected]<O 9 J? KDML M=8NFO DK;>HJI8S y7 KO <; 2OUP 8QR=8STP MN OU=8N
ic=8P I8NF;>S]; 8Q –L S];>NFOaPTS y7dP]OUL Z= :E<;>H{;>LF;>I8P h cLUN X ML X ic=8P KOUL
Σ
S P]9 cLUPTL OU=8N Σ XV V
;>PTL α = |LUNUPT? [email protected]]O[@BLUP]; mDK;>HJI8S y7 ~9 TM ic=8P<I8NF;>S]; 8Q (X ) ; JI M=8L
OF=8I8I€;RO MNYic= α DK;>NUN KOUI€; <9 s OU=8N I × D V Z;>LF; <O α [email protected] N KOFLUNUP]DMLUP]; 29
9 y5 8PTL~= [email protected]]O[@BLUP]; `DK;>HJI8S y7 9 ic=8P I8NF;>S]; 8Q
α
M α
; <OUP]9 MNF; <O[email protected] 4j ?>@BNUP ML I × M V e;>=<OqI€;>=8?u; <OMDK; <OULUNU=8PTN e= eO XKDMLUP]; V m9d=
5<X8N P(I × TM ) 9 [email protected] [email protected] 8P]žMN YOF=8PT?>@ cL
SU (3)j
cLUN X ML X P <9d=8PTLš= {E<;>H{;>LF;>I8P h –LUN 0S KO O KD j
S E<;>H{;>LF;>I8P
h
LUP]; <O s ML s V MLUL 9 MN 8P]žMN I M=8L MLUN š? = nDK;>HJH = O KDMLUP]; _9
@B= j 9 KOFOF=<O\9 I × I × D V
P(I × TM )
K
U
O
w
L
K
D
;
<O
U
[
L
@
–L
e
S aS]; 8Qn9
MLqI M=8L MLUN aN KDK;>STS [email protected]:O KDMLUP];
h
9 (P(I × TM ) )| DKI; <O×UL[[email protected] ×cL ∂D
YS fS]; 8Ql9 I MLaP <9d=8PTL [email protected] α V
W"@{O KDMLUP]; `;>X8L –= nN KOULUN MP –L
r<ic= :S ; <;>L MN[@ rGP <9d=8PTL\=
LUNUPT? [email protected]]O[@BLUP]; `9 TM ic=8PsI8NF;>S]; 8Q (X{1})× MV W KO~LUNUPT? [email protected]]O[@BLUP]; <OeαDK;>HJI8S y7 KOZ; –L
@BS];>NFO H MH σj P –[email protected]@ –LN [email protected] x ML <;>=<Oq;>X8L <; <O S [email protected][@BHJH \DK;>HJHn=8L[@BLUP x"9
M
M
0
1
3 -
X
' -
0
g,1
0
0
0
C
X1
1
M
2
2
0
0
C I 0 ×M
0 M 0
0
0
0
C
X0
2
0
X
0
3
1
1 M
3 0
1
X8P KDMLUP]; <O\[email protected] <; 8P]ic= KO
/ P (M , X )
1
su
o
o
o
oo
ooo
w oo
o
Psu (M , X0 )
OOO
OOO
OOO
OO'
Psu (M, X)
ic=8P KOUL\@ [email protected] = j 9 KO[OU=<O\9 KOaP]O[;>H{;>NUI8E<OUPTH KO\[email protected] <; 8P]ic= KO
-
id
H 3 (M , ∂M ; Z)
/ H 3 (M , ∂M ; Z)
lll
lll
l
l
ll
l
v ll
RRR
RRR
RRR
RRR
R(
H 3 (M, ∂M ; Z)
g ^@BI8NFžKO\[email protected]šI8N M=8? Y9 :[email protected]šI8NF;>IŸ;ROUPTLUP]; V V dr <;>=<O\@K?u; <O\S [email protected]~OF=8PT?>@ cL
(') ;>P –L X ML X 9 M=d7 [email protected]]O}@BLUP]; <OmDK;>HJI8S y7 KO 9 TΣ
MLnOF;>PTL = 2DK;>X€;>NF9dP]OFH l9 E<;>H{;>S];>QRP mOU=8N ; <OUP]9 MNF; <O N KOUI =
LUNUPT? [email protected]]O[[email protected]]; 1DK;>HJI8S y7 9 TM ic=8P I8NF;>S]; 8Q ΣXV N KOUI V X αV V α
W KO*DK;>=8I8S KO α) ML (M, α ) ; –LtHMH σj P –[email protected]@ –L N [email protected] x€OUP ML*O M=8S MH cL
OFP PTS y7 P]OUL f= O (M,
KDMLUP]; 19
ic=8PsDK;>NUN KOUI€; <9
Oα F=8N {0} × M r P(I × TM )
OU=8N {1} × M
α
[email protected] Φ [email protected]= YE<;>H{;>LF;>I8P –LUN X ML
MLeic=8P"OF;>PTLeP <9d=8PTL nOU=8N
I × ∂M
U
O
[
L
B
@
8
X
T
P
T
S
]
P
O
–
Q
B
@
<
=
F
D
E
G
I
B
@
~
N
k
S
@
d
9
T
P
N
KDMLUP]; 1I8N yx MN Y9 I V
X
(') W fS MHJH <;>=<O~O MN[@ =8LUPTS [email protected]@šOU=8PTL fIŸ;>=8N <; <D MN
9 KOYN KOU=8STL[@BLFOY9<@ <O:S [email protected]>O:Q MN[@BV SwV 9 Σ = Σ ;>= Σ
I8NF;>[email protected] –Lši–= šS
[email protected]>O Σ = Σ S [email protected]>O Σ = Σ O MN[@J9 K9d=8PTLe9d=bI8N KD K9 –[email protected] N KOULUNUP]DMLUP]; V
`@BP cL [email protected] –L|i–= <;>=<O@K?u; <Oq9 y5 8PG= aX8P KDMLUP]; 0I8N yx MN –LUN P (M, X)
ML r|[email protected] ic= KOULUP]; KOULb9 _O[@A?u;>PTN1OUP MSTS KOFLbDK;>[email protected] [email protected]? [email protected] N [email protected]]; 9
XŸ;>NFZ9dP]OUH
% 3)(') €>„
ƒ #" ’>‰e‹ ”–‚wŽ>$‚ K„“‚ † ‡ ˆ FR‰\‹ -T‚ ‚|ƒ >ƒ ˆy‘ƒ / † - ™‚ ‡ ˆ †c‡
X†
4 G
‚K„ N ˆM>ƒG„*”–‚ † FU ‡ ”>kˆy‰l‚}ˆ:” + " R‰J - 0 /Rz‚ F ‡ ”–’>ƒG„Šˆf’ -  ‡ ˆ\ -€‚ ‘ˆy„“‚ † ƒ€‚
Σ
M
M A$‚ K„z>ƒ F’>ƒ >ƒG ‚ ‚yƒG„ ‚
‚M„
‘ƒ€” † „“‚w‹8’ ‡ - ’ ‡ #‚ - ’u„>ƒ_”–‚
P (M, X) P (N, X)
†
‡
F ”Rkˆy‰{‚
‡
† –‹ ‡ Œ}ˆA‚ ‡ ŽB‚-]‚ σ ‘ƒ<ŽB’ ‡ z’>ƒG„ ‡ ‚#„™-T’R’R„„  4 ”–‚ ƒ0y”,A$‚+ ’ K„† z„ >‡ ‚}ƒ8ˆaˆ „“F’‚ >‡ ƒ ‰lR‚}ƒ<ˆ  -T’ ‚}ˆ‡ ‚ ˆ - ’R„‘zŽ>’Rƒ0ƒG„”c‚
F ”Rkˆy‰{‚ Šƒ ” Š„ -T‚š”>z’ / ’>‰ ‰l‚ F>‰ ‰
‡
†
‡
† † †
2
-
-
0 M
0
M
0
C
0
0
-
g
g,1
g,1
M
g 0
2
su
su
su
Psu (M, X)
Psu (N, X)
σr
σr
/Z
/Z
% g ^@BI8NFž[email protected] N [email protected]= \I8N KD K9 cL RrdPTS <;>=<OOU= {L|9 \I8NF;>=8? MNaD MLUL
I8NF;>IŸ;ROUPTLU5 P]; 1O M=8S MH –Le9<@ <O~S [email protected]>O Σ = Σ V
;>P –L (M, i ) ML (N, j ) 9 M=d7 DK;>XŸ;>NF9dP]OUH KO 9 E<;>H{;>S];>QRP #@B= j 9 KOFOU=<Om9
=8I8IŸ;ROF; <O0ic= PTS]OmOF;>P –LmX€;>NF9<@ cLFO PTS y7dP]OUL `@BS];>NFOl=
?>@BNUP ML 9
Σ V
XŸ;>NF9 ∂W = M ∪ −N ; Σ KOUL|N KDK;>STS :@K? KD −Σ L MSTS Zic= ej S KO|P <DMST=<WOUP]; <O
[email protected] <; 8P]ic= KO M ,→ W ML N ,→ W P <9d=8P]O –Le9 KO|P]OF;>H{;>NUI8E8P]OUH KO E<;>H{;>S];>QRP V
[email protected] MS]; <Oqi–= aS ; {DK; <OFP]98žMN W @K? KDaS KOSTP M=d7{9 ^@BLUL[@>D[E MH –La9 M @K? KD −N
[email protected]]O[OUP]O
∂W = M ∪ (−Σ × I) ∪ −N ∪ (Σ × I)
; KOUL~P]9 –LUP 5 š@K? KD −Σ × {0} r
Σ
@K? KD Σ × {0} r
−Σ
@A? KD Σ × {1} r
−Σ
A
@
?
KD −Σ × {1} V
Σ
g ^@BI8NFžKO[email protected];>I€;ROUPTLUP]; & V V drcPTS y7dP]OUL ~= [email protected] ~? KDML M=8NFO <; OUP 8QR=8STP MN
OU=8N W i–=8P KOUL <;>[email protected]–OF;>NUL[@ cLvOU=8N M ML <;>[email protected] cLUN[@ –L*OU=8N N ML"98; [email protected]
A
[email protected]~9dPTN KDMLUP]; I8N yx MN 9 I S S]; 8Qe9 KO STP M=d7Y9 ^@BLUL[@>DFE MH –L [email protected]]OFOFP]O V S–P <9d=8PTLs=
9 KDK;>HJIŸ;ROUPTLUP]; OF;>HJH w9 E8PTL 9 S [email protected]>D L[@ 8Q cL TW = ν ⊕ζ ; ν ML
OF; –L 9 KOt5<X8N KO ? KDMLF;>NUP MS]O N MS]O;>NUP –L KOw9 |N[@ 8Q ML \N KOFI KDMLUPT? MH cL V
ζ := ν
h [email protected]–=<; <O\i–= ζ @>9dH ML\S KO~I8NF;>I8NUP ML KO\OF=8PT?>@ cL KO g
3 ±
M
0
−
M
±
N
+
M
−
M
+
M
−
M
+
M
+ M
−
M +
N
− N
W
g
±
-
±
-
⊥
ζ|M
= TM,
ζ|N
= TN.
;>PTL α = [email protected]]O}@BLUP];#DK;>HJI8Sy7{9 M ic=8PI8NF;>S];8Q X MSTSnIM=8L
@B= 9 KOFOF=<OZ9 W"@0DK;>E<;>H{;>S];>QRP N
MLUNn? =nDK;>HJHš= OKDMLUP];•9 Rr α O{j I8NF;>S];8Q MWV 9 {[email protected]; = 8P]i–= j
[email protected]? H (W, M ; Z) ML[@cL [email protected](ζ)
E<;>H{;>LF;>I8P NM [email protected]? M I8NFžKO V e;>LF;<O α = LMSTSy7 L<OUP];ML α O[@0NKO j
LUNUP]DMLUP]; N V o =8P]O[i–= ζ| = TN r*[email protected]_OKDMLUP]; α 9 y58PTL{= LUNUPT? [email protected]]O[@BLUP];
DK;>HJI8Sy 7 9 N V [email protected] SK O~P]9 c LUP [email protected]];< O:9 Σ @A?K D −Σ r α I8NF;>S];8 Q X V
P!< OUP™r< ;>=<O\@A?u;< O\= Y @BI8I8STP][email protected]];C X8P1 9 y 58 P
M
M
C
∗
M
W
N
N
±
M
N
±
N
Psu (M, X) → Psu (N, X)
N
N
ic=8"P –?u;>P [email protected]•[email protected]>O[O69 E<;>H{;>LF;>[email protected]?
9 [email protected]`[email protected]>OFO69 E<;
H{;>LF;>I8P{[email protected]? ∂N 9 α V I8NUP];>NUP™rMSTSJ∂M
9 MI<9_α9d=#DFE<;>P 7_9d=•D[[email protected] j
? KDML M=8NFO A V
e;>=<Oa?u;>=8S]; <O I8NKO–LeH{;–[email protected] DK;>HJH =8L[@BLUPT? [email protected][@BHJH
W
M
N
Psu (M, X)
Psu (N, X)
σr
σr
/Z
/Z
;>=8ND [email protected]NM?<;<O 9 [email protected]]DM=8S]Ot9 σj P!–[email protected]@–L ML*DK;<OUP]9 MNF;<[email protected] j R? @BNUPML
o
>
;
X8Lc={9 •P]9 –LUP 5 @–L
@K?KD
Dyx 5<QR=8N V V W W
XŸ;>NF969 W KOUL W∂W = T ∪ −T −ΣV × I Σ × I V
0
0
M
−
M
+
M
0
N
M
W
W’
−N
V ; <OULUNU=<DMLUP];`9 W V
@BNfNKDK;>STSMH–LArŸ[email protected] KDK;>HJIŸ;ROUPTLUP];_9 TW P!<9d=8PTLf= n9 KDK;>HJIŸ;ROUPTLUP];_9d=
o
5<X8N~L[@8Q–L TW = ν ⊕ ζ ; ν ML ζ OF; cLq9 KO*5<X8NKOq?KDMLF;>NUPMS]O N MS]O ;>NUPcLKO
9 fN[@8Q ML P!<[email protected] ν ML ζ N KOUI KDMLUPT? MH–L V [email protected]–=<;<Oei–=
0
0
0
0
0
0
ζ 0 | TM
= TTM ,
0
@BN N KDK;>STS MH cLAruS aD[[email protected] |?KDMLM=8NFO <;lOFP!8QR=8STPMN P!<9d=8PTL = {D[[email protected]
9 Z?o KDMLM=8NFO"<;bOFP!8QR=8STPMN A OF=8N W ML[email protected] OKDMLUP]; α A9 P(ζ) P!<9d=8PTLa= OKDMLUP]; 9d=
5<X8N ) V Z;>LF;<O γ ML γ SKO:[email protected]]O}@BLUP];<O DK;>H j
I8Sy7 KO"9 γ TT SUML"(3)9 j TT P!P(ζ
<9d=8PTLKOv9 γ [email protected]]DMLUP]; V g ^@BI8NFžKO"[email protected]|I8NF;>IŸ;ROUPTLUP];
dr
VV
ζ | TN
W0
0
M
= TTN .
W
0
W
C
M
C
N
N
σ(TM , γM ) = σr (M, αM ),
“S"<;>=<[email protected]];>NFO H{;–LUNMN i–= , γ ) = σ(T , γ ) Vwo ;>=8NmD [email protected]
DK;<OFP]9 MNF;<OY= j [email protected] Z DK;>[email protected]>DML0σ(T
DK; y7J;>NUPcL l9 JXŸ;>NF9 ∂Z = T V
<;>LMN[@ [email protected] [email protected];>X8L–= 6NKDK;>[email protected]–L Z ML W SYS];8Q{9 T Z @
IŸ;>=8NaXŸ;>NF9 TZ V S]j ;>NFO
OUPTQ Z ,
σ(T , γ ) = < p (TZ , γ ), µ > −3
OUPTQ Z,
σ(T , γ ) = < p (TZ , γ ), µ > −3
; N KOFI
DK;>NUN KOUIŸ; <9 S ;>X<OULUNU=<DMLUP]; [email protected] I8NF;>S]p; 8(TZ
Q MN γ, γ N)KOUI V γ V pOF(TZ,
L[@BX8PTSTP]Oγ ) Q–@B=<D[E [email protected] S ~D[[email protected] a? KDML M=8NFO <;>[email protected]
O[;>NUL[@ –L~N [email protected]? MH cL Z N KOUI V Z V
! *, ! 0 ..$ +. .&4+ ( 3 + ."$ 3 .! 0! σ(TN , γN ) = σr (N, αN ).
M
M
N
0
N
0
N
N
0
M
0
1
M
M
!*$ (!
M
M
1
N
N
1
N
0
M
N
1
N 0 0
Z0
Z
/
=8P]OFic= γ OUL[@BX8PTSTP]O [email protected]S eD[[email protected] 9 e? KDML M=8NFO A 9 y58PTL|= eOKDMLUP]; 9 o
ic=8PŸDK;>NUNKOUI€;<[email protected]=d7mOKDMLUP];bOUL[@BX8PTSTP]O KO OU=8N ML OU=8N r
P(TW )
<;>=<Ol;>X8L<;<OArQRN >D [email protected]`LUEK;>NUPb9 6DK;<D cLUN[@BLUγP]; 9 ;>TX<OFLUNU=<DMLUP];γsrS MQ–@BTSTPTL
OF=8PT?>@cL
W0
C
M
M
N
N
W"@[email protected] Z KOULn;>X8Lc=m9 Z ML W (SKOšNKDK;>[email protected]–L S0S];8Q`9 = j
[email protected] YDMS];RO :98;<DRr<[email protected]~SfLUEK;>NFžMH 9^@>989dPT? PTL 9 Z;A? P u;A?Ÿr
OUPTQ Z = OUPTQ W + OFPTQ Z.
W"@1OFPTQ [email protected]=8N 09 W KOFL –=8STS [email protected] W @bS E<;>H{;>S];>QRP 9 {[email protected] j [email protected] T V
GmMLAr–S MLU=<9 f9 [email protected]:xz;>NU
H Z9 !P –L MNFO KDMLUP]; bOU=8N H (W ; Z) = H (T ; Z) IM=8L
98; <D0O JN KOULUN M!P <9dN lS MLU=<9 l9 KO j D DMS [email protected] –Ln9<@ <O = 2DK;>STSTP MNš9d=2XŸ;>NF9 V N
;6IM=8L~LF;>= U;>=8NFOaSKO\OF=8I8I€;ROMN~9dP]O ;>P!cLFO V
o @BNZDK; <O Ki–= cLAr σ(T , γ ) = σ(T , γ ).
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∂M
= T ∪ T ∪ −T
 M„“‚yƒ † ‚\‹d’ ‡n‡ 0‚ F.- -T‚y‰l‚yƒ „\”–‚}ˆ ŽB’ ‡ ™ŒM„“Œ}ˆ I × M.N ‚M„ −I × I × Σ ‚K„|Rœ M.N
”cŒ}ˆ} />ƒ€‚:ˆM>„ M.N ˆy Σ = Σ ˆM>„*-]‚[U ‡ ”>‘ˆ}‰l‚ ”,+ " >‰{.-T0/>™‚ ’ † ”c‚}ˆ[ˆ † ˆn”–‚ Σ
’>ˆ[ˆM MzŒ M.N ˆy Σ = Σ 4
0a’ [>ƒ8ˆy„ K„Rƒ -]‚ F -]‚
‚yˆy„ ‚y‘ƒ „|’ /  ‹8‚{”–‚ ”R eŒUu„™}‹ ™‚}ˆ M(Σ)
[ ‘ƒM”c‡ ‚n’RŽB$‚ ‡y-]† ‚ [ -]‚ 3τ >œ τ bc‚}ˆy„ ‡ -]8‚ ‡[ -]‚J† ”–‚ ‡ •‚† –‚ ‡ 4
X
0
0
0
0
0
M,N
M,N
M,N
M
N
M.N
2
g,1
1
g
g
0
g
−M .N
I × M .N
M
−I × Σ−
N
I × Σ+
[ 38 ; 58 ] × Σ+
M
N
M
I2 × {0} × Σg
−{0} × [ 38 ; 58 ] × Σg
−I2 × {1} × Σg
−I2 × I1 × Σg
V
;<OULUNU=<DMLUP];`9 :[email protected] j R? @BNUPML
MM,N
% S <;>=<O~OU= {L\9 fI8NF;>=8? MNeS fLUE K;>NFžMH 9<@ <[email protected]>O\;
vP 78;5 <Of= š[email protected]]O}@BLUP];#DK;>HJI8Sy7 X 9 TΣ V ;>P–L M ML NΣ9 =M=dΣ7•DKV ; j
XŸ;>NF9dP]OUHKO~9 E<;>H{;>S];>QRPš@B= j 9 KOFOU=<Oe9 Σ V
g ^@BI8NFžKO 7 N ;“r S{DK; D DMS cb @>OFO[;cDMP [email protected]];29 LUP –@BE sb : H → Hb
KOFL~9 y5 [email protected]~S MQ–@BSTPTL g
g
;
g
0
0
sb0 (M )b
s0 (N ) = i(b
c0 (M, N ))b
s0 (M.N )
X
g
g
KOFLSaDK;>X€;>NF9dP]OUH\9 E<;>H{;>S];>QRP~LUNUPT? [email protected] I × Σ H = 8P 9 |S = 8P]ic=
j [email protected]<N[[email protected] STSSMYSTP]QRO[@BNF;>LUP]=8;IC 9 b σj K P!c OULa?>@BP]OFNU;>[email protected]{c L\;>NUNI8M [email protected] LUP @Bx =bM Q–[email protected];>S =8 Icb (M, N ) V H = 8P"9 [email protected]šS];>P
H(Σ) × Z
H
i(b
c0 (M, N ))
SU
g
0
X
g
∀M, N ∈ H(Σ).
(M, m).(N, n) = (M.N, m + n + b
c0 (M, N ))
=8P]OFic= sb KOFLm= •O KDMLUP]; <;>[email protected]]O Rr|PTS KOUL 9 CHMH`IŸ;>=8N cb P V o
W cLUPMN cb (M, N ) KOULmS σj P!c?>@[email protected]@BLUP xš9d=
b
c (M, I) = cb (I, M ) = 0 V
DK;>=8I8S (M.N, α .α ) V
[email protected]<OY= _I8NMHJPMNYLMHJI<OAr<;>=<O @BSTS];<OnDK;<OULUNU=8PTNJ= j [email protected] M 9 [email protected]@ 8P]žMN [email protected] –L *DK; <OUP]9 MNF; <O I × M.N MLJOKO OF;>=<O j [email protected] −I × Σ r
@B!P <OUP"ic= = bDK;>STSTP MN I × Σ 9 {1} × Σ <Dyxv5<QR=8N8 V V
I ×Σ
; <OUP]9 MNF; <O MQ–@BS MH –[email protected] j [email protected] ML −I × I × Σ MLmS KO OF;>=<O j [email protected]
r
ML {0} × [ , ] × Σ 8Dyxt5<QR=8N V V
−I × {0} × Σ I × {1} × Σ
e;>=<O|IŸ;>=8?u; <O\@BS];>NFOaS KO|N KDK;>STS MN~DK;>HJH Y9<@ <O|[email protected]<QR=8N šIŸ;>=8N~;>X8L [email protected]
?j [email protected] ML M V STS @ IŸ;>=8NaX€;>NF9 T ∪ T ∪ −T V V
0
0
0
0
M
0
0
0
N
M,N
+
N
+
M
2
+
M
g
2
2
3 5
8 8
g
1
g
g
M,N
M
N
M.N
−
M
−M.N
I × M.N
M
−I × Σ−
M
N
V [email protected]
I × Σ+
N
[ 83 , 58 ] × Σ+
M
I2 × {0} × Σg,1
I × M.N
−{0} × [ 83 58 ] × Σg,1
−I2 × {1} × Σg,1
−I2 × I1 × Σg,1
V [email protected]
−I2 × I1 × Σg
;>PTL
NKOUI = nLUNUPT? [email protected]]O[@BLUP];#DK;>HJI8Sy7 J9 NKOFI
ic=8P
I8NF;>S];8Q αX NKOUI V V αX ML 98;–[email protected]@>OFOa9<@<O P (M, X)TMNKOFI V PV TN(N, X)
[email protected] SU j [email protected][@BSTSMSTP]O}@BLUP];[email protected]?J9 LUP –@BE α NKOUI V α V€o @BN:NKDK;>STSMHcLAr
MSTS KOw!P <9d=8P]OcLw= \[email protected]]O[@BLUP]; DK;>HJI8Sy7 α .α ic=8P<I8NF;>S];8Q X V [email protected]
OFL[@BX8PTSTP]O[@ –L Q–@B=<DFE [email protected]"[email protected]|9dPTN KDMLUP]; I8N yx MN 9 I r MSTSP!<9d=8PTLv= *[email protected]]O}@BLUP];
DK;>HJI8S y7 Y9 T(I × M.N ) DK; <OUL[@ cL :S :S]; 8Ql9 I V
; <OUP]9 MNF; <[email protected][email protected]]O}@BLUP]; bDK;>HJI8S y7 :OU=8N
DK; <OFL[@ –L :S ZS]; 8Q
9 I × I 98; :[email protected]@ 9dPTN KDMLUP]; bI8N yx MN :9 II rc×[email protected] I 9dPT×N KΣDMLUP]; 1I8N yx MN :9 I r
ML X V
@BNwN KDK;>STS MH –LArd; l;>X8LUP –L= ~LUNUPT? [email protected]]O[@BLUP]; mDK;>HJI8S y7 α OF=8N M i–=8P™r
o
[email protected]DK;<OULUNU=<DMLUP];sr DK; <DMP]9 [email protected]?KD\[email protected]]O[@BLUP];<O|DK;>HJI8Sy7 KOwDK;>NUNKOFI€;<9<@cLKO
OU=8N T r α OU=8N T ML −α .α OU=8N −T V
α
;>P –L Z ML Z 9 M=d7 j [email protected] ML KOfDK;>[email protected]>DML KOZDK; y7 KOZ;>NUP –L KO:9 YXŸ;>NF9
ML T N KOUI KDMLUPT? MH –L V Z;>LF; <O W [email protected] j ?>@BNUP ML •;>X8L c= N KDK;>[email protected] cL
T
S [email protected] ML KO Z S :S]; 8Q09 T ML Z S fS]; 8Ql9 T V S];>NFO
−M
OFPTQ W.
σ (M.N, α .α ) =< p (TW, α .α ), µ > −3
=8P]OFic= O qI8NF;>S]; 8Q rS ;>X<OULUNU=<DMLUP]; I8NF;>S]; 8Q MN α .α
KOFL~oDK;<D –LUNα šOU.α[email protected] Z −M
ML Z ML
M
M
N
N
su
0 M
M
su
0
N
N
M.N
2
2
1
1
g
2
1
M,N
M
M
N
N
M
N
M.N
M
N
M
N
M,N
M
r
M
M
M
M
N
1
N
M
N
W
M,N
M
N
M
M
N
< p1 (TW, αM .αM ), µW >=< p1 (TZM , αM ), µZM > + < p1 (TZN , αM ), µZN > .
g ^@BI8NFžKOmSCLUEK;>NFžMH•9^@>989dPTLUPT?cPTL•9 e;p?cP u;p?€r <;>=<[email protected]?u;<O S MQ–@BSTPTL•OU=8P j
[email protected]–L OUPTQ W = OUPTQ Z + OUPTQ Z + OUPTQ (−M ).
M
N
M,N
−M.N
I × M.N
M
−I × Σ−
M
N
I × Σ+
N
[ 83 , 58 ] × Σ+
M
I2 × {0} × Σg
−{0} × [ 38 , 58 ] × Σg
−I2 × {1} × Σg
−I2 × I1 × Σg,1
V
o @BNeDK;<OKi–=cLAr bc (M, N )
0
;<OULUNU=<DMLUP];`9 :[email protected] j R? @BNUPML
MM,N
= σr (M.N, αM .αN )
OUPTQ 3
= σr (M, αM ) + σr (N, αN ) − 3
=
OUPTQ (−MM,N )
(MM,N ).
2
 † ‡ ™„  † „“‚m„ ‡ ‘ŽAz’.- kˆK’R„z>ƒ [>‰e‹-T‚ ‚ X ”–‚ TΣ H(Σ)
b
‚yˆy„wkˆK>‰{ ‡ ‹"d‚ H(Σ) × Z ‰ † <ƒ  ”–‚-T’ - >q”c‚ / ‡  † ‹8‚
ˆ} />ƒ M )
(M, m).(N, n) = (M.N, m + n + 3
Rœ M ‚M„ N ’y‹u‹d’ ‡ „z‚Mƒ<ƒ€‚yƒG„ H(Σ) ‚M„w>œ m ‚M„ n ˆMRƒG„w”–‚yˆ ‚MƒG„z‚ ‡ ˆ 4
e;>=<O{@A?u; <[email protected];>I8P ML 6OU=8PT?>@ cL S];>NFOFic= 0S DK; D DMS bc KOUL NKOULUNMP!–[email protected]=d7
D dDMST!P <9dN KO\9 E<;>H{;>S];>QRP % 
‚M„
ˆK>ƒG„q”–‚ † .- ‘ƒ ” ‡ ‚}ˆ ,” + " >‰{. -T0 />™‚
(M, i ) (N, i )
’ - ‡ ˆ ˆy /Rƒ M = 0.
% 5 S <;>=<OfOU= {Lf9 šDK
; <OUP]9 MN MNfS [email protected]>O Σ = Σ V ;>P cL (M, i ) ML
9 M=d7 D ST!P <9dN KO 9 E<;>H{;>S];>QRP \OF=8N Z;>=<O <;>=<O O MNU? PTNF; <O 9d=JLUE K;>NFžMH (N, i )
9 [email protected]Ÿ;>[email protected]]DM=8S MN [email protected] [email protected]=8N a9 [email protected] Σj ?>V @BNUP ML M V W |I8NU!P <DMPTI |9 aD MLUL 9 MH{;<OFLUN[@BLUP]; KOFL\[email protected]{OF=8PT?>@cL GSKOe9 [email protected] I × M.N ML I × I × Σ
O[; –L\9 nOUPTQ [email protected]=8N c=8STS V S <;>=<O~N KOUL [email protected]];>NFO [email protected]= MN\S Y9 yx™@B=8L\9 nOUPTQ [email protected]=8N % % X
g,1
M,N
0
±
M
±
N
M,N
±
N
±
M
g
g
M,N
2
1
g,1
D MST=8P DMP KOUL –=8S*[email protected] ML OF;–Lf9 KOZD cSTP!<9dNKOf9 E<;>H{;>S];>QRPMLZ98;<D @BQRP]OFOcL
LUNUPT? [email protected] SMH–LZOU=8N H M(Σ ;NZ) V
C; cLUNF; <[email protected][email protected]=mSm9 yx™@B=8L{9 6OUPTQ [email protected]=8N KOULc=8St=8LUPTSTP]OF;<OJSKO
<;>L[@BLUP]; <Oe9 :S ^@BNULUP]DMS 7 [email protected] ; ML\I€;ROF;<O
ML Y = I × M.N.
Y = −I × I × Σ
e;>LF; <O
1
g,1
−
2
1
+
g
e;>LF;<O
Ia = −I2 × {0}
3 5
Ib = {0} × [ , ]
8 8
Ic = I2 × {1}
3
Id = {0} × [0, ]
8
5
Ie = {0} × [ , 1]
8
Ic = −{1} × I1
S CSTPM=9^@BLUL[@>DFEMHcL19 I × M.N @K?KD −I × I × Σ ML
<;>LMN[@[email protected]];>NFO X := ∂(−I × I × Σ ) \ P!–L (X ) ML X =
Z = ∂X V
P!cL
=8LUNMHcLe9dPTL
∂(I × M.N ) \ (X ) V
X0
2
−
0
2
1
1
0
g
g
+
0
X0 = I a × Σ g ∪ I b × Σ g ∪ I c × Σ g
+
+
= Ia × Σ−
M ∪ Ib × ΣM ∪ Ic × ΣN
Z = ∂Ia × Σg ∪ ∂Ib × Σg ∪ ∂Ic × Σg
X− = I d × Σ g ∪ I e × Σ g ∪ I f × Σ g
>; =8N
ML
r <;>LF;<O
QMo N[@BLM=8NFOZ=9 0,H1 ({ α} =× Σa, b,; Z)c V S];>NFO
X+ = −{0} × M.N ∪ {1} × M ∪ {1} × N.
1
α
H1 (Z; Z) =<
γiα , i = 1, . . . , 2g,
= O dOULFžMH 9 g
γiα , ∈ {0, 1}, α ∈ {a, b, c}, i ∈ {1, . . . , 2g} >
A = Ker(H1 (Z; Z) → H1 (X0 ; Z))
= < γi1α − γi0α , α ∈ {a, b, c}, i ∈ {1, . . . , 2g} >
B = Ker(H1 (Z; Z) → H1 (X− ; Z))
= < γi0b − γi1a , γi0c − γi1b , γi0a − γi1c , i ∈ {1, . . . , 2g} >
C = Ker(H1 (Z; Z) → H1 (X+ ; Z))
1b
1a
0c
− −1 +
−1 +
= < γi0b − [(i−
M )∗ (iM )∗ ]γi , γi − [(iN )∗ (iN )∗ ]γi ,
1c
−1 +
γi0a − [(i−
M.N )∗ (iM.N )∗ ]γi , i ∈ {1, . . . , 2g} >
o =8P]OFic= M ML N OF;cL~9 KO\D STP!<9dNKO~9 E<;>H{;>S];>QRPRr C = B ML
V =
P!<OUPŸ[email protected] OUPTQ [email protected]=8N:9 A ∩ (B + C)
= {0}.
(A ∩ B) + (A ∩ BC)
MM,N
r8i–=8PKOUL"MQ–@BS [email protected] OUPTQ [email protected]=8N:9 V rKOUL c =8STS V
2
! #"$%'&(! )
+*,!-.% /0- ) /123 4%#-(!5% /6; <OUP]9 MNF; <O I8N KO –LfS QRNF;>=8I W X8=8Lf9 šD šD[[email protected] KOULZ9 nI8NF;
S];8QMN SšDK; D dDMSJ9 P!–LMNFOKDMLUP];@B=d7•DKH;>XŸ;>NFV 9dP]OUHKOY9 E<;>H{;>S];>QRP V W P]9 KOUL:9 j
O !P <OUI8PTNMNC9 [email protected] DK;<OULUNU=<DMLUP]; [email protected] @B= [email protected] ;>=8N`DK;<[email protected]
[email protected][@BSTS MSTP]O}@BLUP];[email protected]?_9 `;>NUPTL[@ 9 KO [email protected]!8QuO6DV~cSo TP!<9dNKOAr[email protected](OULUN[@BLMQRP SUKOULj
9 =8LUPTSTP]O MNArDK;>HJH I8N MHJP MN{? KDML M=8N{9d= [email protected] 9 mN MIŸžMN KOAr*S [email protected] 9 m?KD
L M=8NFO I 98; [email protected]@:9dPTN KDMLUP]; mI8N yx MN e9 I I8=8P]Ow9 MLU=<9dP MNwS ^@>DMLUP]; 9d={QRNF;>=8I j
9 :9dnP mK;>LF;>I8P KO M OF=8N KDML (Σ ) V
[email protected] <O S •D[[email protected] _I8N KD K9 –LA"r <;>=<[email protected]?u; <Ob9 y5 8P:[email protected] <;>LUP]; 9 •[email protected] ? KDML M=8NFO\L MHJI<O~OU=8N|S KO~DK;>XŸ;>NF9dP]OUH KO~9 E<;>H{;>S];>QRP V [email protected] MS]; <O~i–= fLF;>=8L~DK;>XŸ;>N j
9dP]OFH e9 E<;>H{;>S];>QRP :OU=8N @>9dH MLa9 KO|[email protected]<O|9 e? KDML M=8NFO|L MHJI<O[email protected]]Oic= PTS]O
:O[; –[email protected]>Oa= 8P]ic= KO E<Σ;>H{;>LF;>I8P :N [email protected]? @B=6XŸ;>NF9 I8NFžKO V W fI8NF;>X8S]žMH KOFL~i–= <;>=<O O[@A?u; <O:[email protected]>O:9 ML MNUHJ!P MN:= #[email protected]#9 š? KDML M=8NFO:L MHJI<OfI8N yx MN JOU=8NfS KO
DK;>XŸ;>NF9dP]OUH KO:9 E<;>H{;>S];>QRP 8P*H MH [email protected][@>DML MNUP]O MN S šD[[email protected]#9 š? KDML M=8NFO:L MHJI<O
OU=8N\S [email protected]!P 8Q6D cST!P <9dN KOZO[@ <Oe=8LUPTSTP]O [email protected]=<DMLU=8N nI8NF;c9d=8PTLfOF;>=<O j [email protected]>D –L š9 I
W P]9 KOUL|98;<D:9 fDFEMNFDFEMN I8NF;>S];8QMNaSfDK; D DMSf9 P!cLMNFOKDMLUP]; I ×Σ V
L [email protected] cL DK;>HJI8L J9 D ML:!P <9 ML MNUHJ!P 8P]OUH Vso ;>=8NYD [email protected] <;>=<O @BSTS]; <Of=8LUPTSTP]O MN:[email protected] <; j
LUP]; 9 ZOULUNU=<DMLU=8N KO|9 =8S MNwN [email protected]? KOpr D ei–=8P€N M? P <9dN[@ DK; <OUP]9 MN MNS KOwD[[email protected]<O
9 f? KDML M=8NFOeL MHJI<O E<;>H{;>S];>QRP YI8NFžKO V
[email protected] <O = I8N MHJP MN6L MHJI<OAr <;>=<O69 y5 8PTNF; <[email protected] <;>LUP]; 9 •OULUNU=<DMLU=8N _9 =
SMN|[email protected]?:9 = 6DK;>XŸ;>NF9dP]OUH:9 E<;>H{;>S];>QRP:OU=8N Σ V <OU=8PTLRr<;>=<[email protected]>OFOF; DMPMNF;<Oj
S <O MHnX8S J9 KOfDK;>X€;>NF9dP]OUH KO:9 E<;>H{;>S];>QRP {H = 8P9 = JOULUNU=<DMLU=8N 9 =8S MN:= QRNF;>=8I i–= qS ; <;>L MN[@ H V e;>=<Ov? MNUNF; <O*i–= D QRNF;>=8I [email protected]=8N H (Σ ; Z) V
Z;>=<O\DK; <OFLUNU=8PTNF; <[email protected]];>NFO\= 1E<;>H{;>H{;>NUI8E8P]OUH DMNF;>P]O k̂ : H → H (Σ ; Z)
IŸ;>=8N:LF;>[email protected](9 n? KDML M=8NFO X 9 š[email protected] OU=8N x™@>D Σ ŸD MST=8P j DMP I8NF;>S]; 8Q MN[@ [email protected] j
LU=8N MSTS MH cL6S E<;>H{;>H{;>NUI8E8P]OUH #DMNF;>P]O k 9 y5 [email protected][email protected] V d5 sr <;>=<O
g,1
g,1
g,1
g,1
g,1
g,1
E
g,1
1
X
g,1
X
E
g,1
1
g,1
g,1
9 y58PTNF;<O = DK;cD[[email protected] 9 i–=8PI8NF;>S];8QMN[@ [email protected]=8NMSTSMH–L
S DK;cD dDMSY9 P!–LMNFj OKDMLUP]; c ĉ9 [email protected]=1([email protected];PTLUZ)N V Z;>=<OaH{;–LUNMNF;<O:@BS];>NFOei–=
D MLUL j DK;cD[[email protected] 6x™@BPTLa= j DK; D DMS V
X
E
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2
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9.a65,7;4,7OQ 9<;4,7V> .BX
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9<3 >CE;^QO %? J_`8M.9 /1 8TX X A9CED9a1>CESCE;?T8M.<O %? 3/CEOUCE0MCEVW8 %.<D;
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