This document discusses vector bundles and characteristic classes. It provides examples of the Euler, Stiefel-Whitney, and Chern classes and their properties. It also discusses the relationship between a non-zero section and the Euler class, stating that if a vector bundle admits a non-zero section, then its Euler class is zero.

1. 2018/4/14
*d¥*±h¥±kte & #X## EPEE sttt
'
EEE' .
slit -
4¥ HE ¥A a K¥+5
@ TakeTo 1024

2. Contents
0 .
t
'
7th # ksuc .
1 .
4¥ ' HtTHE a hat # at #¥x .
2 . Euler ¥ta a A¥# Aiz 'l¥s4t¥a .
3 .
PEEMEIE'EF# rmz .
4 .
4¥ HE #a a
' '
IHI
' '
.
* =a7¥Eirt Ih¥±± F)
"
¥eF±t5
"
2 h#¥El£F .
5*h{ KIBE,
tkukrt # Fta 1- t.EE#TEeIFFET In .

3. O .
t
'
7th # knuz

4. i 7th # sit
¥F⇒sb'ts3#¥h#
a # ¥n±kt9kh# Eat A "¥n2n3ta .
531 M =
s2 I a f¥tt9HlFh E = TM .
(Tangent bundle )
TM
~
# '
* Beta .
I>
y
=
¥µTpM :# '
LIMIT
p 1122 : 7 } hli
'
-
4pm
YY
: EIIERIF
( pkaF¥isHvtEp¥l )
←
n7HH¥aaF#±⇒
'
(
ninth '¥ X e * an a × :
t.Y.IT#ptas
# HE .
)

5. "
Etants
"
n' 9th #
E¥4¥a '
EMIT E = Mx
Rr
it €53hMknt9Hv#n .
= th 2 As #± A t
'
9th # z Eta # At 9th
'
# said .
^3Ni'
.
4¥ , , A¥# Egg ;qµ*z .
' trivial Vector bundle )
By M =
s
'
In ELKE # .
FZTHTEAKH F¥e*U×R' at 's .
FT IT # RA '
# rtn9HlFkath¥kE£h3 .
RR
-
R
d
s
' -
•
•
•
•
•
€1- - -
4) E = S
'
x IR 2) E :
open
Mobius band
C trivial ) ( non . Trivial )

6. 2) to Etna Itau tea IEIAA .
i ) El E = s
'
x IR tiztnktt
' '
.
3.
'
E. Ii
'
n' 7Hubt -
ha ( to ) { 53
'
'Ei7Hv±¥s to 2123 ttf
lb 't FEE # a z #
2324M¥
the a
'k#¥ b 's HF
"
0 { he ,2lF 's .
qp¥.
" rahman
• n' 9th # A 3F÷kh# e i7HvBE¥n
"
tslikk TELEFE
"
.

7. 95k¥ HE M a
"
#'s
"
a .
Zn F¥ Fk K Ls ERk2a3 .
4¥ K TM A
'
LFAA Azi 's b'
IF ,
M 2 'E#i3 It E¥#a9 .
931 .
s
'
a F¥ # a # a A .
*
•
•
¥ "
Fit 'U
•
~
•
=
•
•
•
•
- - -
TS
'
5×121
'

8. 931 .
5 a t¥ # T5 is Etna itsu .
→ →
tl #aAts5 ,
5 Ii non -
zero then 7khE¥ b"
h¥k3 rtd .
→ Is
→
I→I ( b 'l
'
# 1£ zfnbt i Etsu ! ( taei 'EEAAF3 )
→
€ a 4- It fi -
N KH AT
'
7ft (£EHBHA=) b "
ZEZLF 5 .
a # in 'a#⇒
-15¥ 5×1122

9. Kd .
K - 52 T
'
a ¥¥ # It Etaa .
T2 = s
'
× s
'
,
TT2= Ts
'
×Ts
'
= -12×(112×112) F) ,
¥,
¥
% "
€a4=L⇐k .
574£ EIEHZFFHL 5123 .
T -12=5×1122

10. ( £4 e :X na £¥X )
-
n' 9th Eh it 3F¥eh# ± a n' 7th ±¥s * HYEXBFKEl
'
{ FR 's than ¥¥¥ .
- ni 9th # on # AA i tahr he ¥Eti¥ K FR 23 .
-
A- a A 1 ¥ F Eta A I n 9th # 2 EB 't It " .
A #¥ His Hv In a H hit
"
Al
"
he
"
{
'
¥ a 3 ft hi Makin .
it 'E±¥ )
¥ t Et a A Tt a 7th # it suzrt . EnKEEFE FE ( Bikaata ) t ÷s¥t# tkn .

11. 1 .
4¥44 * A ah #¥ as tE±¥
"
HE # A f
"
Tsai
'
Fti tEF¥5 It
43$33 s ( II ( 2 k£ £ .

12. HE # East .
3. THE # M tansth # t : E ¥ M KKHZ .
Mn ITHEIII -
¥¥ a "¥tEi CCE ) e HIM ; @ ) { t.tk#2FhF5az
"
.
Az#± to A7Hv # lift hit Ali h#Lz It 'Fi+t3ta .
input output
E
| µ
B
"
) W
°
DE ) e HIM ; %)
n'
'
7th # .
IKEA Ii Fs¥a¥t¥a .

13. ( "FEFE ) KERI - .
ITHEDI -
rmz
FYFE # M a Th Eat
'
-
¥¥ elf ,
Ma IKE K
£H3A#
Eta th 'm { 5-23 A' ¥2 at # FE .
BI .
Hals
'
) =
|
.
( k=° ) HKCT ) =
|
•
( k=o )
0 ( k = 1 ) g
( k=l )
.
( k = 2) ;
( k= 2)
,
-
- -
-
,
•
it.
52 -12

14. ( "FEFE ) Thea I - .
IF # a I -
rmz
FYFE # a ] THE DI
'
-
¥¥ e it ,
M a TMNK Hk hT# z
t.tk#3FE7*tixti5Ai3ht*ExAS2FFx
.
HE ' '
HE a
iktt Hit a Ifk k£3 bags a :c :
/
IK , <
sd .
this) =
|
. →
2 h=o )
( ±*YYY*Yn⇒¥IFIt¥¥I*F¥*n#⇒
0 ( k = 1 )
.
→ 2 ) ( k = 2)
] # EDI -
#¥ k If
Cup
45¥ K JR I'¥a tt3 Eh ,
t
At2 K
Then I -
¥4 zy t±# bits F¥a4⇐ z to .

15. 4¥ # ¥57 sit ( # HE )
Fy THE # M Ian "9th # E httlz .
Mn ITHEDI -
¥¥ a "¥tEi CCE ) e HEM ; @ ) { t.tk#2FhF5az
"
.
Az#± to A7Hv # lift hit At ht# z It 'Fi±t3€a .
input output
f
Ka'
tkn
E
( µ
"3
"
) W
°
DE ) e HIM ; @ )
t.IE#iazY
( Helen )
i 7th # .
ITHERI
'
-
¥¥a¥t¥a .

16. 1) Euler #a
FYFE # Mta A±HH5hEi7Hv # t : E ¥ M lift .
E a Euler ¥57 It ,
ECE ) e
HTM
; 7L )
zi .
4nF {
IEAEFEN
.
D e ( ft E) =
fte (E) I AYE 't )
2 e ( EDF ) =
ec E) .
e (F )
3 E : A- a A ⇒ EC E) =
0
4 E : E ATEGTE ⇒ ec E) = -
ECE ) .

17. 2)
Stiefel -
Whitney #A
FYFE # Mta i7Hv # t : E ¥ M KFH ,
E a Stiefel -
Whitney # A { Witt
)3F=
.
H .
WICE ) E H
:(
M ; 22 ) ( i. 0,1 ,
...
,r
)
zi .
4nF { is
'¥kF€n.
00 Wo( E) =
1 .
1
Wic# E) =ftw:( E) l⇐7*H± )
2 W ( EDF )
=W(E) -
WCF) ( w=1 + wit . - .
+ Was )
3
W
,(8 ,
"
) = L ( Le HYRPIZ. ) =2z
,
4=AiE )

18. 3)
Chern #I
FYFE # Mta
kFE¥ai7Hv
# t : E #M kttl .
E a Chern ¥52 {
Cite
)
}F=°
H .
CICE
) E H
"
( M ; 2 ) ( i. 0,1 ,
...
,r
)
zi .
4nF {
IEAEFEN.
@ Co (E) = 1 .
D
Cicft E) =ftc:(E) IEMIIHE)
2
c(EDF ) =
CCE) .
CCF) ( c=nta+ ... +
cn )
3
c.( 8 ,
'
) =
d ( de H4 # 2) a 2 ,
EIAIE )

19. l 7¥¥uEgttkR¥ .
)
75k¥ # MKIH . Mta nsth # a
R#±¥k¥¥¥
{
tfk±t2B¥I
:
Vectt ) : Mtd →
Sets
a a
M i→ Vea (M ) =
{ E → M }/±
4¥ '
Kt ¥2 et, =aF¥t 't $5 IAEA 's
"
.
P¥f4 na FFYEI '¥## a :{ :
µf+E
E
a
f*
a
Vectcm ) Vect CN )
f : M → N tr c c
3- A¥t#E¥n¥h÷k
f*
HTCMIA ) HTCN ;A )
6
c # E) =
ftctt )
"
CCE )
( ETYIIHI )

20. 2 .
Euler #E a t¥¥Ai z
'
pasta .
* .
) ] I It Bbl 55kt ¥EE¥ Hani Fta
.

21. Euler #aa#I¥Ai .
Rr
FFEHHSKK A 9th # TL :
E M K # L ,
Eo :=
E -
Mx { o }
SEE
,
Mta # Eva
"
a 7tN±'¥
"
s : M → E Ezy ,
-
Annett -47*4 ( section ) 2n3 .
EaThom*a :
TC
E)
TIE
)
A i* 9
HTE ,Eo ) > HRCE )
= Thom #¥
' I s*
HOCM ) -
HTM )e a
1 ECE ) fruits .
K -2,2 E a Euler #a el E) e HIM ;2 ) { 'E¥xF3 .

22. the# 1 E bl
"
non -
zero section Etosit ,
ECE ) = 0 .
4¥ 's E A
"
Et a A Fsstt
'
ECE ) =
0 .
i
'
) s : M → Eo E non -
zero section
{(2
.
e (E) a #¥aAikikZAn32 . ( E. Eo ) a
cohomology ex ,
seq
.
LT ,
des HE ' °
)A A A A
HTEEO ) HTE ) HMEO )
A
s* s*
HIM )
a
ECE ) =
0
B'

23. # I # 2 77 FE # Ma Euler #x { XCM) 5h32 .
←
Ma¥¥¥a
(
ECTM
) ,
[ M ] > =
XCM ) (
fuetlu
) =
XCM ) )
^Mn¥*
# a Euler #A
MnEler¥I7 EEEAA RB )
97 F¥h# M a Euler #x XCM ) H ,
XCM ) =
Io c- isi rank CHICM ) )
i 'a¥x±k3¥¥#k .
4¥ K M A
"
= # FIB '¥'t HE F¥ # At a ±n¥h± .
(M ) = V -
E + F .
X
aa¥nh##* ) Katana ) M#ah#¥D

24. tz .
KTFIHF ( a # At ) if 8 a I # FK Bas't s 5-23 .
•
XC s
'
) =
4 -
6+4 =
2 .
•
•
•
He I # 2 gy ,
eCTs2 ) =
0 s if 1274¥ An .
tkbtr theft 1 sy Ts2 If FIAA itsu
!
-
to ,
XCT ) =
0 g) ,
t*hL ECTT
'
) =
0 { ta2n3 .
( It ) iaiz $15 TTZ of EFAA iTa3
'
- st kFBt2u .
)
ni 7th # a Et AAHI a TXFTEH't HE a- a A # at ,A¥h4 ) 2£23
AE'¥E A H¥skt= !

25. ( Fun { Hina Eatx )
-
7¥ 'HE ¥k ,
E. a ! EID
# In it to { ÷zGH=n izt '
teeth h3 . . . ( Ftxk Ethic ,zH2 ! )
-
cxl
A¥¥ Ai I ÷*d ... ( SW #a .
Chern#aaA¥# Air Er Hit )
he
"
k¥57 a chtttn { R > It 5k£ ?

26. HEE )

27. 3. PEE '¥FE¥÷t¥ i= suz

28. per
¥ Et night # IT : E → M A
"
# AA b 'i5b '
# 2312 ;
non -
zero section s : M → Eo I zkzdlz # HE .
#
E
Rr i
:
-
Mx{o } : 11h ftp.lntznjk -23 .
it
?
If :
S
µ
EH
#aAt£5€53h
s It FTFSFZ ( A=aAita3t=aaXT¥FHt )

29. # EE As,
r=2 I M HE# FIFA'¥'t
Ikzeisza
.
sz ¥¥th#lz2£ 's .
( o )
Fai Ma Jake. As a # 95k M → Eo zhF3 .
;•,⇐#
EPi
, 1 1
'
-
Mx{ 0 }
"Ytiltii'
'
l '
.
' ' ' '
,
1
.
1
i
, ,
i t
i
, I I
1
itufAsI
ML,
'
,
1 1 1
1
1
1
•
' 1
*.
:-/,
=kE #a5¥u Its 's a # both M
"
'→Eo
{ 94-3 .
t.sk the #a3⇐acz A±A5n. - .

30. # baa,
#a5Eu it YBK Faa Eb ' ?
t#•
⇐
•
s
( )
¥¥#
" " " }
•
OK
of )
" "
a
FX
.
Fy.
• •
IFFFH, a TIEKF * Mx { o } e # Litt 's e .
¥42 it IHNI't KFFAFE it An !

31. # EE { it
'
Ma # IFAFI's little ,
EE'Re bi Mx{o } a As DE
FI # F3 A { EAF P¥f#£ c : cz ( µ ) →
2 2¥23 .
i.
⇐ . as
E
• • 2
T- • 1
¥14×{0 } . °
• - 1
sf
• :
•
PH. a•
c = o 135 it
"
Ehn FEEEHFEAE !
at 0 not ,
Eitik s z FEBEAFZ PEETE ( obstruction ) 4
'
'ta3 .

32. # EE in F '¥#x c : GCM ) → 2 IF 2- cochair 2ta3 .
±5k
Sc(D3
) =
cc
D8
) =
0 ztay ,
2-
cocyde 2ta3 .
i.
€ . as
E
• • 2
T- • 1
;t.to::
•
< •
YT#ML•
Letsie [ c ] EHYM ;2 ) an 's cohomology '¥kbtek3 .
c it s NEK '
) 'FkhFc3bt , [ c ] rshfustan ! ( EaAbI¥± )

33. -
#k¥F# the Hit 5kt a 9th # a : E # M re HL ,
E n non -
zero section s : M → Eo It
M a v. 1
t.hn#a=tiaEeFhh4i=FFtathtiiI3
.
stir
#aFE
£iF¥aAi
it 3M£ a¥s 1 PEE '¥¥k
[ c ] e HRCM ; 2)
k£2 K£3 .
> FT ,
Is : Mcr ,
→ Eo [ c ] =
0
T c=o tss { a££FEBEaiE3 .
A
'
ht
'
'
) 'I > ! [ c ] =o 175 ,
-
¥4Ey£iE3izz
'
( r¥#a£ifEiEiE3 ! cost # au )

34. 4 .
4¥ 't'¥¥B a
"
I #
"

35. Euler #A a Ih#
HE ## A -±hfH5hk n' 7th # t : E
*
M KJHL .
E a Euler ¥72
ECE ) e HRCM ; 2)
IF ,
E a non .
zero section E hF3Eh a
* np¥a'¥¥A e -
Exa3 !
CEIIAAIIE )
= KK £ , 2 Euler ¥52 { hE¥x at En .
f3{ . ( E :#a A ⇒ ECE ) .
.
0 ] if ha¥xb 's AA
5101473.

36. EL F¥k
r=dimM
at¥h¥ .
s : M → Eo e (E) =0
'
Ee !
.
:) Ma ✓ Manta it M { atafib 's . B
[ TI En # kktitvkt Xtistutsti 'E3 .
: ) rank ( TM ) = dim M F) ,
In # HA
'
As '
I 'Is .
XCS
'
) = 2 F) easy to Ita 's ,
5 I KIF non -
zero vector field ttzktsu . DK

37. Stiefel .
Whitney #ZaIh#
a-
EL
FEEE ninth # E * M ⇐ HL ,
f , T
•
. - -
→
Ea k -
FFKYE VKCE ) E it E In k ] a
!
12k$ '&'IHA9kka¥E $ 's # 373mi
'
.
# .
I
i M
p
E a
kikStiefel -
Whitney ¥57,
WRCE ) e HKCM ; 22 )
If ,
Vr.
kt , (E) a MK 't a section ehF3Eh a
* 1p¥IH÷**k ( mod 2) s -
ZQFZ .

38. % WICE ) = 0 c⇒ E it A IHTHFIFAE
.
:) W , (E) =
0 # M
' ' '
± zi Vr .
it , (E) = VRCE ) a
section lot 2123 .
c ⇒ M
" '
s F¥aAih3E±a sink -
7012€ ,
A⇐4eEs⇒. i. ¥hEI3 Ea ¥± # A ek3 .
←
AEHAFEKXZHE's .
( ⇒ Mta KEE. ah .
s°±i .
ftp.g#$ygy,y*⇐&*
⇒ " ¥453 Ea¥± # signs . #

39. Chern #2aIh*
4ft # KEEFE ninth # E
Is
M EHL ,
E a kBa¥,
k -
HER VKCE ) at ,
E In k ] a
1 -
12km , 'ItaA9Hua4E a 's Ai 3 73mi
'
-
# .
E a kihChan ¥57,
CKCE) e
HYM ; 2)
It ,
Vr
.
kt , (E) a MMI a section ktthf
* 1 EE '¥*k e -
2ixF3 .

40. ( £4 c ¥iF¥5 )
-
4¥ HE # a it n' 7th # a Btk¥E .
> Fy
' '
EFAAHEFZEEVH
'
'
{ tttttu ⇒ ,
Ali 's { FEI ,
-43 FED #x state
!
-
ZEZE 3 Th Eat
'
-
EE
Ka
kER¥7 At A 31 , by "
) E #E4IF3
F¥f**k # took hEhia3 Fkttai ,
{ it 4¥ 'tE#I ol
'
ta5hn3 at 22€ AEK .
-
n' 7th 5ha Bnikb
"
It #sFaA a ] # Eaii -
EF a 4h Task R3 aii
FIFA tabspart at kit ¥-445 A 9th # c b '
Ft # Liliana €45613 .
-
# in 114+714 Blitz " 3 AIFAF a Iitt .
"
h543 faith
"
to "¥nERit3 .
→
6^(18 ) :
F¥PQzkE Grossmann 957.3kt# ,
A 7th Tan B¥A # Part

41. ¥,¥¥e
Characteristic Classes
Milnor, Stasheff
微分位相幾何学
田村一郎
Lecture Notes in Algebraic Topology
Davis, Kirk
available online!

42. Thanked
@ Take To 9024 .