Connection :
G. + ✓ →
V represeut .
of a we
group .
↳ +
V* V
-
(g. b) f) =
Ilg
I
) FGEG ,
KIEW
HUE ✓ .
→
gxv
"
- v .
(x. d) ( r) = Jku
) .
-
IIBUNDESWEshall work in the Smooth
category ,
i e
.
we will
6ns: der Smooth fiber bundles ,
vector unedles and Principe bundles .
2.1Fiberbundlesdef.INSupport is a smooth mtd .
A (smooth) fiber bundle with standard fdsw F is a
Smooth uvp p
: E - M between Smooth mfds .
E and M
s - t .
for any
× e- M 7 an
open neighbhd . U of x in M and a
diffeouu .
¢ :p
-
'
Lu ) → Ux F s . t
(*)
P
"
( U ) ) Ux F
P
↳Lpr,
Gwen tes .
e -
•
E is called the total Space and M the hate
of p : E → M .
•
¢ is called a fieser bundle Chart or load trivial ize treu
of
P : E → M .
• for any × e- M
, Ex :
=p
"
k ) is called the
Reinen lf WEM is
open ,
den p
Yu)
= : E
µ U
tweet "
.
is a
the Knolle wihshaderdf.hu F over U .
Def.2.2-supp.ae p
: E - M is other bundle .
• A (Smooth ) oectieu of p is a Smooth uop s : M → E
Ey
Sir.
p! ,
sk) E Ex .
¥|• A Loud ( suwote) reden o
) P
-
µ
is a sedan of Elw for open zehrt UEM .
×
Y
p
We wie TLE ) resp .
TLEIU ) for hue set
of seetiers
resp . Loud zechieus an U
of p .
ETM .
Due to (* ) ,
a fiber bundle has many
Loud seitens
UXF
( by K ) they done sp .
houeops U → F x # K Ak) ) seciee +
Em u
f- : U →
F Sunde
- s
but global sectreus
night U ,
E- IR
hat * ist .
Er
Melkt
Exitteuce
of Loud sauren , ( er 4) ) im plies that
MT DREHER"
"zu
→
p : E - M is a
surjediwe Subversion .
=) Fox e- M
,
E. =p-1k) e- E Smooth SabuJet .
ef E
and it is diffeouu .
to F .
Det
A morphin
( resp .
isomorph in) between two the bundle
p
: E - M and E :
E → NT is a Smooth uvp ( resp .
adiffeau .
)
f- : E →
E tweet von fires to the ( ft) c
EI her
Jene FEAT
i. e .
7 a
man f- : M - Ä St
'
E
der .ae
)
f w
E -
E
"
ty ¥ ¥5
bauen Les .
=
By the universal propwtyofsurj.subwesieus , f- is automat
iudlgSmooth .
( If f is an
isauarpu .
of the bundles
,
tun f- 1
is too ) .
Erahnt
① M and F mtds .
prs : MXF → M trivial
f-ihn handle over M wir skoudod f.her F .
A fieser bundle is called trivial zahle ( or trivial) , if
it is is and place to M × FAM .
Wehe that H) sag , that any
the bundle is
locallytniidizable .
② Vector bundles (see Global Analysis )
Vector bundles ore Jeher bundles p
: V → M witn shaolerel
Juhu a
never space
'
Ne IR
"
s . h .
for any
× EM
4 is avedar Space
and there existtoud trivialitaeten on a
neigen mhd .
U of ×
,
⑦ :p-40) → Ux IV s .
L .
→
011 :
Vg 544 × N = N
Y
is linear
isomorph im .
tty EU -
Er ⑨ Tangente handle TM → M
of a mhd .
M .
~ -9ns
⑤ Any Lauer bundle TMQ - QTM
⑦ TMQ . .
⑦ Tre → µ
→
IT
M
- M
We how see that ⑥ auch ③ how
always many global
seclieus . Seetiers
of ⑥ are never Leeds and of ④
(9) Jensens
resp.
K farms
.
If E is a hie
group ,
we home seenland TEIG
"
otrivializeble (
by left and
right in verrat neuer Held> • G) .
TG → Gx g. ( ging) '→ los , Tgsfig
)
P ↳ GL Prs ↳Lg)
← ( y ,
x )
③ Sur rose
G is a hie
group
and H E G- a Closed
kusgnap .
Then GIH i ) o smooth mfd und p
:
§ →
GIH
•
surjeaivezibmersiee.by Then .
^ - 42 -
*,
n , +,
expkih
In the proof we uaestnched a at Hear .
F :
Wxlt → U
E t
g
=
G ⑦ k oha neige .
Genuege .
a) Oek
oftpoexp × ich
sie u
Welt → plo) - H
c is
odifteou .
↳
u
% ( plo) EGIH
P)
is
open )
=) p
:
G- →
% is a
Lbv handle wie skandal 2W H .
=
Suppen p : E
IM fiber bundle and kt
da :p
"
( Ua ) → ↳ + F and % :p-1L ! ) UßXF betwo
Charts over
open zebshs Ua .
Us EM 9. L .
Was : =
Was ↳ Ff .
Transition uiop / vorlaege is
givae of tue tou :
% °
%
"
.
Was x F ↳x
F fosaly , _
) : EF
is edi )Kan
↳ it ) - ly ,
ßaly, f ) ) for
Smooth
mops 01¥ :
Um × F → F s .
t . for my yt Ua)
Im Msa ( y ,
_
)
Was → Biff (F) .
Local triviolizotreusg.ve
rise to tue netten other bundle
athos A- =
{ ( Ua , a ) : ×
EIG hier p :
ET M :
•
.
[
=
! a 01 ,
:p
"
( Ua ) → ↳ x F
I d
↳
Und Inwdizatieu .
Prop.IN Suppose E is a Set
,
Monat Fore Snooker
infos and
p
: E. → M a
more ( hetweeu Sds ) .
Assuue there is
• I } open
Law
of M
• for any a EI
,
a bijectieu da :
p
"
( Ua ) →
↳ F
Str .
p
)
=
pr,
o
a
JUL Prs
Flux)
• for a
, es c- I with Uaj=
Was Us # ¢ the tnensitrawop
is
a) huhdl
.mg/Oao/0j1:UapxF→
↳ × F farm
i ( y ,
f) klug, daslat))
for Smooth mops % :
Was ×
F → F .
EM
Then E war he uuiquely werde Ihnen o smooth mfd.sk .
p :
E → M is a the bundle with the bundle atlo) A-=
{ ( Ua ) :<
EY.
Proof ( et . Global Analysis ,
when we used this for veaar
hundes) .
•
Without has
of geuwoiky we mag ossuue { ( Y.
. 4) ist ]}
i ) o neun tobte also, for M and that there one bijedons
Hj :
p
-
1
( Vj ) →
Vj × F KJE ) wh. n one tendiere ,
of tue
gian %
'
ß ,
x EI -
•
Equipp E with the
holloway toplogy :
the subsels UEE s.tn Ol;
( Un p-1W;
)) is
J 2
•
neu in
Vj,
x F
VIE#]
olefine u topdogy an
E .
If VEM is
open ,
then ↳
Vj is open in Vg Vj E I
and so
p
-
'
(V) is open in E
,
i.e . p , E. → M i ) Continuous-
Fu
any WE F open , 0151 (Vj × W ) EE is open VIE ] .
( check this ) .
• The hopology an E is
Hausdorff ,
zince pauls indifferent
fies war he seperded by Hen Sunset s
of M and point)
in the some hier neu he sereohed hy gaukelnd of F .
Since M and F one Seeed Cannhoble ,
so is E .
•
Fix { (Wu , wa) : ke k } an alles Leer F .
Then the sets
01514; × W
.
) for DEI ,
KEK
form an
open come
of E and
G- x w
. obj :
51W;) → vjlvj ) xw.tw )
is a how eonearphizn anno the
open gebaut vjlvj ) xwr.tw)
C- IRA
im IM) tot im (F)
One Keri fies directly theol transition
maps of here mops
one smooth .
So , they nehme a Smooth uteos an E
wir vdues in 112ohm (MIT dein (F)
.
In here warts
, p
Comebacks to Prs and Lance
p
: E-
M
is Smooth :
Moe one
,
A- =
{ lvj , 01;
) : je ]} is a the buchte
etwas treu p : ET M .
D .
2.2 Bundles with Structure group
Def.2.5-supr.ae
p
: E- M is a lauerte ) fiber bundle
with Shared end Gw F Lev
Gte a hie
group octing
suootuly (trau tue Left ) an F .
① A E ottos
fer p : E - M (or a reauctieee of structure
group Ko G of p
: ETM
Corvey.
GXF→
F)
is a fieser bundle athos A- { Na
, da) : a- I} her p: E> M
sie .
for (E. da ) , Yo,
ots ) C- lt the transition uop
is
of the form . Gratien
% -
%
"
. Was XF → Ua,
×
F
"
F
4. f) i-ly.de/--/y,Ydy).tf)
-
-
for Smooth Ichs Y . .
:
Usa→ G .
(W a fiber handle
② A fiber handle wie structure
grau,
G is a
" " G -
Strecke
)
f iser handle p
: E → M ↳
gehen wir a luoxiweoh
( or
eguivoeeuce) G- athos for seine E action
au its
Standard her .
Without has
of generality ,
we
mag
reshict hohe lose
w here Gx F - F is effective ,
i. e
.
G- → DifflF)
was trivial Kernel .
If the action is und effea.ve ,
then the Kernel K
of E- →
Biff(F)
is a Closed normal
sungnap of G and the the Knolle
wir streusel
gap G- war he also wen es one Win
Stunde
gar % .
③ Two G bundles
one isouorneic it they are itauornwc
°
' hm hmdles
by an
isauorwanideuhrhyuugtueG-hudeoes-Examples.IOA (smooth ) vector bundle of rank K is the game as
a fiber bundle with standard Jeher u k dein
.
weder
space
F - N -
IR
"
with Structure GLIN) - GLLK, IR)
( when GUN) is
actrvg an IV by standard tepres .
) .
② p
: G →
% G- hie
group ,
HEG closed
suibgr .
is a fieser bundle with Structure
group It
actiug on It
by left umltopliuehiae .
For
any
× E % ,
J an gen neigte .
Uofx and a
I
Loud sedieu s : U - G
of p ( Sly ) H =
y VIEW
S
Dives nik ho Loud trrvidizeetrae
g p ,
PYSLY))
<
¥10:p
"
Lu) → Ux H
folg) =
(Ng), sfplgj.bg)
+
u
Lpr, d-
"
(y ,
h ) =
sly) h
~) H atlas
{ ( Ua . da ) } ze,
in this
way -
„
9
④ °
%
'
: Was × H Ups
× H plsalg ) )
t
( g. n ) -5 ( sig ) h ) ! (pisdglh ) ,
:b
!!;))=
( b , sslg )
{
(g)h )
Sxly ) =
ssly ) . h
'
home
F-
-
"
'
EH .
% 9) e H
since plsaly ) ) =
P Holy) )
Suppose p
: E → M an effecliue f. her knolle wir Structure group
G- .
Then we have
given :
jene
(i ) effea.ve G- actien of G on the standard 2W F
(ii ) open love
hey
U {
Ua } ,
of M
liii ) smooth mops %.
. Was ↳ → E ka
,
C- I
related by tue
pnenerwy that ,
ihr Was Von Oz FH,
= :
Wasgtun
¥) Yards ) .
Ysaly ) -
4µg) ↳ eVarg
.
(
* ) is called the
wcyae identity / egnatia
{=)
4dg) Yaaly)
= 4dg ) = -
4. aly) e
V-yc.ua
C- G
4g ßly) Ysgly) =
Ygglg ) = e =
,
4µg) =
.
Ulsglg)
-
1
4 gß
:
Ugnß →
G- ←
HYE
%g .YSI :
Urs → G
tag)
Prop.2.G-supr.ae M and F are smooth uutds .
,
G
aliegr .
and we are
Give the data d) Xiii ) .
Let E : =
! Vax F)µ =
{ G. x. f) ixtt ,
x Eva ,
f cµwluere
(d. x. f) n ( ß , x
'
,
f
'
) if x - x '
and
N
f- '
=
Yak ) .
f .
→
Then the natural project.eu p :
ETM ( PLE
II
)
)
harte mode in ho o knienden her kuehle wir skaderd fuhr F
and Shane G .
×
Prod da :
FILL ) → Uaxt 544 ) =
%f¥f )
{ E. JA ] :
94,4}
di
"
( y , f) =
[ a. y , f ] long egniv . das re
p
-
14) Leos aneigne
hijecliouss.tn .
p =
pr, opa .
nespresanhathewih
↳( ↳ )
am the first
camp .
Transition maps
:
% .
Ola
-
^
:
Was ×
F Was
×
F
( y , f) 1Tiff
= [ P , y , Yeats)f
]
-
ly.ys.ly/.f)ByProp.
2. U
, p
: ETM Can he made into a Kunde
J.hu wedle wir f.her kuehle otto ) { Wa .
Dabei ,
heute
• f-aber handle wih stand .
fiho F and Structure
group G .
D .
If we a
place { yßa} by a
cohoueohogaesß-
LEIXI
↳
cycle { 43,
} * < ⇐ * [
lassen (i ) end (ii ) oe
:
Leo Ieee
)
-
.
yaesk) fßk ) - KK ) Yislx) Hapert
✓ htt C- Was
for Snooker Ids fa : E KEI .
Then the uaestructreu in the proof of Pvp .
6.2 Ieod)
ho an isomorphe frhu handle wir buchen
group
Gau M
. .
Carver>
ely , any sie Aber handle wie studiere
greys
E
definiere a cohouudogy Class of cocycles .
Let n ) wow
↳ ok at two Imad Coles
of such bundles ,
weder hindley and principe kuehles .
2.3 Vector bundles ( recall f- neue Global
Analysis)
-
p
: V → M vector bundle
It is a fieser bundle with > hand .
LW a never
Space N
|and structure
group GLLN) .
• Fer ung x EM , Vx is a weder
Ipoe :
y ,
y
'
Elfe ,
TEIR
ytty
'
: =
9-
'
(vttv ' ) 10 :p
Yu)
- Ux N
WW Olly) =
(x
- g) = (×
, ✓
,
)
loudtrividiz. .
,
+ e- U .
-
Note that this is well def
.
,
i e
. independent of here kehrt .
← Heule
,
TW ) =
Space of social is a we do Irene
and a modul over the
ring
( MdR) ( s ,
s
'
e- TW)
( Sts
'
)G) =
stets
'
k)
C-
4T C-
¥
=
fs (x ) =
fk) Sk)
HXEM.
• As we know frau heuser fields ,
ve cheer bundle how
many global seciiens ,
since Load
zectieus heute extended
by zero vie bump functions .
Def.2.jp
: V - M and
q
: W → N two neuer bundles .
①
w.tn shaderd 2W , IV and 1W
Then a vector bundle morphin ( resp .
isomorph ihn )
hetweeu p
und 9 is a
Jeher bundle morrison
( resp .
isomorph .
) f : V - W wir under
Gay way I : M → N
sie .
for any
+ am
f)
µ
:
K -
Kfz,
is linear .
② In the special Lease M =
N and f- =
idee ,
a Uecker
handle morphin f : V - W iuducos a linear luop
f-*
: TW ) → TIW ) .
f- * Ist =
fos 1M - W
Prop.2.tl p :
VTM
, q : W - M Vector bundles and
IT : TW ) → TIW) a linear
weg .
Then OI =
f- *
fw a weder handle vorhin f : VTW
⇐ ¢ is linear 0hm .IR ) .
Prod (f .
analog ums statement for teuer finds in GA -
Cbs)
Exercise .
Tl
TM → TN
ix. tEia f : MAN 4ueop
hetweeu mfds .
,
denen
Tf : TM - TN is a neuer nude
warrior
louuiugf
Utiweg Prop .
2.4 .
( af .
Global Analysis ) ,
we love
. v - M ,
4 :
=
Yuki - M
is a vector handle and it is called the dual of V .
•
VTM
,
W -1M weder kuehle
,
than their Lenker
Product
✓ ① W : Hauk
① We
:
IS also a vector bundle aus M <
.
1
"
V
.
S
"
V one also
notudly woher kuehles .
Rennen k -
Theory
-
=
Continuous woher kuehles ( usuolly langte weder kuehles )
bevor some nice homolog .
X .
V - X
.
W → ×
V W → X
Set of isomorphem nasses ef neuer unedles der X
is a
www.hahvlsemi-gneupw.r .
ho ⑦
( weit 0 , gruen by id
: #→ MX viewed es weder bd .
du X wir 0 dim
.
Jshos ) .
¥ → 2
One war laustrudr en one lieu
group
KH )
Otte t
of this semi -
group
( Grothendieck geup ) .
2.4.Princiralfiserbundledef.INLet E be a hie
group .
A principaliser
bundle with structure
group
G (or principe Gbundle
)
is o fieser bundle p : P - M with Standard Eher G
and structure E
octrmgonitselfbyleftmultpucah.eu .
By definitiv ,
then means one ho ) an atlo ) l ( 4. %) REIS
her
ns.t.gs.o/a-1lx.g)--lx.4saH:g )
Yßa :
Usa- E Junk
mops -
The fires of a priuard E - bundle one
difteau .
ho G
,
hat they don't have a natural group
Strecke , since
left
multiplikatives in a
group
are
not
your
honour reims
!
But
eam fiber I has a naturel transitive night
• den
of G ,
which is free i e
.
dl rsohsory geup
=
"
Eu =
{ es tut Px
( u -
gu
→
g- e
In particulier ,
for any
tixed Pa ,
iuoeuae,
VIE )
•
einen .
%;
Eku E
-
Px
=
Ses
How is
this night
lecken
o
) C- an ten there of p
detiued ?
leuma2.10-supr.ae p
:P- M is a principle -
bundle
Fur ne P
, gt
G- seh
g
rlu.gl : - u
:3
: =
Hä King ) EP
-
=
Wwe
! :p
"
( 4) → ↳ × E is n
paarother laude war
wie + e- Ua and Halle ) = E Ua x
G. .
Then r : Px G- → P dehnen o devote
right
active
of E an P ,
called the
priucipd right action
e) Gon P .
More over
,
this actien preserve, tue there of p
auch testen d)
to a free and Transit.ve
rigu .
neuen af G au each the Px .
PRI
r : Px C- → P is well de
find ,
i. e .
Independent
of the Choreo of Chart
,
since left and
right multiplen hier
umlenken ' n E Px
Summe % is mehr Chart • and x and % Lu) - K ,
h
'
)
Thai h
'
=
Yak) h und
0/5^4 ,
h
'
g) =
0/54, Ysaklhg )
-
=
Ohio;D: King )
=
Fg) -
( leorly ,
ris Smooth and since
right
-uralt -
of
G an G
is
right
-
achten
,
r is a Smooth
right
action
.
By laustnectieu ,
r :
II x E- →
?
V - EM .
and it is transitive .
It is also Lee .
( n g-
n ⇐ h -
g
- h ⇐ e =
g )
Kdu ) =
( x
,
h )
D -