5.3Manifddswithboundary-Def.5.CAn dimensionen
(smooth) moni total with
boundary
is a Hausdorff ,
sehend cauntable topdog .
Space M
aguippad
with a maximal atlas
of Charts with Value s in
_
⇐
"zz
the half
space H
"
:
{
41, . .
.
×
"
EIR"
: x
'
EO} .
=
• # Aatw.EE}
of M wie
✓du es in H "
is a Collection I of homeauarpwsu
"
a
'
Ua →
Udk) EH
"
E IR "
, Wwe Ua EM and UdK)>H"
are
open Sunset g. \.
.
• M =
Vater
•
hä nö :
na ( Van Us ) → adh G ) .
• ne smooth ,
www. means that they can he extended
to devote mops definad on
open Subset
of IR
"
Conboin
]
Ua ( Van Up) .
. A point x EM is called a
boundary point of M
,
if
I a Choir (Ua , ha ) s .
t . × E Ua and Uak ) E hat 4)n 304×151
=
↳ LU )n OH "
whve OH
"
= { 41, . .
.
xu ) c. Hh : +1=0 } .
We wrik IM =
{ × EM :
× is a
boundary point } .
Note that x e- DM ⇐
V Charts ( Ua .ua ) EA weh XE !
Leak ) e OH"
.
.
Pointe × E MIOM one Called interior point .
xe MIOM ⇐ Udk ) E HHIDHU H Chart ( Ua , Lea )
weh XE Wa .
Prop.5.7-Mn-diuu.lu/dwihbowedaydM-to/ .
Then OM is a (n s)
-
dim . mhd -
without
boundary .
Prod An athos for DM is
give by
{ ( Van DM ,
klang:*
( Unna) EA} , whee
A is out otto, for µ .
( udk ) n { OSXIR
" "
is open ink"
!
Ex .
MFB"
=
{ × EIR
"
: IKHE 1
] n = 2
0¥DM = S
" -
1
=
9 × EIR
"
:
IKII =
1} .
B2
E. Rotatatieu of 132 brand a Circle Leeds to a
lufd wie
boundary M s .
L . DM =
T
?
Ah
Concepts such suwok
mops , weder fields ; differential
fan) etc . Wo Kos also Leute for Kidd .
Wik
boundary .
•
If i : DM ↳ M is the natural indien
,
then it is
Smooth and ter any K - fern war M
,
Tw is a kfernen
DM .
•
+ E 0M ,
M
EIN .
An oriente time on M ( defrued es
tarnt .
what
boundary ) .
iudnces an orientalen an OM :
Support B {
( Ua .ua) , < EI } is on aneuteidatlos for M
and Laesion
uoouj
'
: 6. (Ua. ) ↳
( Ua.
) for two
Charts in B .
As dsservld
, Walks )s So SXIR " -1 is
luopped to
UßLOAßISSOSXR"
-1 .
At a
pour + =
(O ,
Xl
,
.
.
, x
"
) E
Ua ( Un ) the
der vdive Dfu, nie ) ↳ tue
tdlouuy fan
D. um:) =
) IIIa.
.
AE
Mux ! R ) .
Ugo Ua
- 1
mops in Wien Pauls ↳ Tukur
raus
,
i. e .
pauls wir
hegen +
^
ho paule wie
negativ +1
,
whin in pures b > 0 .
Sauce del ( Delusoui)) >
!this im
plies der (A) > 0
,
when de Scriba die derNolde
of
the transitiv mops of tueottos
on 0M iuduceol by B .
Hence ,
tue athos an OM Medical
by B is also brechend Suppose
M is a mhd .
Wir
boundary of dim . u .
If WE R
(M ) ,
then w Vennes an
den open Kessel
Mlsupplw) and so does dw ( by Then -4.28
)
www.iuplies that a
supp Cdu) csupplw).
In
partei War
,
dw E RI ( re) -
Thin ( Stokes Theorem) .
Suppose M is an
oriented n dim
-
mhd .
wie
boundary DM .
For
any
we R ( M ) we home :
%- =
!: t.fi;) .
In partikular ,
it M is a maitddwihobut bei
ndeoy ( 0M¥
then
fdw = 0 .
M
Proof Assuue WE
RI
-
^
( M )
•
het ( Ui ,
u
;
) i - 1
, . .
,
l he war
of on areuhed athos of M
Sir.
supplw) ⇐ U
,
u . . u We und fi : M → [0,1] i - 1 . . .
,
l
Sunde tds s . l .
Suppen;) CO ;
and
%)i
)
=
1 .
Supplw)
• Then I Vis DM
,
uilu.fm and
film war
↳
und
www.uefmw :
µ -
Ishii•
Also
,
w =
:[tiwiupiiesdw-I.attiw)
und supp (d # iw ) ) c
supplfiw) EUI
Heule ,
{dw =
II !!
Hiw )
It
suthces to how that
[dlfiw ) =
Gut;
u t i
.
Without less
of geeerdihg ,
we Lance ossuue
scyplw )
is uaioiued in daran
of one
Single Avv ( U
, n ) .
Then w =
Efi drin . .
nduin . . n du"
←
toi Smash fcts W
:
:
Mt IR weh
Corral superb
in U .
" the
taugen Space IDM her + e- DM
is
spannend ¥ ,
i > 2 .
=) du
'
10µF
0 and so
wfj-pyw-duh.indan
Iw =
!!
= Shou"
←IM
T lolx IR
"
-1
W
,
ho ,
haipvct Support In U .
'
By Then .
4.18
,
dw %fdjui-duindcinr.netin
-
sehen=
=
§!
n
)
" "
du ! . . nduu .
=
{ dw =
Ist-
e)
" "
/ Hui =
ulo )
Oxi
n
Olwi .
un)=
E ) G
)
i i - 1
↳ .am
!"
Wi ho ) "
lourod Suppen
Hn µ
in U Fu bin i Then for integrals allows
to delaware 4) in
toihtegnds der
the individual Coordinator iwhee tee oder
of integrative
doesn't matter .
o
d
dw
-
{"
( / d) du . .
die
-
a
+
Ii"
"
!"
!!! dei )- äi:*
=
!!!
"
-
1)(O
, ¥ .
.
,
xh ) dx ? .
den
FTC
+
wihove
=
{MW -
gg
.
loupocl Support
5.4DERhamwlomology.az(M ) : = ⑦ R
"
(M ) SEIM) los
for KS die (re) n
µ KEIN
grddad neuer
space
•
gmdud-lauuhd.ve algebra w.r.to ^ :
S2
"
(m ) n helm ) E R
"
"
( m )
Why = L-
1)
"
ey 1W WERK (m) , ze Relay)
Moreau , we home a Linear
mgs el : RIM ) → RIM)
While is o
grad deinetien of degree 1 of Chloe) ,
1) .
O_0, sik ) d-, Ryu) → .
.
.
-7 szd" '
"
(re ) 3 o
By tun .
4.18 :
=
O P
Det we ULM )
① w is closed , if dw = 0
② w is exact ,
it 7
yen
SELM ) s.tw =
dz .
d ?
= 0 =)
any exod form is Closed .
.
Key,
(d) =
: ZLM) c RIM ) subspace of Closed oliff-
focus
{ WEAR TM ) : dw - O } i We wie ZKIM) = ZLM ) ne
"
(re)
It is a
subalgenraog@lMl.n ) i Snce
d (way
) =
dwnztl- 1) "
Wendy ter WERK IM) .
• lmld) : BLM ) EZIM) E RIM )
is u sub
Space .
It is a two sidud ideal in ZCM) :
F- die
'
y
'
En
"
Im ) , we ZIM )
d hin !
) =
t.dk/swtl-s)Ydw--qsw--=Y-
O
=
→ HIM) =
ZLMYßIM)
=
q.HN/BryMjTYM)
is o
groded
www.hd.ve
( uurtid , ossoa.de/olgebruawlR .
It is called tue de Rlaam
www.logyalgenra of M
and HIM ) tue k the
de Rhea
cdauology (Space argrap
)
of M .
For WE Z
"
(m ) we warte [ WIE H
"
(M ) for its
loluauology Closs .
Rein , [ w] ^ Ey] : = [
way]
[ w ] +
Ey] , = [
Wtq ]
JEW] , # [du ] J ER -
Klein. If M is Lear
pod ,
H (M ) is find -
dimensionale
Also time for way ohne non wurde
uebels. but not for all .
• f :
MAN Guuop
kehren nfds .
=
, f-
*
:
RLN ) → RIM) dg .
warum .
Since F. d = d. f. ( Tun . 4.18) ,
f- LZLN ) ) ( ZH)
und f- YBLN))( BLM) and here f
'
Indices
um
degen ne luarphlkn :
f
#
: HIN ) → HIM
) ( f
#
( H
"
IN ) )
[ w ] IN [für] .
c HMM) ) .
• ( got )
#
=
f
#
.
g
# for ander Gup
g
: Nt } P
µ
between wfds .
lf f is a
diffeaurpn.sn ,
then f#
: HEN ) > HIM)
is an
isomorphismen wie muerte (f
#
)
-
I
= (f)
#
.
( So di ffeau .
mfds love isomorphie de Rhea
hdouudogy).
M and N
In fach , Smootnly hauohoprcmfd-isauernh.ch
de Khan Lobo
uiology ( 70mops
f. MAN and
9 : NTM sie fog and got
are
zueootnly kauohgsrc tollen
identity ) .
•
. In fach , von hinaus
leg hawhonic Centos home
isouuerpvc
der war
Coleonulogy ( in
Partido , heaueaudperc
①mhd,
love isomorphe de Rue
cdouedogy ) .
.
de Rhein Thun : de Rhone
cohomology =
Singular cohauology of M wie red coefhicieubs
~,
Kon use tools freu
algennaictor.bgy
to
leugnete de Rhoen
lokaudogy .
¥ M -
IR
"
Holm ) =
IR
An, ↳ seid fan is doch an IR "
H
"
(M) =
Sos K > 0
=) Poiucore Laune : On
any mhd .
,
ony •seid
form is ↳
cdly exod .
\