space of any countable - Mathmilicic/Math_6240/LG3.pdfLemuria. Let G-be a connected Lie group. Let U...
Transcript of space of any countable - Mathmilicic/Math_6240/LG3.pdfLemuria. Let G-be a connected Lie group. Let U...
A topological space X is53
a Baire space if theintersection of any countablefamily fun , n E Nn } of
open dense sets Un is
dense in X ( i.e.M Un is
dense in X).new
Lemuria (category theorem)locally compact space Xin aBaine space .
Proof . Let (Un ; ne N ) bea family of open dense subsetsin X
.
Let V--V,be a nonempty
open set in Xmuch that
I in compact -IX is locallycomply
Consider,V
,n U
,- this is
54'
a nonempty open set in X sinceU
,is dense in X
.
Therefore , there bustsa
nonemptyopen
set V,such that
VacVI e V, n U ,and VT is compact 1 sinceX is locally compact)
u,
continue inducting
'
The compact⇒ UntieVI
,KYnun
⇒ VI.,
aVic. . .
a VTSince these are compactsets ⇒ when VI =W is
55
compact and nonempty .
This implies that VV cUnfor all me N . ⇒ Wc n Un
MEN
⇒ v nn?*on to ⇒
¥h . is dense in X.TX
F-
Manifolds are Baine spaces !
#
A locally compact spacex iscountableatiufiuityifit is a union of
countably many compactsets .
56
temping .
Let G- be a locally compactgroup countable at infinity .
Assume that A- acts
continuously on acompact space M , andthat the action is transitive .
Then the orbit map Wm : G-→M
(given by gi-g.nu) is open -
Lemmy .Let G- be a Lie
group .Then
,the following
are equivalent :e: ) G is countable atinfinity
Iii ) A- has countably57
marry components .
Throat . fi)⇒ Iii ) Assume thatG- is countable at is . ⇒
G- = U km , km are compactMEN
(Gi j i c-I) are open and
disjoint ( Gi n kn ; i c-⇒is an open cover of km .
It
has a finite subcover ⇒
kn intersects finitelyMaury Gi , iEI . ⇒
Union of km = G- intersect
countably many Gi⇒ Iis countable
.
58
Iii)⇒ Ci) a- has countablymany components .It is enough to show that Gois a union of countably marryCampaet sets .
This follows from the lemma.
Lemuria . Let G- be a connectedLie group .
Let U be a neigh .
of 1 in G . Then G- = U unMEN
"
Proofs .Can assume that
U is symmetric .Then
Hµ
U"
is a subgroup .
It is also open .
he It⇒
h E U"
for some n e N59
h .U c Uhtt
€gh. of h ⇒ h . U c H
H is open . ⇒ G is union
of H co sets = open .G-
is connected ⇒ It =G-. TH
Gt is a Lie group with countablycomponent ,
G is countableat cs
.
G- acts differentially onmanifold M . If the actionis transitive
,for me M,
Wm : G → M is open .
The orbit map Wm :G → M60
hasto have constant rank
(sub immersion) .It is open
only if the rank is maximali.e
. com is asubmersion .
P.noafofopeumappin-gthoreur.ltU be an open neighborhoodof l e G .
We claim first that
Wmu) is a neighborhood of m .
Assume that V in a compactsymmetric neighborhood of I
/
-
such that v2 u.
61
Existence: malt ,
in continuous.
F V,z I open such that YZ U .
Can shrink it to Vaal whichis a neigh . of l and compact .
VI c U . Van Y"
is compactand neigh of l l Va is a neigh .
and VI'
is a neigh .)⇒ V =Van Va
- '
satisfies our
assumption .
Lgi g←G) is a cover of G .
D
since a- = U K ⇒n= I
( g . int(V) ; g c- Ct) is a cover62
of K n ⇒ 7 finite subcover
( g n . int IV) ; ne N) is anopen cover of Ct .
Cgn .Vj ne N) is a cover off.
Un -- M - w Cgi V ) == M - gu .V -m
rcompact⇒ closed
Un is open
It,
Um = ( M-gnv:m) =
= M - guv.m = M- G. me =
= $ .
Since M is a Baise space,63
'
at least one Un is notdense in M .
the = M - gn .V. m not dense
⇒
grill . m has nonemptyinterior-
V - m has nonempty interiorV.m is a neigh of g.m
⇒ g-' V.m is a neigh . of m
g- !V.m c V?m c U . m
⇒ U . me is a neigh . of me .
Can complete the proof .0 open in G, ye 0 .
g-' O is a neigh . of I
64
g-! O .
'm is a neigh . ofM
⇒ am is a neigh of g. we .
=) O. m is open !
Universal covering spaces65
eo#
xnaamauifold , x . - base pointX t
.CI,
is aumiversaltr t coveringsX Xo ifitisaeomneebed manifoldsuch that p :X → Kina coveringpail - Xo
,and for any other covering
space (Yy . )
is'
anise amustbe
~. a identity !--
→ X⇒ Xx - ← another-¥4 universal
must be diffeo . cover
66Universal covering space is
unique up to an isomorphism .
Universal covering spacesare simply connected ; i.e .
it,CE,E) ⇒ B .
I d c- p-
'
exo) (I,d) is a
PI universal covering spaceX F unique Td 'I→ IIFI Tako) =D
on
Htp - deck transformation×Td ( p
-'
exo)) =p- '
G.)Deck transformations forma group
⇐ IT,CX,xd
G- a connected Lie group67
⇐ yCE
,T ) universal covering
F unique lie group structureon E ( compatible with manifoldstructure on E) such thatT in the identity and
p :E-→ G- in a lie group
morphism .
E- - universal covering group#I
hiftimgprperty(X mo) connected manifold
(Y, p , yo ) a covering speciesF
'
G,yo) e - - n
Pf←¥
Gao)
Cx,xo)(Z , z . )
connected and simplyconnected manifold
,Fdiff .
map .
Then flare bursts the
unique diff . map F ! Z →YF' fzo) = yo such that F '
ft ) = yo .
Constructionalpstructure on E-
.
-
'
Ex E is simply connected'69
⇐ x E,TXT ) -→ LE , i )im
pxp I I(GxG , I xD =3 (G
,i )
im in unique .
In .Ex E → E
is a diff . map .
- binaryoperation .
Have to show
that it defines a group structurewe have
p .mi = mo Cp xp) .