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) .