COVERING SPACES Let p E B be continuous

and surjective Then a nbd U is evenly covered it p U is a disjoint

union of open sets Vd Such that restricted to eachV2 p is a homeomorphism to U

p is a coveringmap if every point in B has an evenly covered nbdE is a covering space

MIDTERMEXAMPLES 2 28

asuniversal cover

GROUP ACTION 52 CX.y.aeR3 X2ty2 cE I

4 S2 5h given by Cay 27 C X y z

x y v 4 Ky H C Y Ya Z antipodal Rgp

IRP 5hThe map p S2 IRR is a covering map

Note that 4 id

Need to find evenly covered abds for everyequivalence class cx.y.DE RP2

let U be a ball of radius K catered atn n

x.y.tl c 5 Then Urdua

locale putr

Let V Uns let V be the image ofF in IRR Then p V T u Cv4 so V

is evenly covered

Similar argent work for 5 covering IRP

S Cx xnfelR X t x 7 1

4 42 t 112given by 4ha xa.at C X i ka

s cz.wic.cc lzf lwl2 1

4mm tw e iz e in

4mm E 14 win dncz.at E S

Iff Czw C S

2 xtiywutirlzlX.gl lvl2 42eq2

1217142 2 is thesame as Tty th tu 1

p 5 s Ln m

p is a cowering napp choose a small ball 9 as 1 G w

s t all dn.fm lui n duffel pif Kfj

04 ok.acal

0 044,1410

2 w dnk.nlz.ch Hk

LIFTING LEMMAS P E B a covering SpacesGiven f X B when can we find a

lift I X E e

n

f tp fr p ofX f p

The identity nap id S as doesn't lift to CR

IR

tas id S

why not I IR

idfo D f

LITs s

I is unique Ifa Fff But ffohf.ccso indoffkidoffD Contradiction

A constant nap S codes will 1 ftConstant maps always 1 ft

g S s b gCcey Ct ostekCc Ed kata 1

Sg Sn

5 It Otts is a lift of g42C 41

Lifting criteria cannot just depend on X

BASIC LIFTING LEMMA Let P E B bea covering space EEE a basepointbo p c B Let f coD Eos Bbe a continuous map Then 3 lift

I co.is son Eia

E

Jp999

0So p 9

Eat _5in