i4

for pix X a covering isomorphism I I are

p Ifpcalled decktations or g X

trainstheyform a group Deck I undercomposition

byunique lifting property adecktask is term edby where it sends a singlepointassuming IT is PC

only theidentitytransf can fix apout n TfA covering p X X is normal for regular if Deck I actstransitively in thefibers of pPCAVG covering of X a normal covering with Deck G

PropositionHatcher1.3g P71

Let p I E X X be a PCcovering of a PC regular Xlet H in p Thena Thiscovering is normal elf H is a p of IT CX xob Deck Xn IT X x H if the covering isnormal forIronnormalDede NCHhg

PC thismapif I nei.e forX normal one has a Shortexactsequence ofgroups is not surjective

Ti X Io IT X x Deck I 71I I Fa

ID X Lpp X x

Ex au C ab a

normal a o.ch bnon normal

p 1 cover b acover

P ta b DedeIa Dede L acting

actingtransitivelya b nontransitively

n p fibers

Seifert vankampen for reasonablespacesLet Xbe top space assume X X Xz YX X Uk X.nl z Y

p p oare patted andreasonable

open Ko

Then ii Yao Tick Xo

j 6 is a pusbutdia

grum.ITXz X X Xo Here ja Xake t Ie ya X

inclusionssketch 2

Proof Needto showthat gives a group G and mars fi fi I Xia o Gagreeing on IT CY Xe we can construct I mmap fi it CX o GIT Y no IT Xo

St L t µ commutes

T.cl z Xo Ti X xo fal a u

yGFzIrfByThinCH f f giveGcoverings p k

2whichrestrict to isomorphic

C ouerngsp.fi 7xEpEicys.xi1Rly7 x Using4 we engine

XTyUXI

iX E

P

this is a G averyX Ko

byTHMCH it induces a homomorphismf.ITCX Gmaking commute

Uniqueness follows from uniqueness of the construction head step

Smoothmanidsmotivation M spaceof configurations of a physicalsystem e.g Anglepoiselampparticle movingon52 want to solvean ODEthere diet equationofmotionspacetime generalrelativitywant tobe able to do calculus on M differentiate integratefunctions

what are smoothfunctions whichobjects canbe integrated over Mwhatkindofobjectis dfdef Let1Mbe a topological n manifold A coordinate chart for M

Tr

is an open subset U C M and a homeomorphism 9 U HU CIRopen

AnAtlas on M is a collection of coordinatecharts U2 G a I s t

M is coveredby Ua e I

IieEEP apes ceacuanup icepcuanup is C

Rn open openrecallthat I b R is smooth on C if ithasPaditivative ofall

X Xn Fka Xn orders

likewise Fi Rn is cos if each component hasfellderivatives

Examples charts atlases

U CIR 4 d tautologicallyopen

M S covered by Ui Go Xn CS l Xi o i o nTwit spherein IR Ui Xo Xn CS l Xeco

6 u RGo Ko x

in Kit openunitBall Deco CR

M S withE U Sh Iko o Il IR stereographicprojections

NorthlSouth poles

transitive E EI Cu ulh plu VDfunction

M IRP 112 631 Ui Exo Xa ERP IX tox XX

4 Ui IRdekko

xD homeomorphism

Ce3Ev I Vil Vi Un inverse

Ti thplace so u.q atlas for RIP 6i Vin

Vo Va

44 Ix EIR IX i x to HEIR 1Xi 1 Xj63 with u o

u 17 V

The defnition of a manifold

Tue atlases Ua G 4,4 on M are ibk if theirunion isanatlasi e extramaps 4,42 mustbesmooth

A e on M is an equivalenceclass ofatlases

def An n dimensional smooth manifold is a topological n men bCd witha smooth structure

fi M IR is a smoothfunct if V 02,92 cordchartfOEI oftheatlas

IR 392ND U2 Ris a smooth feet of n variables

nm

Note on an overlap UanUp fog is smooth

at eacxft to9J is smooth at qc Varus

7 7 Cfoei ca.gstransitionnap112 Rn

continuous

Armap F M N of manifolds is asmoothnap

ifeahkemaydu.HUied chartof M and

44 cha tof N

the function 4 of oeI GCUan F Cvn IR

is a Nato Cx FRM F N

Vi

aon

Remer it is enough to dede I for one atlas it is then automatically true inanycompatibleatlas since 4,4J

ar cos

A diffeomorphism F M Nis a smooth map with smooth inverse