Introduction to Algebraic Topology

67 Pages·2011·0.38 MB·English

Introduction to Algebraic Topology
MitschriderVorlesungvon
Dr.M.Michalogiorgaki
TobiasBerner
UniversitätZürich
Frühjahrssemester2009
Contents
1 Topology 4
1.1 Topologicalspacesandcontinuousfunctions. . . . . . . . . . . . . . . . . 4
1.1.1 Topologicalspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.2 Continuousfunctions . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Conectedness&Compactness . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Connectedspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Algebraictopology 16
2.1 Fundamentalgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.1.1 Pathhomotopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 efundamentalgroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Coveringspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Liingproperties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5 efundamentalgroupofthecircleandapplications . . . . . . . . . . . . 27
2.5.1 eFundamentaleoremofAlgebra . . . . . . . . . . . . . . . . 28
2.5.2 Deformationretractsandhomotopytype . . . . . . . . . . . . . . 29
2.6 SeifertvonKampeneorem. . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6.1 Directsumsofabeliangroups . . . . . . . . . . . . . . . . . . . . . 35
2.7 Freeabeliangroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.8 Freeproductsofgroups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.8.1 Freegroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.8.2 eSeifert-vanKampentheorem. . . . . . . . . . . . . . . . . . . 43
2.9 CWcomplexes(cellcomplexes) . . . . . . . . . . . . . . . . . . . . . . . . 47
2.10 Surfaces(two-dimensionalmanifolds) . . . . . . . . . . . . . . . . . . . . 49
2.10.1 Fundamentalgroupofsurfaces . . . . . . . . . . . . . . . . . . . . 49
2.10.2 Homologyofsurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 50
2.10.3 Classi(cid:277)cationofsurfaces . . . . . . . . . . . . . . . . . . . . . . . . 51
2.11 Knottheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.12 Classi(cid:277)cationofcoveringspaces . . . . . . . . . . . . . . . . . . . . . . . . 58
Index 66
2
CONTENTS
Literature
• Muncres:Topology2ndedition
• Jänich:Topologie
• Massey:AlgebraicTopology:AnIntroduction
• Stöcker,Zieschang:AlgebraischeTopologie
• Lickorish:Anintroductiontoknottheory
Introduction
Incalculus,youhavestudiedRn,n∈N,aswellasfunctionsf ∶Rk →Rℓ,k;ℓ∈N. You
havestudiednotionssuchasneighbourhoodofapointx∈Rn,aswellasconvergenceand
continuityofafunctionf atapointx∈Rk. Forthisstudy,youhaveusedtheEuclidean
metric.
Forinstanceiff ∶R→R,wesaythatf iscontinuousatx∈Rif∀">0∃(cid:14)>0suchthat
if∣x−x ∣<(cid:14)then∣f(x)−f(x )∣<".
0 0
Weusedthemetricd∶R×R→R,with(x;y)↦∣x−√y∣.IngeneraltheEuclideanmetric
d∶Rn×Rn→Rwith(x1;:::;xn);(y1;:::;yn)↦ ∑ni=1(xi−yi)2.
Intopology, westudynotionssuchasneighbourhoodofapointx ∈ X, convergence,
continuityforageneralsetX.Forthisstudyametricisnotnecessary.Whatwewilluse
isopensetsinX.
Inalgebraictopology,weuseabstractalgebratostudytopologicalproperties.
3
1 Topology
1.1. Topologicalspacesandcontinuousfunctions
1.1.1. Topologicalspaces
ConsiderasetX andP(X)∶={U ∣U ⊆X}.
De(cid:277)nition1.1 AtopologyonX isacollectionT ⊂P(X)suchthat
1. ∅;X ∈T.
2. IfUi∈T ∀i∈I,then⋃i∈IUi∈T.
3. IfUi∈T,i∈{1;:::;n},then⋂i∈{1;:::;n}Ui∈T.
(X;T)iscalledatopologicalspace(sometimeswewillonlywriteX).
U ⊆X iscalledopen,ifU ∈T.
U ⊆X iscalledclosed,ifX/U ∈T.
Example1.2
1. X someset.
• Td={U ∣U ⊆X}discretetopology.
• Tt={∅;X}trivialtopology.
2. X ={x ;x ;x }.enthethreecollections
1 2 3
• {∅;X}
• {∅;{x };{x ;x };X}
1 1 2
• {∅;{x };{x ;x };{x ;x };X}
2 1 2 2 3
aretopologiesonX.
3. X isaset.enthecollections
• Tf ={U ⊆X ∣X/U is(cid:277)niteorX/U =X}(cid:277)nitetopology.
• Tc={U ⊆X ∣X/U iscountableorX/U =X}countabletopology.
aretopologiesonX.[Homework]
4
1.1Topologicalspacesandcontinuousfunctions
De(cid:277)nition1.3 SupposethatT andT′aretopologiesonX.IfT′⊇T,thenT′iscalled
(cid:277)nerthanT. IfT′ ⊋ T,thenT′ iscalledstrictly(cid:277)nerthanT. IfT′ ⊇ T orT ⊇ T′,
thenT andT′arecomparable.
De(cid:277)nition1.4 Let(X;T)beatopologicalspaceandA⊆X.
einteriorofAis
intA∶= ⋃ U.
U∈T;U⊆A
eclosureofAis
A∶= ⋂ U.
X/U∈T;A⊆U
Basisforatopology
De(cid:277)nition1.5 AbasisBforatopologicalspace(X;T)isacollectionBofopensub-
setesX (i.e.B⊆T),suchthat∀U ∈T ∃{Bi}i∈I,Bi∈B,∀i∈I,withU =⋃i∈IBi.
Remark1.6 Abasisisnotunique.
Properties:Bbasisfor(X;T)thenBhasthefollowingproperties
1. ∀x∈X ∃B∈Bwithx∈B.
P X ∈ T andBisabasis(cid:212)⇒X = ⋃i∈IBi,Bi ∈ B,i ∈ I. Sox ∈ Bj for
somej ∈I.
2. Ifx∈B ∩B ,B ;B ∈Bthen∃B ∈Bsuchthatx∈B ⊆B ∩B .
1 2 1 2 3 3 1 2
P B ;B ∈ B ⊆ T (cid:212)⇒B ;B ∈ T (cid:212)⇒B ∩B ∈ T (cid:212)⇒B ∩B =
1 2 1 2 1 2 1 2
⋃i∈IBi,…
Conversely: IfacollectionBofsubsetsofX satis(cid:277)esproperties1. and2. thenthereisa
uniquetopologyT forwhichBisabasis.
T iscalledthetopologygeneratedbyBanditconsistsofallunionsofelementsofB.
P WeprovethatT isatopology
1. X ∈T?
Property1(cid:212)⇒ifx∈Xthen∃Bx∈Bwithx∈Bx⊆X.ereforeX =⋃x∈XBx,
i.e.X ∈T.
∅∈T?
∅istheemptyunionofelementsinB,so∅∈T.
2. IfUi∈T ∀i∈I,does⋃i∈IUi∈T?
Ui∈T (cid:212)⇒Ui=⋃j<inJBij.So⋃i∈IUi=⋃i∈I⋃j∈JBij ∈T.
3. IfUi∈T fori∈{1;:::;n},is⋂i∈{1;:::;n}Ui∈T?
WewillshowthatifU ;U ∈T thenU ∩U ∈T.
1 2 1 2
U1=⋃(cid:21)∈(cid:3)B(cid:21),U2=Uk∈KBk.Considerx∈U1∩U2.enx∈B(cid:21)xandx∈Bkx,
forsome(cid:21) ∈(cid:3)andk ∈K. usx∈B ∩B (cid:212)⇒[byproperty2.] ∃B ∈B
x x (cid:21)x kx x
withx∈B ⊆B ∩B ⊆U ∩U .
x (cid:21)x kx 1 2
ereforeU1∩U2=⋃x∈U1∩U2Bx∈T.
5
Topology
Example1.10
1. X =R,B ={(a;b)∣a;b∈R; a<b}. Bsatis(cid:277)esproperites1. and2. thereforeit
generatesatopologyonR.isisthestandardtopologyonR.
2. X =R,B={(a;b)∣a;b∈Q; a<b}
Bsatis(cid:277)esproperties1.and2.
Claim[Homework]:etopologyT generatedbyBisinfactthestandardtopol-
ogyonR.
Lemma1.11 LetB;B′ bebasesforthetopologiesT andT′ onX. enthefollowing
statementsareequivalent:
1. T′⊇T.
2. ∀x∈X,∀B∈Bwithx∈BthereisB′∈B′suchthatx∈B′⊆B.
P
1.⇒2.
Considerx ∈ X andB ∈ B withx ∈ B. B ∈ T ⊆ T′ (cid:212)⇒B ∈ T′ (cid:212)⇒B =
⋃(cid:21)∈(cid:3)B(cid:21)′,B(cid:21)′ ∈B′.
Sox ∈ B,thereforex ∈ B′,forsome(cid:21) ∈ (cid:3). i.e. wehaveB′ ∈ B′,suchthat
x∈B′ ⊆B. (cid:21) x (cid:21)x
(cid:21)x
2.⇒1.
Csoomnesi(cid:21)der∈U(cid:3)∈.2T.iamndplxies∈tUha.ttheernexex∈isUts=B⋃′ (cid:21)∈∈(cid:3)BB′(cid:21)su,cBh(cid:21)th∈aBtx. ∈Bat′isx⊆B∈B(cid:21).x,for
(cid:212)⇒enUUx∈⊆T⋃′.x∈UeBre(cid:21)′fxor⊆eUT (cid:212)⊆⇒T′U. =⋃x∈UB(cid:21)(cid:21)′xx (cid:21)x (cid:21)x
Producttopology
Let (X;Tx), (Y;TY) be two topological spaces. e producttopology is the topology
withbasisthecollectionB={U ×V ∣U ∈T ; V ∈T }.
X Y
Metrictopology
X isaset.
De(cid:277)nition1.13 Ametriconthissetisafunctiond∶X×X →Rwith
1. d(x;y)≥0,andd(x;y)=0iffx=y.
2. d(x;z)≤d(x;y)+d(y;z)∀x;y;z∈X.
3. d(x;y)=d(y;x),∀x;y∈X.
WecallthesetBd(x;")={y∈X ∣d(y;x)<"}the"-ballcenteredatx.
{Bd(x;")}x∈X;">0isabasisforatopologyonX,themetrictopology.
6
1.1Topologicalspacesandcontinuousfunctions
Example1.14 Rwiththestandardtopology(Tstand).
ConsiderR×R=R2.
Claim:eproducttopologyonR2andthemetrictopologyarethesame.
Forthis,onehastoshow
• T ⊆T
pr m
Bylemma1.11onehastoshow∀x ∈ R2,∀B ∈ B ,x ∈ B ,∃B ∈ B such
pr pr pr m m
thatx∈B ⊆B .
m pr
• T ⊆T
m pr
…
Subspacetopology
De(cid:277)nition1.15 (X;T)isatopologicalspaceandY ⊆X. ecollectionTY ∶={U ∩
Y ∣U ∈T}isatopologyonY,calledthesubspacetopology.
Example1.16 Y =[0;1]∪{2}⊆X =Rwiththestandardtopology.
enthesets
• (a;b),witha;b∈[0;1],
• [0;b),withb∈[0;1],
• (a;1],witha∈[0;1],
• {2},
• [0;1]
areopensetsinthesubspaceY.
1.1.2. Continuousfunctions
De(cid:277)nition1.17 Let (X;TX), (Y;TY) be topological spaces and f ∶ X → Y. f is
calledcontinuousiff−1(V)∈TX ∀V ∈TY.
Claimf ∶R→R(withstandardtopology).e"–(cid:14)de(cid:277)nitionofcontinuityisequivalent
tothede(cid:277)nitionabove.
P
“⇐” Supposethatf ∶R→Riscontinuouswiththede(cid:277)nitionabove.
Consider x ∈ R and " > 0. en V = (f(x )−";f(x )+") ∈ T , so
0 0 0 stand
f−1(V)∈T byourde(cid:277)nition.
stand
Nowx ∈f−1(V),sothereexists(a;b)∈B withx ∈(a;b)⊆f−1(V).
0 stand 0
Take(cid:14) =min{x −a;b−x }. Clearly(cid:14) >0. en∣x−x ∣<(cid:14)(cid:212)⇒x∈(a;b)⊆
0 0 0
f−1(V)(cid:212)⇒f(x)∈V
(cid:212)⇒f(x)∈(f(x)−";f(x)+")(cid:212)⇒∣f(x)−f(x )∣<".
0
7
Topology
eorem1.19 LetX;Y betopologicalspacesandf ∶X →Y. efollowingareequiva-
lent
1. f iscontinuous.
2. foreveryclosedsubsetof Y theinverseimageofitisaclosedsubsetofX.
P
1.⇒2.
Let B be closed in Y, then X/f−1(B) = f−1(Y/B) which is open in X, i.e.
f−1(B)isclosedinX.
2.⇒1.
…
Lemma1.21 (thepastinglemma)
Let X;Y be topological spaces and A;B closed subsets of X with X = A∪B. Let
f ∶A→Y,g ∶B →Y becontinuouswithf(x)=g(x)∀x∈A∩B. enh∶X →Y
with
f(x) x∈A
h(x)={
g(x) x∈B
iscontinuous.
P LetCbeaclosedsubsetofY.enh−1(C)=f−1(C)∪g−1(C)wheref−1(C)
isclosedinAandg−1(C)isclosedinB.
(a) f−1(C)isclosedinA(cid:212)⇒f−1(C)=A∩G,whereGclosedinX.
(b) g−1(C)isclosedinB(cid:212)⇒g−1(C)=B∩H,whereH closedinX.
(a)(cid:212)⇒f−1(C)isclosedinX,
(b)(cid:212)⇒g−1(C)isclosedinX.
(cid:212)⇒h−1(C)=f−1(C)∪g−1(C)isclosedinX.
De(cid:277)nition1.23 LetX;Y betopologicalspaces.f ∶X →Y iscalledahomeomorphism
iff isbijective,andf;f−1arecontinuous.
BijectivecorrespondencenotonlybetweenX andY butalsobetweenthecollectionof
opensetsinX andthecollectionofopensetsY.
us any property of X that is expressed in terms of its open subsets yields via f the
correspondingpropertiesforY.Suchapropertyiscalledatopologicalproperty.
Quotienttopology
Example1.24 Torus
De(cid:277)nition1.25 LetX;Y betopologicalspaces,p∶X →Y asurjectivemap.emap
piscalledaquotientmapifasubsetU ofY isopeninY,ifandonlyifp−1(U)isopen
inX.
8
1.2Conectedness&Compactness
De(cid:277)nition1.26 LetXbeatopologicalspace,Y besomesetandp∶X →Y asurjective
map.equotienttopologyonY inducedbypisde(cid:277)nedasfollows:
AsubsetU ⊆Y isopenifandonlyifp−1(U)⊆X isopen.
efactthatthisisatopologyfollowsfromp−1(∅)=∅,p−1(Y)=X,p−1(⋃a∈JUa)=
⋃a∈Jp−1(Ua),p−1(⋂ni=1Ui)=⋂ni=1p−1(Ui).
Remark1.27 equotienttopologyonY isthe(cid:277)nesttopologythatmakespcontinuous.
Example1.28
1. X = {(x;y) ∈ R2 ∣ 0 ≤ x ≤ 2(cid:25); 0 ≤ y ≤ 1} = [0;2(cid:25)]×[0;1] ⊆ R2. Y =
{(x;y;z)∈R3∣x2+y2=1; 0≤z≤1}.
f ∶X →Y,with(x;y)↦(cosx;sinx;y).
f issurjective.Usingf wecande(cid:277)nethequotienttopologyinY.
2. p∶R→{x ;x ;c }
1 2 3
⎧
⎪⎪⎪ x1 x>0
p(x)=⎨ x x<0
⎪⎪⎪⎩ x23 x=0
equotienttopologyon{x ;x ;x }inducedbypis{∅;{x };{x };{x ;x };{x ;x ;x }}
1 2 3 1 2 1 2 1 2 3
1.2. Conectedness&Compactness
Incalculusyouhavestudiesfunctionsf ∶[a;b]→Randyouhaveprovedthreetheorems
forcontinuousfunctionsf ∶[a;b]→R.
1. IntermediateValueeorem(IVT),
2. MaximusValueeorem(MVT),
3. UniformContinuityeorem(UCT).
esetheoremsrelyonthecontinuityoff andsometopologicalpropertiesof[a;b] ⊂
R. InparticularIVTreliesontheconnectednessof[a;b]. MVTandVCTrelyonthe
compactnessof[a;b].
1.2.1. Connectedspaces
De(cid:277)nition1.29 Consider(X;T)topologicalspace. AseparationofX isapairU;V
whereU;V ∈T,U;V ≠∅,U ∩V =∅,andX =U ∪V. X iscalledconnectedifthere
isnoseparationofX.
Remark1.30 Connectednessisatopologicalproperty.
Example1.31
1. R,T ,
stand
Q=((−∞;a)∩Q)∪((a;∞)∩Q),a∈R∖Q.
2. R,[a;b],[a;b),(a;b],(a;b)areconnected.
9
Topology
Claim:XisconnectedifftheonlysubsetsofXthatarebothopenandclosedare∅and
X.
P
“⇒” Suppose, thatA ⊆ X, A ≠ ∅, A ≠ X, andA ∈ T , X ∖A ∈ T . enX =
X X
A∪(X∖A)(cid:212)⇒A;X∖AisseparationofX,contradiction.
“⇐” IfU;V isaseparationofX. enU ∈ T , X ∖U = V ∈ T , U;X ∖U ≠ ∅,
X x
U ∩(X∖U)=∅.V ∈T ,X∖U ∈T ,U ≠∅,U ≠X,contradiction.
X X
ereforeX isconnected.
Lemma1.33 IfC;D areaseparationofX andY isaconnectedsubspaceofX, then
Y ⊆CorY ⊆D.
P C∩Y,D∩Y areopensubsetsofY. Inaddition(C∩Y)∩(D∩Y) = ∅. If
C∩Y andD∩Y werebothnonempty,thentheywouldformaseparationofY.ButY
isconnected.SoC∩D=∅,orD∩Y =∅.
(cid:212)⇒Y ⊆CorY ⊆D.
eorem1.35 Let{Ai}i∈I beconnectedsubspacesofX andp∈∩i∈IAi. en∪i∈IAiis
connectedsubspaceofX.
P AssumethatUi∈IAi isnotconnected,and∪i∈IAi = C∪D,C;Daseparation
of∪i∈IAi. enp ∈ C ∪DandWLOGwecanassumethatp ∈ C. Ai isaconnected
subspace.
(cid:212)⇒[Lemma]Ai⊆CorAi⊆Dforanyi∈I.p∈∩i∈IAi.
(cid:212)⇒p∈A ∀i∈I.
i
(cid:212)⇒A ⊆C∀i∈I.
i
(cid:212)⇒∪i∈IAi⊆C.
(cid:212)⇒D=∅.Contradiction.
eorem1.37 Let A be a connected subspace of X and A ⊆ B ⊆ A. en B is also
connected.
P AssumethatC;D isaseparationofB. ByLemmaA ⊆ C orA ⊆ D. WLOG
A⊆C.
A⊆C(cid:212)⇒A⊆C.
D⊆B⊆A⊆C(cid:212)⇒D⊆C.
Claim:D∩C =∅.
Proof: eclosureofC inBisC∩B,whereC istheclosureofC inX. C =C∩B =
C∩(C∪D)=(C∩C)∪(C∩D)=C∪(C∩D).
(cid:212)⇒C∩D=∅.
WehavethatD⊆CandweshowedthatD∩C =∅.ereforeD=∅,contradiction.
ereisnoseparationofB,i.e.Bisconnected.
eorem1.39 Letf ∶X →Y beacontinuousfunctionbetweentopologicalspacesXand
Y.IfXisconnected,thenf(X)isconnected.
10