Algebraic topology

553 Pages·2002·3.363 MB·English

by  Allen Hatcher

Allen Hatcher
Copyright c 2000 by Allen Hatcher
Paper or electronic copies for noncommercial use may be made freely without explicit permission from the author.
All other rights reserved.
Table of Contents
Chapter 0. Some Underlying Geometric Notions . . . . . 1
HomotopyandHomotopyType1. CellComplexes5.
OperationsonSpaces8. TwoCriteriaforHomotopyEquivalence11.
TheHomotopyExtensionProperty14.
Chapter 1. The Fundamental Group . . . . . . . . . . . . . 21
1. Basic Constructions . . . . . . . . . . . . . . . . . . . . . . . 25
PathsandHomotopy25. TheFundamentalGroupoftheCircle28.
InducedHomomorphisms34.
2. Van Kampen’s Theorem . . . . . . . . . . . . . . . . . . . . 39
FreeProductsofGroups39. ThevanKampenTheorem41.
ApplicationstoCellComplexes49.
3. Covering Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 55
LiftingProperties59. TheClassificationofCoveringSpaces62.
DeckTransformationsandGroupActions69.
Additional Topics
A.GraphsandFreeGroups81.
B.K(G,1)SpacesandGraphsofGroups86.
Chapter 2. Homology . . . . . . . . . . . . . . . . . . . . . . . 97
1. Simplicial and Singular Homology . . . . . . . . . . . . . . 102
(cid:209) Complexes102. SimplicialHomology104. SingularHomology108.
HomotopyInvariance110. ExactSequencesandExcision113.
TheEquivalenceofSimplicialandSingularHomology128.
2. Computations and Applications . . . . . . . . . . . . . . . 134
Degree134. CellularHomology137. Mayer-VietorisSequences149.
HomologywithCoefficients153.
3. The Formal Viewpoint . . . . . . . . . . . . . . . . . . . . . 160
AxiomsforHomology160. CategoriesandFunctors162.
Additional Topics
A.HomologyandFundamentalGroup166.
B.ClassicalApplications168.
C.SimplicialApproximation175.
Chapter 3. Cohomology . . . . . . . . . . . . . . . . . . . . . 183
1. Cohomology Groups . . . . . . . . . . . . . . . . . . . . . . 188
TheUniversalCoefficientTheorem188. CohomologyofSpaces195.
2. Cup Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
TheCohomologyRing209. AKu¨nnethFormula215.
SpaceswithPolynomialCohomology221.
3. Poincar´e Duality . . . . . . . . . . . . . . . . . . . . . . . . . 228
OrientationsandHomology231. TheDualityTheorem237.
ConnectionwithCupProduct247. OtherFormsofDuality250.
Additional Topics
A.TheUniversalCoefficientTheoremforHomology259.
B.TheGeneralKu¨nnethFormula266.
C.H–SpacesandHopfAlgebras279.
D.TheCohomologyofSO(n)291.
E.BocksteinHomomorphisms301.
F.Limits309.
G.MoreAboutExt316.
H.TransferHomomorphisms320.
I.LocalCoefficients327.
Chapter 4. Homotopy Theory . . . . . . . . . . . . . . . . . 337
1. Homotopy Groups . . . . . . . . . . . . . . . . . . . . . . . 339
DefinitionsandBasicConstructions340. Whitehead’sTheorem346.
CellularApproximation348. CWApproximation351.
2. Elementary Methods of Calculation . . . . . . . . . . . . . 359
ExcisionforHomotopyGroups359. TheHurewiczTheorem366.
FiberBundles374. StableHomotopyGroups383.
3. Connections with Cohomology . . . . . . . . . . . . . . . . 392
TheHomotopyConstructionofCohomology393. Fibrations404.
PostnikovTowers409. ObstructionTheory415.
Additional Topics
A.BasepointsandHomotopy421.
B.TheHopfInvariant427.
C.MinimalCellStructures429.
D.CohomologyofFiberBundles432.
E.TheBrownRepresentabilityTheorem448.
F.SpectraandHomologyTheories453.
G.GluingConstructions456.
H.Eckmann-HiltonDuality461.
I.StableSplittingsofSpaces468.
J.TheLoopspaceofaSuspension471.
K.SymmetricProductsandtheDold-ThomTheorem477.
L.SteenrodSquaresandPowers489.
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
TopologyofCellComplexes521. TheCompact-OpenTopology531.
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 540
Preface
This book was written to be a readable introduction to Algebraic Topology with
rather broad coverage of the subject. Our viewpoint is quite classical in spirit, and
stays largely within the confines of pure Algebraic Topology. In a sense, the book
couldhavebeenwrittenthirtyyearsagosincevirtuallyallitscontentisatleastthat
old. However, the passage of the intervening years has helped clarify what the most
importantresultsandtechniquesare. Forexample,CWcomplexeshaveprovedover
time to be the most natural class of spaces for Algebraic Topology, so they are em-
phasizedheremuchmorethaninthebooksofanearliergeneration. Thisemphasis
alsoillustratesthebook’sgeneralslanttowardsgeometric,ratherthanalgebraic,as-
pectsofthesubject. ThegeometryofAlgebraicTopologyissopretty,itwouldseem
apitytoslightitandtomissalltheintuitionthatitprovides. Atdeeperlevels,alge-
bra becomes increasingly important, so for the sake of balance it seems only fair to
emphasizegeometryatthebeginning.
Letussaysomethingabouttheorganizationofthebook. Attheelementarylevel,
Algebraic Topology divides naturally into two channels, with the broad topic of Ho-
motopy on the one side and Homology on the other. We have divided this material
into four chapters, roughly according to increasing sophistication, with Homotopy
split between Chapters 1 and 4, and Homology and its mirror variant Cohomology
in Chapters 2 and 3. These four chapters do not have to be read in this order, how-
ever. One could begin with Homology and perhaps continue on with Cohomology
before turning to Homotopy. In the other direction, one could postpone Homology
and Cohomology until after parts of Chapter 4. However, we have not pushed this
latterapproachtoitsnaturallimit,inwhichHomologyandCohomologyarisejustas
branches of Homotopy Theory. Appealing as this approach is from a strictly logical
point of view, it places more demands on the reader, and since readability is one of
ourfirstpriorities,wehavedelayedintroducingthisunifyingviewpointuntillaterin
thebook.
There is also a preliminary Chapter 0 introducing some of the basic geometric
concepts and constructions that play a central role in both the homological and ho-
motopicalsidesofthesubject.
Each of the four main chapters concludes with a selection of Additional Topics
thatthereadercansampleatwill,independentofthebasiccoreofthebookcontained
in the earlier parts of the chapters. Many of these extra topics are in fact rather
importantintheoverallschemeofAlgebraicTopology,thoughtheymightnotfitinto
the time constraints of a first course. Altogether, these Additional Topics amount
to nearly half the book, and we have included them both to make the book more
comprehensive and to give the reader who takes the time to delve into them a more
substantialsampleofthetruerichnessandbeautyofthesubject.
Not included in this book is the important but somewhat more sophisticated
topic of spectral sequences. It was very tempting to include something about this
marvelous tool here, but spectral sequences are such a big topic that it seemed best
tostartwiththemafreshinanewvolume. Thisistentativelytitled‘SpectralSequences
inAlgebraicTopology’andreferredtohereinas[SSAT].Thereisalsoathirdbookin
progress,onvectorbundles,characteristicclasses,andK–theory,whichwillbelargely
independent of [SSAT] and also of much of the present book. This is referred to as
[VBKT],itsprovisionaltitlebeing‘VectorBundlesandK–Theory.’
In terms of prerequisites, the present book assumes the reader has some famil-
iaritywiththecontentofthestandardundergraduatecoursesinalgebraandpoint-set
topology. One topic that is not always a part of a first point-set topology course but
which is quite important for Algebraic Topology is quotient spaces, or identification
spacesastheyaresometimescalled. GoodsourcesforthisarethetextbooksbyArm-
strongandJ¨anichlistedintheBibliography.
Abooksuchasthisone, whoseaimistopresentclassicalmaterialfromafairly
classicalviewpoint,isnottheplacetoindulgeinwildinnovation. Neverthelessthereis
onenewfeatureoftheexpositionthatmaybeworthcommentingupon,eventhough
in the book as a whole it plays a relatively minor role. This is a modest extension
of the classical notion of simplicial complexes, which we call (cid:209) complexes. These
have made brief appearances in the literature previously, without a standard name
emerging. The idea is to weaken the condition that each simplex be embedded, to
require only that the interiors of simplices are embedded. (In addition, an ordering
of the vertices of each simplex is also part of the structure of a (cid:209) complex.) For
example, if one takes the standard picture of the torus as a square with opposite
edgesidentifiedanddividesthesquareintotwotrianglesbycuttingalongadiagonal,
thentheresultisa (cid:209) complexstructureonthetorushaving2triangles,3edges,and
1 vertex. By contrast, it is known that a simplicial complex structure on the torus
must have at least 14 triangles, 21 edges, and 7 vertices. So (cid:209) complexes provide
a significant improvement in efficiency, which is nice from a pedagogical viewpoint
since it cuts down on tedious calculations in examples. A more fundamental reason
for considering (cid:209) complexes is that they just seem to be very natural objects from
the viewpoint of Algebraic Topology. They are the natural domain of definition for
simplicialhomology,andanumberofstandardconstructionsproduce (cid:209) complexes
ratherthansimplicialcomplexes,forinstancethesingularcomplexofaspace,orthe
classifyingspaceofadiscretegrouporcategory.
Itistheauthor’sintentiontokeepthisbookavailableonlinepermanently,aswell
aspublishitinthetraditionalmannerforthosewhowanttheconvenienceofabound
copy. Withtheelectronicversionitwillbepossibletocontinuemakingrevisionsand
additions, so comments and suggestions from readers will always be welcome. The
webaddressis:
http://www.math.cornell.edu/˜hatcher
Onecanalsofindherethepartsoftheothertwobooksthatarecurrentlyavailable.
Standard Notations
Rn: n dimensionalEuclideanspace,withrealcoordinates
Cn: complex n space
I (cid:131)(cid:134)0;1(cid:135): theunitinterval
Sn: theunitspherein Rn(cid:130)1,allvectorsoflength1
Dn: theunitdiskorballin Rn,allvectorsoflength (cid:20)1
@Dn (cid:131)Sn−1: theboundaryofthe n disk
11: theidentityfunctionfromasettoitself
q: disjointunion
(cid:25): isomorphism
Z : theintegersmodn
n
A(cid:26)B or B (cid:27)A: set-theoreticcontainment,notnecessarilyproper
Theaimofthisshortpreliminarychapteristointroduceafewofthemostcom-
mon geometric concepts and constructions in algebraic topology. The exposition is
somewhat informal, with no theorems or proofs until the last couple pages, and it
shouldbereadinthisinformalspirit,skippingbitshereandthere. Infact,thiswhole
chaptercouldbeskippednow,tobereferredbacktolaterforbasicdefinitions.
To avoid overusing the word ‘continuous’ we adopt the convention that maps be-
tweenspacesarealwaysassumedtobecontinuousunlessotherwisestated.
Homotopy and Homotopy Type
Oneofthemainideasofalgebraictopologyistoconsidertwospacestobeequiv-
alent if they have ‘the same shape’ in a sense that is much broader than homeo-
morphism. To take an everyday example, the letters of the alphabet can be written
either as unions of finitely many straight and curved line segments, or in thickened
formsthatarecompactsubsurfacesoftheplaneboundedbysimpleclosedcurves.
Ineachcasethethinletterisasubspaceofthethickletter,andwecancontinuously
shrinkthethicklettertothethinone. Anicewaytodothisistodecomposeathick
letter,callit X,intolinesegmentsconnectingeachpointontheouterboundaryof X
toauniquepointofthethinsubletter X,asindicatedinthefigure. Thenwecanshrink
2 Chapter0. SomeUnderlyingGeometricNotions
X to X by sliding each point of X−X into X along the line segment that contains it.
Pointsthatarealreadyin X donotmove.
We can think of this shrinking process as taking place during a time interval
!
0(cid:20)t(cid:20)1,andthenitdefinesafamilyoffunctions f :X X parametrizedby t2I (cid:131)
t
(cid:134)0;1(cid:135), where f (cid:132)x(cid:133) is the point to which a given point x 2 X has moved at time t.
t
Naturallywewouldlike f (cid:132)x(cid:133) todependcontinuouslyonboth t and x,andthiswill
t
be true if we have each x 2 X−X move along its line segment at constant speed so
as to reach its image point in X at time t (cid:131) 1, while points x 2 X are stationary, as
remarkedearlier.
These examples lead to the following general definition. A deformation retrac-
!
tion of a space X onto a subspace A is a family of maps f :X X, t 2 I, such
t
that f (cid:131) 11 (the identity map), f (cid:132)X(cid:133) (cid:131) A, and f jjA (cid:131) 11 for all t. The family f
0 1 t t
shouldbecontinuousinthesensethattheassociatedmap X(cid:2)I!X, (cid:132)x;t(cid:133),f (cid:132)x(cid:133),
t
iscontinuous.
Itiseasytoproducemanymoreexamplessimilartotheletterexamples,withthe
deformation retraction f obtained by sliding along line segments. The first figure
t
belowshowssuchadeformationretractionofaM¨obiusbandontoitscorecircle. The
other three figures show deformation retractions in which a disk with two smaller
opensubdisksremovedshrinkstothreedifferentsubspaces.
In all these examples the structure that gives rise to the deformation retraction
!
canbedescribedbymeansofthefollowingdefinition. Foramap f:X Y,themap-
pingcylinder M isthequotientspaceofthedisjointunion (cid:132)X(cid:2)I(cid:133)qY obtainedby
f
identifyingeach (cid:132)x;1(cid:133)2X(cid:2)I with f(cid:132)x(cid:133)2Y.
X
X£I
M
f (X) f
Y Y
Intheletterexamples,thespace X istheouterboundaryofthethickletter, Y isthe
!
thin letter, and the map f:X Y sends the outer endpoint of each line segment to
its inner endpoint. A similar description applies to the other examples. Then it is a
general fact that a mapping cylinder M deformation retracts to the subspace Y by
f
slidingeachpoint (cid:132)x;t(cid:133) alongthesegment fxg(cid:2)I (cid:26)M totheendpoint f(cid:132)x(cid:133)2Y.
f
Not all deformation retractions arise in this way from mapping cylinders, how-
ever. For example, the thick X deformation retracts to the thin X, which in turn