Fundamentals of Algebraic Topology

169 Pages·2014·3.564 MB·English

by  Steven H. Weintraub (auth.)

Graduate Texts in Mathematics
Steven H. Weintraub
Fundamentals
of Algebraic
Topology
Graduate Texts in Mathematics 270
Graduate Texts in Mathematics
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
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AlejandroAdem,UniversityofBritishColumbia,Vancouver,BC,Canada
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RogerE.Howe,YaleUniversity,NewHaven,CT,USA
DavidJerison,MassachusettsInstituteofTechnology,Cambridge,MA,USA
JeffreyC.Lagarias,UniversityofMichigan,AnnArbor,MI,USA
JillPipher,BrownUniversity,Providence,RI,USA
FadilSantosa,UniversityofMinnesota,Minneapolis,MN,USA
AmieWilkinson,UniversityofChicago,Chicago,IL,USA
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Moreinformationaboutthisseriesathttp://www.springer.com/series/136
Steven H. Weintraub
Fundamentals
of Algebraic
Topology
123
StevenH.Weintraub
DepartmentofMathematics
LehighUniversity
Bethlehem,PA,USA
ISSN0072-5285 ISSN2197-5612(electronic)
ISBN978-1-4939-1843-0 ISBN978-1-4939-1844-7(eBook)
DOI10.1007/978-1-4939-1844-7
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Preface
The title of this book, Fundamentalsof Algebraic Topology, summarizes its aims
verywell.
Inwritingthisbookwehaveattemptedtoprovidethereaderwithaguidetothe
fundamentalresultsofalgebraictopology,butwehavenotattemptedtoprovidean
exhaustivetreatment.
Our choice of topics is quite standard for an introductory book on algebraic
topology,butadescriptionofourapproachisinorder.
We begin with a short introductory chapter, with basic definitions. We assume
the readeris alreadyfamiliar with basic notionsfrompoint-settopology,and take
thoseforgrantedthroughoutthebook.
WethendevoteChap.2tothefundamentalgroup,includingacarefuldiscussion
ofcoveringspaces,vanKampen’stheorem,andanapplicationofalgebraictopology
to obtain purely algebraic results on free groups. In general, algebraic topology
involves the use of algebraic methods to obtain topological information, but this
isoneinstanceinwhichthedirectionisreversed.
We then move on to discuss homology and cohomology. Here we follow the
axiomaticapproachpioneeredbyEilenbergandSteenrod.InChap.3weintroduce
thefamousEilenberg-Steenrodaxioms.Indeed,inthischapterweconsiderarbitrary
generalized homology theories and derive results that hold for all of them. Then,
in Chap.4, we specialize to ordinary homology theory with integer coefficients,
andderiveresultsandapplicationsinthissituation,stillproceedingaxiomatically.
Of particular note are such important results as the Brouwer fixed point theorem
andinvarianceof domain,which followfromthe existenceofa homologytheory,
not from the details of its particular construction. Also of particular note is our
introduction of CW complexes and our development of cellular homology, again
fromtheaxioms.
Of course,atsomepointwe mustshowthatahomologytheoryactuallyexists,
and we do that in Chap.5, where we construct singular homology. We deal with
thefullpanoplyhere–homology,cohomology,arbitrarycoefficients,theKünneth
formula,products,andduality.
Manifoldsareaparticularlyimportantclassoftopologicalspaces,andwedevote
Chap.6totheirstudy.
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viii Preface
Finally,inChap.7wegiveashortintroductiontohomotopytheory.
Argumentsinalgebraictopologyinvolveamixtureofalgebraandtopology.But
some arguments are purely algebraic, and, indeed, algebraic topology spawned a
new branch of algebra, homological algebra, to deal with the algebraic issues it
raised.Whileitisnotalwayspossibletocompletelyseparatethetopologyandthe
algebra,inmanyinstancesitis.Inthoseinstances,wefinditadvantageoustodoso,
as it better reveals the logical structure of the subject. Thus we have included the
basicalgebraicconstructionsandresultsinanappendix,ratherthanmixingthemin
withtherestofthetext.
In our discussion of Poincaré duality on manifolds, we need some basic facts
aboutbilinearforms,andwesummarizetheminasecondappendix.
Algebraic topology also spawned the language of categories and functors. We
havetendedto avoidthislanguageinthetext,asitismostly(thoughnotentirely)
superfluousforourpurposeshere.Butitisilluminatinglanguage,andessentialfor
students who wish to go further, and so we have included a third appendix that
introducesit.
Thereareseveralpointswewishtocallthereader’sattentionto.Thefirstisthat
we have not felt compelled to give the proofsof all the theorems. To be sure, we
give most of them (and leave a few of them as exercises for the reader), but we
haveomittedsomethatareparticularlylongortechnical.Forexample,wehavenot
provedvanKampen’stheorem,norhaveweprovedthatsingularhomologysatisfies
the excision axiom. The second is that we have not always stated results in the
maximumgenerality.Forexample,indevelopingproductsinhomologyandcoho-
mologywe haverestrictedthepairsofspacesinvolved,andthecoefficientsofthe
(co)homologygroups,tothesituationsinwhichtheyaremostoftenusedinpractice.
Thethirdisthatwehavehewedtotheaxiomaticfoundationsofthesubject.Since
its inceptionovera centuryago,algebraictopologyhasbuilta vastsuperstructure
onthesefoundations,asuperstructurewehopethestudentwillgoontoinvestigate.
Butwedonotinvestigateithere.Forexample,wesayverylittleabouttechniques
forcomputinghomotopygroups.
However,weprovideashortbibliographywherethereadercanfindthematerial
we omit, as well as materialthat is beyondour scope. (In other words, the reader
whowantstoseethis,orwantstostudyfurtherinalgebraictopology,shouldconsult
thesebooks.)
Our notationandnumberingschemehereareratherstandard,andthereislittle
tobesaid.ButwedowanttopointoutthatweuseA⊆BtomeanthatAisasubset
ofBandA⊂BtomeanthatAisapropersubsetofB.
Lemmas, propositions, theorems, and corollaries are stated in italics, which
clearlydelimitsthemfromthetextthatfollows.Similarly,proofsaredelimitedby
thesymbol(cid:4)(cid:5)attheend.Butthereisusuallynothingtodelimitdefinitions,examples,
andremarks,whicharestatedinroman.Weusethesymbol♦attheendofthesefor
thatpurpose.
Bethlehem,PA,USA StevenH.Weintraub
October2013
Contents
Preface............................................................................. vii
1 TheBasics..................................................................... 1
1.1 Background............................................................. 1
1.2 Homotopy............................................................... 2
1.3 Exercises................................................................ 4
2 TheFundamentalGroup.................................................... 5
2.1 DefinitionandBasicProperties........................................ 5
2.2 CoveringSpaces........................................................ 8
2.3 vanKampen’sTheoremandApplications ............................ 13
2.4 ApplicationstoFreeGroups........................................... 15
2.5 FreeHomotopyClasses................................................ 19
2.6 Some“Bad”Spaces.................................................... 20
2.7 Exercises................................................................ 21
3 GeneralizedHomologyTheory............................................. 23
3.1 TheEilenberg-SteenrodAxioms....................................... 23
3.2 ConsequencesoftheAxioms.......................................... 25
3.3 AxiomsforCohomologyandTheirConsequences................... 30
3.4 Exercises................................................................ 32
4 OrdinaryHomologyTheory................................................ 35
4.1 HomologyGroupsofSpheres,andSomeClassicalApplications ... 35
4.2 CW-ComplexesandCellularHomology.............................. 38
4.3 RealandComplexProjectiveSpaces.................................. 49
4.4 Exercises................................................................ 52
5 SingularHomologyTheory................................................. 55
5.1 DevelopmentoftheTheory............................................ 55
5.2 TheGeometricMeaningofH andH ................................ 61
0 1
5.3 HomologywithCoefficients........................................... 66
5.4 TheKünnethFormula.................................................. 69
ix