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CompactTextbooksinMathematics Thistextbookseriespresentsconciseintroductionstocurrenttopicsinmathematics and mainly addresses advanced undergraduates and master students. The concept is to offer small books covering subject matter equivalent to 2- or 3-hour lectures orseminarswhicharealsosuitableforself-study.Thebooksprovidestudentsand teachers withnew perspectives and novel approaches. They may feature examples andexercisestoillustratekeyconceptsandapplicationsofthetheoreticalcontents. The series also includes textbooks specifically speaking to the needs of students fromotherdisciplinessuchasphysics,computerscience,engineering,lifesciences, finance. • compact:smallbookspresentingtherelevantknowledge • learning made easy: examples and exercises illustrate the application of the contents • useful for lecturers: each title can serve as basis and guideline for a semester course/lecture/seminarof2–3hoursperweek. Holger Kammeyer Introduction to Algebraic Topology HolgerKammeyer MathematicalInstitute HeinrichHeineUniversityDüsseldorf Düsseldorf,Germany This textbook has been reviewed and accepted by the Editorial Board of Mathematik Kompakt, the Germanlanguageversionofthisseries. ISSN2296-4568 ISSN2296-455X (electronic) CompactTextbooksinMathematics ISBN978-3-030-98312-3 ISBN978-3-030-98313-0 (eBook) https://doi.org/10.1007/978-3-030-98313-0 MathematicsSubjectClassification:55-XX,55-01,18-XX ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerland AG2022 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered companySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Thisbookintroducesthereadertothetwomostfundamentalconceptsofalgebraic topology:thefundamentalgroupandhomology.Weshalltakeamodernviewpoint so that we begin the course by studying basic notions from category theory. The fundamental group is afterwards treated as a special case of the fundamental groupoid. Accordingly, we first prove van Kampen’s theorem in a categorical versionduetoR.Brownandthenexplainhowtoactuallycomputethefundamental groupofanattachingspace.Wemoveontopresenthomology.Toconveytheidea, we construct simplicial homology and motivate the Eilenberg–Steenrod axioms of a homology theory. Next, we construct singular homology and show that it satisfies the axioms. Afterwards, we develop machinery for computing homology theories,giveexamplecalculations,andseesomeapplicationssuchastheBrouwer fixed point theorem and the Borsuk–Ulam theorem. Finally, we introduce cellular homologyforCWcomplexes.Astheconcludingresult,weshowthattheEilenberg SteenrodaxiomsdetermineordinaryhomologyonCWcomplexes. We assume the reader has taken an introductory course on topology and is familiar with point set topological concepts, the definition of the fundamental group,andcoveringtheory.Suchacoursewillcertainlyhavecoveredthequotient topology,butsincethisconceptisoffundamentalimportanceforourpurposes,we have decided to include a recap in Appendix A. Elementary algebra will likewise be applied throughout the text. In particular, we will work with modules over commutative rings and on rare occasions also with their tensor products. Each chapterendswithacoupleofexercises.Someofthemarenotonlymeanttoprovide a test ground for working with the new concepts but they also establish additional factsandterminologywhicharehelpfultoknowinthegivencontext. A myriad of excellent books on algebraic topology are available in the market. Some texts, for example, [29], choose a formal presentation and are well suited to continue one’s curriculum with a course on homotopy theory. Others, like [8] are more geometrically minded and might be a better choice for subsequent specialization in low-dimensional topology. What this text tries to accomplish is toneithershyawayfromabstractconceptsnorfromprovidinggeometricintuition or doing easy calculations, and at the same time do justice to the series and be a compact textbook: We present only as much material as we found reasonable to cover in a first semester graduate course on algebraic topology. The six chapters are divided into five sections each, so that in a typical 15-week semester with two v vi Preface meetings a week, one should cover one section per lecture on average. This is howeveraroughestimateassomesectionsaremoresubstantialthanotherssothat theblackboardpresentationwillneedsomeshortcuts.However,weurgethelecturer nottotryandskipthetechnicalappearingSect.2.3oncofibrationsandhomotopy pushouts.Itsecretlyprovidesasmuchhomotopytheoryaswedeemnecessaryfor thecorrectpresentationoftheresultsintheensuingchapters. Asanotherremarktothelecturer,letmepointoutawell-knowndilemmawhen aiming for the uniqueness theorem of ordinary homology. At some moment, one will have to know that π (Sn) ∼= Z or more precisely that [Sn,Sn] ∼= Z meaning n homotopy classes of maps Sn → Sn are classified by degree. This theorem just has no quick and easy proof. It is interesting to see how other introductory texts on algebraic topology circumvent this problem. For example, tom Dieck in his monograph [29] develops homotopy theory first, before introducing homology, so that the fact π (Sn) ∼= Z is available once it is needed. Hatcher in [8] takes the n more classic route of treating homology first and simply waits with the proof of theuniquenesstheoremuntilafterdevelopinghomotopytheory[8,Theorem4.59]. Lück advances quickly to the uniqueness theorem [18, Satz 3.53] by taking the Freudenthal suspension theorem for granted. In this text, we suggest yet another road to resolve the issue and prove the simplicial approximation theorem as part of the chapter on simplicial homology which allows us to show π (Sn) ∼= Z by n a lemma given in [4, Lemma 11.13]. This has the virtue that the introduction of simplicialcomplexesservesmorethanonlyadidacticpurpose. Theusedbackgroundsourcesareasfollows:Chaps.1and2looselyfollowthe presentationin[19,Chapter2],thoughChap.1givesamoreextensiveintroduction tocategoricalconcepts,andSect.2.3incorporatesmaterialappearingin[4,26,29]. Chapter 3 is partly based on [8] with the section on simplicial approximation drawingfrom[20].ThematerialofChap.4followsthedefaulttreatmentandcanfor examplebefoundin[8].ReferencesforChap.5areagain[4]and[8],thoughsome proofs are adapted to the more formal notion of cofibration instead of Hatcher’s “goodpairs.”ThemainreferenceforthefinalChap.6is[18],butsomeproofshave beenrevisedconsiderably. Additionally,IwanttothankRomanSauerforprovidingmewithhishandwritten lecture notes [21] which have served as an overall fundament of the course. I am moreover indebted to David Bückel for taking live LATEXnotes when I first taught thematerialofthiscoursesothatIcouldthereaftersimplyextendhisnotestothe text at hand. Finally, I am grateful to Moritz Kerz for suggesting the inclusion of thesenotesintotheBirkhäuserCompactTextbooksinMathematicsseries.Without anyoneofthesethree,thebookwouldnothavecomeintobeing. Düsseldorf,Germany HolgerKammeyer December5,2021 Contents 1 BasicNotionsofCategoryTheory ......................................... 1 1.1 Categories ............................................................... 1 1.2 Functors ................................................................. 4 1.3 NaturalTransformations................................................ 6 1.4 Adjunction............................................................... 9 1.5 LimitsandColimits..................................................... 11 Exercises....................................................................... 29 2 FundamentalGroupoidandvanKampen’sTheorem................... 33 2.1 TheFundamentalGroupoid ............................................ 33 2.2 VanKampen’sTheorem ................................................ 35 2.3 CofibrationsandHomotopyPushouts.................................. 41 2.4 ComputingFundamentalGroups....................................... 53 2.5 HigherHomotopyGroups.............................................. 56 Exercises....................................................................... 58 3 Homology:IdeasandAxioms .............................................. 59 3.1 TheIdeaofHomology.................................................. 59 3.2 SimplicialHomology ................................................... 61 3.3 RelativeSimplicialHomologywithCoefficients...................... 66 3.4 TheEilenberg–SteenrodAxiomsforHomology ...................... 71 3.5 SimplicialApproximation.............................................. 73 Exercises....................................................................... 77 4 SingularHomology .......................................................... 79 4.1 TheDefinitionofSingularHomology ................................. 79 4.2 TheLongExactSequenceofaPairofSpaces......................... 81 4.3 HomotopyInvariance................................................... 84 4.4 Excision ................................................................. 88 4.5 SingularHomologyinDegreeZeroandOne.......................... 96 Exercises....................................................................... 100 5 Homology:ComputationsandApplications.............................. 103 5.1 Relativevs.AbsoluteHomology....................................... 103 5.2 SimplicialandSingularHomologyAgree............................. 110 5.3 TheMayer–VietorisSequence ......................................... 114 vii viii Contents 5.4 Degree................................................................... 120 5.5 Applications............................................................. 126 Exercises....................................................................... 132 6 CellularHomology........................................................... 135 6.1 CWComplexes.......................................................... 135 6.2 CellularHomologyandEulerCharacteristic .......................... 146 6.3 ComputingCellularHomology ........................................ 155 6.4 UniquenessofOrdinaryHomology.................................... 160 6.5 HowtoProceed ......................................................... 164 Exercises....................................................................... 167 A QuotientTopology ........................................................... 169 Bibliography...................................................................... 173 ListofNotation................................................................... 175 Index............................................................................... 179 1 Basic Notions of Category Theory Let us start the text with an easy question: Is the two dimensional sphere S2 homeomorphic to the two dimensional torus T2? Both spaces are connected, compact 2-dimensional manifolds, and hence indistinguishable from the point set topologicalpointofview.Buttherearecompellingintuitiveideaswhythesespaces shouldbedistinct:ConsiderarubberbandinT2fixedatsomepointx ∈T2.Ifthis 0 rubberbandisembeddedinsuchawaythatitwindsoncearoundthe“hole”inthe torus, as pictured on the right in Fig. 1.1, there seems to be no way whatsoever to continuouslydeformthisbandwithinT2 tothepointx .Inthesphere,however,it 0 appearstobeaneasytasktocontractarubberbandtoapoint,nomatterhowitis initiallyembedded. Algebraic topology is the art of making these thoughts precise. If S2 was homeomorphic to T2, then we would have π (S2,x ) ∼= π (T2,x ). However, 1 0 1 0 π (S2,x )={1}andπ (T2,x )∼=Z×Z.SoS2isnothomeomorphictoT2.Here, 1 0 1 0 thefundamentalgroupdefinesafunctorfromthecategoryof(pointed)topological spaces to the category of groups. To a large extent, algebraic topology is about constructingfunctorsfromcategoriesoftopologicalspacestoalgebraiccategories likegroups,abeliangroups,K-vectorspaces,andR-modules. Category theory provides vocabulary to formulate the transition of topological questions into algebraic problems in a precise and consistent manner. This is why ithaslongbecomethegoldstandardtodevelopalgebraictopologyinitsterms.In thisfirstchapterwepresentpreciselyasmuchofthislanguageasweshallemploy inthecourse. 1.1 Categories In the first few semesters of studying math, one realizes that many constructions and arguments pop up repeatedly in different contexts. For instance, products are definedinvirtuallythesameway,nomatterwhetherwearedealingwithproductsof ©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2022 1 H.Kammeyer,IntroductiontoAlgebraicTopology,CompactTextbooks inMathematics,https://doi.org/10.1007/978-3-030-98313-0_1