Avishek Adhikari Mahima Ranjan Adhikari Basic Topology 2 Topological Groups, Topology of Manifolds and Lie Groups Basic Topology 2 ProfessorMahimaRanjanAdhikari(1944–2021) · Avishek Adhikari Mahima Ranjan Adhikari Basic Topology 2 Topological Groups, Topology of Manifolds and Lie Groups AvishekAdhikari MahimaRanjanAdhikari DepartmentofMathematics InstituteforMathematics,Bioinformatics, PresidencyUniversity InformationTechnologyandComputer Kolkata,WestBengal,India Science(IMBIC) Kolkata,WestBengal,India ProfessorMahimaRanjanAdhikariisdeceased. ISBN978-981-16-6576-9 ISBN978-981-16-6577-6 (eBook) https://doi.org/10.1007/978-981-16-6577-6 MathematicsSubjectClassification:22-XX,51Hxx,55-XX,55Pxx,14F45 ©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature SingaporePteLtd.2022 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,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional claimsinpublishedmapsandinstitutionalaffiliations. ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSingaporePteLtd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore Dedicatedto mygrandparentsNabaKumarandSnehalataAdhikari, mymotherMinatiAdhikari whocreatedmyinterestinmathematics atmyveryearlychildhood: —AvishekAdhikari Dedicatedto mymotherSnehalata Adhikari(1921–2007), onoccasionofherbirthcentenary,whogave methefirstlessonofmathematics: —MahimaRanjanAdhikari Preface TheseriesofthreebooksonBasicTopologyisaprojectbookfundedbytheGovern- ment of West Bengal. It is designed to introduce many variants of a basic course in topology through the study of point-set topology, topological groups, topolog- icalvectorspaces,manifolds,Liegroups,homotopyandhomologytheorieswithan emphasis on their applications in modern analysis, geometry, algebra, and theory ofnumbers.Topicsintopologyarevast.Therangeofitsbasictopicsisdistributed amongdifferenttopologicalsubfieldssuchasgeneraltopology,topologicalalgebra, differential topology, combinatorial topology, algebraic topology, and geometric topology. Each volume of the present book is considered a separate textbook that promotes active learning of the subject highlighting the elegance, beauty, scope, and power of topology. Volume 1 studies metric spaces and general topology. It considersthegeneralpropertiesoftopologicalspacesandtheirmappings.Volume2 considersadditionalstructuresotherthantopologicalstructuresstudiedinVolume1. ThecontentsofVolume2areexpandedinfivechapters.Thechapterwisedescription ofVolume2runsasfollows: Chapter1providesabackgroundonalgebra,topologyandanalysisforasmooth study of this volume. Chapter 2 studies topological groups and topological vector spacesusingtopologyandalgebra.Itprovidesmanyinterestinggeometricalobjects ofinterestandrelatesalgebrawithgeometryandanalysis.Chapter3studiestopolog- icalproblemsofthetheoryofdifferentiablemanifoldsanddifferentiablefunctionson differentiablemanifolds.Theconceptsofdiffeomorphisms,embeddingsandbundles playimportantrolesindifferentialtopology.Themainaimofthischapteristoconvey anelementaryapproachtodifferentialtopologyavoidingalgebraictopology,which is studied in Volume 3. Chapter 3 starts with topological manifolds followed by smoothmanifoldsandsmoothmappingsleadingtodifferentialtopology.Themost basicresultsembedded inthetheoryofdifferentialmanifoldsisessentiallydueto theworkofasingleman,H.Whitney(1907–1989).Whitney’sembeddingtheorems saythateverymanifoldcanbeembeddedinaEuclideanspaceasaclosedsubspace. It implies that any manifold may be considered as a submanifold of a Euclidean space.Sard’stheoremisstudied,anditisappliedtoprovethefundamentaltheorem ofalgebra,andtheBrouwerfixed-pointtheoremwithoutusingthetoolsofalgebraic topology. vii viii Preface Chapter 4 starts with Hilbert’s fifth problem and discusses a study of Lie groupsandLiealgebra.Chapter5presentsabriefhistoryoftheemergenceofthe conceptsleadingtothedevelopmentoftopologicalgroups,andalsoLiegroupsas mathematicaltopicswiththeirmotivations. Thisvolumeiscompleteinitselfanddealswiththeintroductoryconceptsoftopo- logicalalgebra,differentialtopologyandLiegroupsanddiscussescertainclassical problemssuchasthefundamentaltheoremofalgebra,Brouwerfixed-pointtheorem, classification of closed surfaces and the Jordan curve theorem without using the formaltechniquesofalgebraictopology,whicharestudiedinVolume3. Anyone who will study Volume 2 and solve exercises in the book will earn a thoroughknowledgeintopologicalgroupsandvectorspaces(whichformanintegral part of topological algebra general topology) and also knowledge in topology of manifoldsandLiegroups(whichformanintegralpartofdifferentialtopology). The book is a clear exposition of the basic ideas of topology and conveys a straightforward discussion of the basic topics of topology and avoids unnecessary definitionsandterminologies.Eachchapterstartswithhighlightingthemainresults ofthechapterwithmotivationandissplitintoseveralsectionswhichdiscussrelated topicswithsomedegreeofthoroughnessandendswithexercisesofvaryingdegrees ofdifficulties,whichnotonlyimpartadditionalinformationaboutthetextcovered previouslybutalsointroduceavarietyofideasnottreatedintheearliertextswith certain references to the interested readers for more study. All these constitute the basicorganizationalunitsofthebook. TheauthorsacknowledgeHigherEducationDepartmentofGovernmentofWest BengalforsanctionoffinancialsupporttotheInstituteforMathematics,Bioinfor- matics and Computer Science (IMBIC) toward writing this book vide order no. 432(Sanc)/ EH/P/SE/ SE/1G-17/07 dated August 29, 2017, and also to IMBIC, University of Calcutta, Presidency University, Kolkata, India, and Moulana Abul Kalam Azad University of Technology, West Bengal, for providing infrastructure towardimplementingthescheme. The authors are indebted to the authors of the books and research papers listed in the bibliographies at the end of each chapter and are very thankful to Profs. P.Stavrions(Greece),ConstantineUdriste(Romania),AkiraAsada(Japan)andalso tothereviewersofthemanuscriptfortheirscholarlysuggestionsfortheimprovement ofthebook.WearethankfultoMd.KutubuddinSardarforhiscooperationtowards the typesetting of the manuscript and to many UG and PG students of Presidency UniversityandCalcuttaUniversity,andtomanyotherindividualswhohavehelped inproofreadingthebook.Theauthors’thanksarealsoduetoIMBIC,Kolkata,for providingtheauthorswithalibraryandotherfacilitiestowardthemanuscriptdevel- opmentworkofthisbook.Tothosewhosenameshavebeeninadvertentlynotentered, the authors apologize. Finally, the authors acknowledge, with heartfelt thanks, Preface ix the patience and sacrifice of the long-suffering family of the authors, especially Dr.ShibopriyaMitraAdhikariandMasterAvipriyoAdhikari. Kolkata,India AvishekAdhikari June2021 MahimaRanjanAdhikari A Note on Basic Topology—Volumes 1–3 Thetopic“topology”hasbecomeoneofthemostexcitingandinfluentialfieldsof study in modern mathematics because of its beauty and scope. This subject aims tomakeaqualitativestudyofgeometryinthesensethatifonegeometricobjectis continuously deformed into another geometrical object, then these two geometric objects are considered topologically equivalent, called homeomorphic. Topology startswheresetshavesomecohesiveproperties,leadingtodefiningthecontinuity offunctions. The series of three books on Basic Topology is a project book funded by the Government of West Bengal, which is designed to introduce many variants of a basiccourseintopologythroughthestudyofpoint-settopology,topologicalgroups, topologicalvectorspaces,manifolds,Liegroups,homotopyandhomologytheories with an emphasis on their applications in modern analysis, geometry, algebra and theoryofnumbers. Thetopicsintopologyarevast.Therangeofitsbasictopicsisdistributedamong differenttopologicalsubfieldssuchasgeneraltopology,topologicalalgebra,differ- entialtopology,combinatorialtopology,algebraictopologyandgeometrictopology. Eachvolumeofthepresentbookisconsideredasaseparatetextbookthatpromotes active learning of the subject highlighting elegance, beauty, scope and power of topology. BasicTopology—Volume1:MetricSpacesandGeneral Topology This volume majorly studies metric spaces and general topology. It considers the generalpropertiesoftopologicalspacesandtheirmappings.Thespecialstructureof ametricspaceinducesatopologyhavingmanyapplicationsoftopologyinmodern analysis, geometry and algebra. The texts of Volume 1 are expanded into eight chapters. xi