Basic Topology 2: Topological Groups, Topology of Manifolds and Lie Groups

385 Pages·2022·3.267 MB·English

by  Avishek Adhikari, Mahima Ranjan Adhikari

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
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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