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1、IntroductiontoDi erentialTopologyMatthewG.BrinDepartmentofMathematicalSciencesStateUniversityofNewYorkatBinghamtonBinghamton,NY13902-6000Spring,1994Contents0.Introduction......................21.Basics........................22.DerivativeandChainrule
2、inEuclideanspaces.........73.Threederivatives...................134.Higherderivatives...................155.Thefullde nitionofdi erentiablemanifold.........176.Thetangentspaceofamanifold.............187.TheInverseFunctionTheorem..............22r8.The
3、Ccategoryanddi eomorphisms...........309.Vector eldsand
ows.................3110.ConsequencesoftheInverseFunctionTheorem.......3711.Submanifolds....................4012.Bumpfunctionsandpartitionsofunity..........43113.TheCmetric...................491
4、4.Thetangentspaceoveracoordinatepatch.........5315.Approximations...................5416.Sard'stheorem...................5517.Transversality....................5718.Manifoldswithboundary................5810.Introduction.Thisisaquicksetofnotesonbasicd
5、i erentialtopology.Itgetssketchierasitgoeson.Thelastfewsectionsareonlytointroducetheterminologyandsomeoftheconcepts.ThesenoteswerewrittenfasterthanIcanreadandmaymakenosenseinspots.WereItodothemagain,the rstfewtopicswouldberearrangedintoadi erentorder
6、.Iamtoldthattherearemanymisprints.Thenotesweredesignedtogiveaquickanddirty,halfsemesterintroductiontodi erentialtopologytostudentsthathad nishedgoingthroughalmostallofTopology:A rstcoursebyJamesR.Munkres.TherearereferencestothisbookasMunkres"inthese
7、notes.Thenoteswerewrittensothatallofthematerialcouldbepresentedbythestudentsinclass.Thisexplainsvariousexhortationstopresenters"thatoccurperiodicallythroughoutthenotes.Icribbedfromthreemainsources:(1)SergeLang,Di erentialmanifolds,AddisonWesley,1972
8、,(2)MorrisW.Hirsch,Di erentialtopology,Springer-Verlag,1976,and(3)MichaelSpivak,Calculusonmanifolds,Benjamin,1965.Thelastisaparticularlyprettybookthatunfortunatelyseemstobeoutofprint.Ialsostolefromafewpagesin(4)JamesR.Munkres,Elementarydi erentialtop
