On Introduction to Algebraic Topology

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1、LecturesonIntroductiontoAlgebraicTopologyByG.deRhamNotesbyV.J.LalNopartofthisbookmaybereproducedinanyformbyprint,microfilmoranyothermeanswith-outwrittenpermissionfromtheTataInstituteofFundamentalResearch,Colaba,Bombay5TataInstituteofFundamentalResearch

2、,Bombay1969PrefaceThesearenotesofapartoflectureswhichIgaveattheTataIn-stituteofFundamentalResearchin1966.TheywereintentedasafirstintroductiontoalgebraicTopology.MythanksareduetoV.J.Lalforhiscarefulpreparationofthesenotes,andtotheTataInstituteofFundamen

3、talResearchforitskindinvitation.G.deRhamContents1Definitionandgeneralpropertiesofthefundamentalgroup12Freeproductsofgroupsandtheirquotients.......73Oncalculationoffundamentalgroups..........184Examples.........................265Thegroupofatamelinkgive

4、nbyagoodplaneprojection316Antoine'sNecklace....................347Elementaryideals-Alexanderpolynomials........398Constructionof3-manifolds...............449InvolutionsofS4.........................55iii1.Definitionandgeneralpropertiesofthefundamentalgr

5、oup11Definitionandgeneralpropertiesofthefunda-mentalgroupWecallacontinuousmapsimplyamap.WedenotethereallinebyR1andtheunitintervalbyI:I={t

6、t∈R,0≤t≤1}.WeconsidermapsofIintoatopologicalspaceXandsaythatamapf1isequivalenttoamapf2ifthereexistsanautomorphismϕ

7、ofIfixingendpointssuchthatf1=f2◦φ.ThisisanequivalencerelationinthesetofallmapsofIintoX.AnequivalenceclassunderthisrelationiscalledapathinX.WesaythatamapfdefinesapathWiffbelongstotheequivalenceclassW.Clearlyanytwomapsdefiningthesamepathmap0ontothesamepoin

8、tandalso1.Theimagesof0and1arecalledtheinitialandtheterminalpointofthepath.Wesaythatapathconnectsapointxtoapointyifxandyaretheinitialandterminalpointsofthepath.SupposethatamapfdefinesapathWconnectingx0tox1.Thenthemapf−1(t)=f(1−t),0≤t≤1definesapathW−1whic

9、hcanbeseentobedependentonlyonW.Thispathconnectingx1tox0iscalledthepathWdescribedintheoppositesenseorthereversedpathofW.IfCisaJordanarc,i.e.asethomeomorphictoI,therearetwopathsdefinesbyhomeomorphismofIontoC,eachreversedtotheother,andcorrespondin