[Botvinnik]notes on the course algebraic topology

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1、NOTESONTHECOURSE“ALGEBRAICTOPOLOGY”BORISBOTVINNIKContents1.Importantexamplesoftopologicalspaces51.1.Euclidianspace,spheres,disks.51.2.Realprojectivespaces.61.3.Complexprojectivespaces.71.4.Grassmannmanifolds.81.5.Flagmanifolds.91.6.ClassicLiegroups.91.7.Stiefelmanifolds.101.

2、8.Surfaces.112.Constructions132.1.Product.132.2.Cylinder,suspension132.3.Glueing142.4.Join162.5.Spacesofmaps,loopspaces,pathspaces162.6.Pointedspaces173.Homotopyandhomotopyequivalence203.1.Definitionofahomotopy.203.2.Homotopyclassesofmaps203.3.Homotopyequivalence.203.4.Retrac

3、ts233.5.Thecaseof“pointed”spaces244.CW-complexes254.1.Basicdefinitions254.2.SomecommentsonthedefinitionofaCW-complex274.3.OperationsonCW-complexes274.4.MoreexamplesofCW-complexes284.5.CW-structureoftheGrassmanianmanifolds285.CW-complexesandhomotopy335.1.Borsuk’sTheoremonextens

4、ionofhomotopy335.2.CellularApproximationTheorem355.3.CompletionoftheproofofTheorem5.5375.4.Fightingaphantom:ProofofLemma5.6375.5.BacktotheProofofLemma5.6395.6.FirstapplicationsofCellularApproximationTheorem40Date:9-14-2010.12BORISBOTVINNIK6.Fundamentalgroup436.1.Generaldefini

5、tions436.2.Onemoredefinitionofthefundamentalgroup446.3.Dependenceofthefundamentalgrouponthebasepoint446.4.Fundamentalgroupofcircle446.5.FundamentalgroupofafiniteCW-complex466.6.TheoremofSeifertandVanKampen497.Coveringspaces527.1.Definitionandexamples527.2.Theoremoncoveringhomot

6、opy527.3.Coveringspacesandfundamentalgroup537.4.Observation547.5.Liftingtoacoveringspace547.6.Classificationofcoveringsovergivenspace567.7.Homotopygroupsandcoveringspaces577.8.Lensspaces588.Higherhomotopygroups608.1.Moreabouthomotopygroups608.2.Dependenceonthebasepoint608.3.R

7、elativehomotopygroups619.Fiberbundles659.1.Firststepstowardfiberbundles659.2.Constructionsofnewfiberbundles679.3.Serrefiberbundles709.4.Homotopyexactsequenceofafiberbundle739.5.Moreonthegroupsπn(X,A;x0)7510.SuspensionTheoremandWhiteheadproduct7610.1.TheFreudenthalTheorem7610.2.F

8、irstapplications8010.3.AdegreeofamapSn→Sn8010.4.Stablehomotopygroupsofspher