a concise course in algebraic topology

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1、AConciseCourseinAlgebraicTopologyJ.P.MayContentsIntroduction1Chapter1.Thefundamentalgroupandsomeofitsapplications51.Whatisalgebraictopology?52.Thefundamentalgroup63.Dependenceonthebasepoint74.Homotopyinvariance75.Calculations:π(R)=0andπ(S1)=Z8116.TheBrouwerfixedpointtheorem107.Thefundamentaltheore

2、mofalgebra10Chapter2.CategoricallanguageandthevanKampentheorem131.Categories132.Functors133.Naturaltransformations144.Homotopycategoriesandhomotopyequivalences145.Thefundamentalgroupoid156.Limitsandcolimits167.ThevanKampentheorem178.ExamplesofthevanKampentheorem19Chapter3.Coveringspaces211.Thedefi

3、nitionofcoveringspaces212.Theuniquepathliftingproperty223.Coveringsofgroupoids224.Groupactionsandorbitcategories245.Theclassificationofcoveringsofgroupoids256.Theconstructionofcoveringsofgroupoids277.Theclassificationofcoveringsofspaces288.Theconstructionofcoveringsofspaces30Chapter4.Graphs351.Thed

4、efinitionofgraphs352.Edgepathsandtrees353.Thehomotopytypesofgraphs364.CoversofgraphsandEulercharacteristics375.Applicationstogroups37Chapter5.Compactlygeneratedspaces391.Thedefinitionofcompactlygeneratedspaces392.Thecategoryofcompactlygeneratedspaces40vviCONTENTSChapter6.Cofibrations431.Thedefinition

5、ofcofibrations432.Mappingcylindersandcofibrations443.Replacingmapsbycofibrations454.Acriterionforamaptobeacofibration455.Cofiberhomotopyequivalence46Chapter7.Fibrations491.Thedefinitionoffibrations492.Pathliftingfunctionsandfibrations493.Replacingmapsbyfibrations504.Acriterionforamaptobeafibration515.Fiber

6、homotopyequivalence526.Changeoffiber53Chapter8.Basedcofiberandfibersequences571.Basedhomotopyclassesofmaps572.Cones,suspensions,paths,loops573.Basedcofibrations584.Cofibersequences595.Basedfibrations616.Fibersequences617.Connectionsbetweencofiberandfibersequences63Chapter9.Higherhomotopygroups651.Thedefin

7、itionofhomotopygroups652.Longexactsequencesassociatedtopairs653.Longexactsequencesassociatedtofibrations664.Afewcalculations665.Changeofbasepoint686.n-Equivalences,weakequivalences,andatechnicallemma69Chapter10.CWcomple