algebraic topology a computational approach - kaczynski , mischaikow , mrozek

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1、AlgebraicTopology:AComputationalcApproachT.KaczynskiUniversitedeSherbrookeK.MischaikowGeorgiaInstituteofTechnologyM.MrozekJagellonianUniversityApril18,20002Contents1Introduction71.1BasicNotionsfromTopology...................91.1.1TopologicalSpaces....................91.1.2ContinuousMaps.....

2、................161.1.3Connectedness.......................221.2LinearAlgebra..........................241.2.1Fields...........................251.2.2VectorSpaces.......................271.2.3LinearMaps........................321.2.4QuotientSpaces......................332MotivatingExamples372

3、.1Topology..............................372.1.1Homotopy.........................382.1.2Graphs...........................412.1.3APreviewofHomology.................432.1.4ZHomologyofGraphs.................5022.2ApproximationofMaps.....................562.2.1ApproximatingMapsonanInterval........

4、...562.2.2ConstructingChainMaps................622.2.3MapsoftheCircle....................683AbelianGroups773.1Groups...............................773.2ProductsandSums........................833.3Quotients.............................873.4Homomorphisms.........................893.5MatrixAlgeb

5、raoverZandNormalForm............9434CONTENTS3.6DecompositionTheoremforAbelianGroups..........1013.7HomologyGroups.........................1054CubicalHomology1094.1CubicalSets............................1104.1.1ElementaryCubes....................1104.1.2RepresentableSets....................113

6、4.1.3CubicalSets........................1184.2TheAlgebraofCubicalSets...................1224.2.1CubicalChains......................1224.2.2TheBoundaryOperator.................1284.2.3HomologyofCubicalSets................1324.3H(X)...............................13704.4ElementaryCollapses......

7、.................1404.5AcyclicCubicalSpaces......................1474.6ReducedHomology........................1554.7ComparisonwithSimplicialHomology.............1584.7.1Simplexesandtriangulations...............1584.7.2SimplicialHomology......