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Algebraic Topology


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Table of Contents

1 Set theory.- 2 General topology.- 3 Group theory.- 4 Modules.- 5 Euclidean spaces.- 1 Homotopy and The Fundamental Group.- 1 Categories.- 2 Functors.- 3 Homotopy.- 4 Retraction and deformation.- 5 H spaces.- 6 Suspension.- 7 The fundamental groupoid.- 8 The fundamental group.- Exercises.- 2 Covering Spaces and Fibrations.- 1 Covering projections.- 2 The homotopy lifting property.- 3 Relations with the fundamental group.- 4 The lifting problem.- 5 The classification of covering projections.- 6 Covering transformations.- 7 Fiber bundles.- 8 Fibrations.- Exercises.- 3 Polyhedra.- 1 Simplicial complexes.- 2 Linearity in simplicial complexes.- 3 Subdivision.- 4 Simplicial approximation.- 5 Contiguity classes.- 6 The edge-path groupoid.- 7 Graphs.- 8 Examples and applications.- Exercises.- 4 Homology.- 1 Chain complexes.- 2 Chain homotopy.- 3 The homology of simplicial complexes.- 4 Singular homology.- 5 Exactness.- 6 Mayer-Vietoris sequences.- 7 Some applications of homology.- 8 Axiomatic characterization of homology.- Exercises.- 5 Products.- 1 Homology with coefficients.- 2 The universal-coefficient theorem for homology.- 3 The Kunneth formula.- 4 Cohomology.- 5 The universal-coefficient theorem for cohomology.- 6 Cup and cap products.- 7 Homology of fiber bundles.- 8 The cohomology algebra.- 9 The Steenrod squaring operations.- Exercises.- 6 General Cohomology Theory and Duality.- 1 The slant product.- 2 Duality in topological manifolds.- 3 The fundamental class of a manifold.- 4 The Alexander cohomology theory.- 5 The homotopy axiom for the Alexander theory.- 6 Tautness and continuity.- 7 Presheaves.- 8 Fine presheaves.- 9 Applications of the cohomology of presheaves.- 10 Characteristic classes.- Exercises.- 7 Homotopy Theory.- 1 Exact sequences of sets of homotopy classes.- 2 Higher homotopy groups.- 3 Change of base points.- 4 The Hurewicz homomorphism.- 5 The Hurewicz isomorphism theorem.- 6 CW complexes.- 7 Homotopy functors.- 8 Weak homotopy type.- Exercises.- 8 Obstruction Theory.- 1 Eilenberg-MacLane spaces.- 2 Principal fibrations.- 3 Moore-Postnikov factorizations.- 4 Obstruction theory.- 5 The suspension map.- Exercises.- 9 Spectral Sequences and Homotopy Groups of Spheres.- 1 Spectral sequences.- 2 The spectral sequence of a fibration.- 3 Applications of the homology spectral sequence.- 4 Multiplicative properties of spectral sequences.- 5 Applications of the cohomology spectral sequence.- 6 Serre classes of abelian groups.- 7 Homotopy groups of spheres.- Exercises.

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