Algebraic Topology

Algebraic Topology I

I Homology Theory

• Singular homology -- definition, simple computations
• Cellular homology -- definition
• Eilenberg-Steenrod Axioms for homology
• Computations: Sⁿ, RPⁿ, CPⁿ, Tⁿ, S²^S³, Grassmannians, X*Y
• Alexander duality -- Jordan curve theorem and higher dimensional analogues
• Applications: Winding number, degree of maps, Brouwer fixed point theorem
• Lefschetz fixed point theorem

II Homotopy Theory

• Homotopy of maps, of pointed maps
• The homotopy category and homotopy functors --examples
• π₁(X, x₀)
• Van Kampen's theorem
• Higher homotopy groups and the Hurewicz theorem
• π₃(S²)
• Higher homotopy groups of the sphere

III Covering Spaces

• Definition of a covering projection
• Examples -- Coverings of S¹, Sⁿ covering RPⁿ, Spin(n) covering SO(n)
• Homotopy path lifting
• Classification of coverings of a reasonable space

IV Homology with Local Coefficients

• Local coefficient systems
• Relation with covering spaces
• Obstruction theory
• The Alexander polynomial of a knot

Algebraic Topology II

I Cohomology

• Cup products
• Pairings homology
• Cohomology and homology with coefficients
• Universal coefficient theorems

II Cech Cohomology

• Open coverings and Cech cochains
• The coboundary mapping
• Cech cohomology
• Comparison with singular cohomology

III Selected Topics

• Group Cohomology
• Sheaf Cohomology
• de Rham's theorem
• Morse functions and Poincaré duality for manifolds
• Thom Isomorphism Theorem and cohomology classes Poincaré dual to cycles