September 9: Class policy, Singular Chains
September 11: The singular chain complex. Categories, functors, natural transformations.
September 14: Functoriality of singular homology. Relative homology. Contravariant functors and singular cohomology.
September 16: Cohomology of pairs. Eilenberg-Steenrod axioms for homology and cohomology. Based spaces. Reduced homology and cohomology.
September 18: Comments on the exactness and homotopy axioms. The homology and cohomology of a sphere. Stability. CW complexes.
September 21: CW homology and cohomology. The CW structure on a complex projective space..
September 23: The differential in CW homology and cohomology. The homology and cohomology of the real projective space..
September 25: The universal coefficient theorem.
September 28: The CW structure of a complex quadric. Adjoint functors. Derived categories..
September 30: Dearived Category of a small category. Derived functors..Left Kan extension.
October 2: The homology of a complex projective quadric II. Riiht Kan extensions.Categorical limits and colimits.
October 5: Ends and coends. Simplicial sets. The derived category of R-modules I.
October 7: Derived categories.
October 9: :Dearived functors from "resolultions"
October 12: Whitehead theorem for chain complexes, Tor and Ext: :
October 14: Right derived funtors via injective resolutions: The tensor product and Hom of chain complexes.
October 16: Mapping cylinder of spaces and chain complexes. The co-Whitehead theorem for chain complexes. Homotopy pushout..
October 21: Symmetry of Tor and Ext by resolving in either variable. The derived category of spaces..
October 23: Mapping cocylinder and homotopy pullback of spaces. Whitehead theorem for spaces.CW approximation of spaces...
October 26:Weak equivalences preserve singular homology. Hurewicz Theorem...
October 28:The long exact sequence of homotopy groups..
October 30:Cofibrations and fibrations. Mapping cone. Relative cohomology and cofibrations...
November 2:Fibrations. Examples of homotopy groups of spheres. Simplicial sets...
November 4:Gluing fibrations. Products of simplicial sets and Milnor's theorem..
November 6:The Kan condition. Approximation and co-Whitehead theorem for simplicial sets...
November 9:Eilenberg-Zilber lemma. Co-Whitehead for simplicial sets. Simplicial abelian groups. Dold-Puppe theorem and Eilenberg-MacLane spaces..
November 11:Eilenberg-Zilber Theorem. Acyclic models.
November 13:Kunneth Theorem. Exact couples and spectral sequences. Kunneth spectral sequence..
November 18:Singular cohomology of a locally contractible space via sheaves..
November 20:Sheaves of simplicial sets. Grothendieck topology. .
November 23:Symmetric monoidal category. Strong duality. .
November 25:Spanier-Whitehead duality..
November 30:Cup product.Alexander duality..
December 2: Poincare duality..
December 4: Comments on Poincare duality and its applications..Vector bundles.
December 7: Comments on vector bundles. Spectra.
December 9: Approximation and Whitehead theorem for May spectra.
December 11: Stability. Smash product of spectra. Spanier-Whitehead duality in spectra. Generalized homology and cohomology of based spaces.
December 14: Examples of generalized homology and cohomology theories.