## Fall 2015; Math 695 Course Notes

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 16:Sheaves..

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.