## Winter 2014; Math 592 Course Notes

January 8: Class Policy. Categories

January 10: Categories, Functors, Natural Transformations

January 13: Typing Diagrams in latex. Equivalence of Categories. Groupoids.

January 15: The Method of Algebraic Topology. The Fundamental Groupoid.

January 17: The Fundamental Groupoid. The Effect of Homotopy.

January 22: Based Spaces. Pushout, Limits and Colimits.

January 24: Seifert-Van Kampen Theorem for Fundamental Groupoids. 2-Pushout.

January 27: Pushout vs. 2-pushout of Groupoids. Fundamental group of $S^1$

January 29: Seifert-Van Kampen Theorem for Fundamental Groups. Free Groups.

January 31: Free Groups. Generators and Defining Relations. Algorithmic Aspect. Knots.

February 3: Invariants of Knots. Non-triviality of the Trefoil Knot. Compact Surfaces.

February 5: The Colimit Argument. Fundamental Group of Graphs. Fundamental group of Surfaces.

February 7: Adjoint Functors. Forgetful Functors of Universal Algebras. Abelianization.

February 10: More on Adjoint Functors and Abelianization. Classification of Finitely Generated Abelian Groups.

February 14: Covering Spaces. Homotopy Lifting Property for Fundamental Groupoids.

February 17: Lifting Maps into Covering Spaces with Local Conditions.

February 19: The Category of Covering Spaces and Deck Transformations. The Orbit Category. Fiber Functor.

February 21: Equivalence of Categories of Covering Spaces and $\pi_1$-sets via Fiber Functor.(Classification of Covering Spaces.)

February 24:Classification of Covering Spaces - Finishing the Proof. Geometrization of Surfaces (Statement). Subgroup of a Free Group is Free.

February 26:Free Generators of Subgroups of Free Groups.

February 28: CW Complexes and their Fundamental Groups.

March 10: Singular Homology. Definition, $dd=0$.

March 12: Chain Complex. Chain Map. Exact Sequence.

March 14: Relative Singular Homology. Long Exact Sequence (Exactness Axiom.)

March 17: Eilenberg-Steenrod Axioms. Compactness of Homology.

March 19: Compactness Continued. Reduced Homology. The Homology of a Sphere.

March 21: Chain Homotopy. The Homotopy Axiom.

March 24: The Homotopy Axiom Continued. The Excision Axiom.

March 26: The Excision Axiom Continued.

March 28: The Excision Axiom - Finishing the Proof. The Mayer-Vietoris Sequence.

March 31: The Homology of a CW Complex.

April 2: CW Homology. An Example.

April 4: The Differential in CW Homology. Degree of a Map. Hopf's Theorem.

April 7: The Degree of a Map. Proof of Hopf's Theorem.

April 9: Concluding the Proof of Hopf's Theorem. The Homology of Projective Spaces.

April 11: Euler Characteristic and Trace.

April 14: Lefschetz Fixed Point and Trace Theorems - Statement. Other Contexts.

April 16: Jordan's Separation Theorem in $n$ Dimensions.

April 18: Invariance of Domain.

April 21: The Derived Category of Spaces.