This semester schedule is from Andrew Snowden’s Undergraduate Seminar in Topology at MIT.
February
W Feb 2 Andrew Organizational meeting F Feb 4 Andrew Introduction to the fundamental group M Feb 7 Scott Paths and homotopies Umut The fundamental group W Feb 9 Kyle The fundamental group of the circle JJ Applications of previous lecture F Feb 11 Marcel Contractible and simply connected spaces Danny The fundamental group of a product M Feb 14 John Functoriality of the fundamental group Noah Homotopy equivalences W Feb 16 Rafael The fundamental group of S1 ∨ S1 Gabriel Amalgamated free products F Feb 18 Aldo van Kampen's theorem Andrew van Kampen's theorem (continued) M Feb 21 President's Day, no class. But there is class tomorrow. T Feb 22 Andrew Introduction to covering spaces W Feb 23 Rafael The universal cover JJ The universal cover (continued) F Feb 25 Noah Lifting properties pdf Gabriel Lifting properties (continued) M Feb 28 Danny Existence of covers Umut The Galois correspondence
March
W Mar 2 Kyle Category theory Marcel The Galois correspondence in categorical form F Mar 4 John Covering spaces of S1 ∨ S1 Aldo Quotients by finite groups M Mar 7 Andrew Overview of homology W Mar 9 Scott Chains and the boundary operator Noah Definition of homology and first calculations F Mar 11 John Chain complexes Rafael Functoriality of homology M Mar 14 Aldo The long exact sequence Danny Relative homology W Mar 16 Scott Excision JJ Homology of a quotient F Mar 18 Umut Proof of Prop. 2.21, part 1 Gabriel Proof of Prop. 2.21, part 2 M Mar 21 Spring break, no class. W Mar 23 Spring break, no class. F Mar 25 Spring break, no class. M Mar 28 Andrew Review Andrew Naturality of connecting homomorphisms W Mar 30 Kyle Axioms for homology Rafael The Mayer–Vietoris sequence
April
F Apr 1 Andrew Homology with coefficients Danny The universal coefficient theorem M Apr 4 Noah CW complexes, part 1 Umut CW complexes, part 2 W Apr 6 JJ CW homology, part 1 Scott CW homology, part 2 F Apr 8 Aldo Definition of cohomology Kyle Overview of formal properties M Apr 11 John The cup product, part 1 Gabriel The cup product, part 2 W Apr 13 Umut The Kunneth formula, part 1 Andrew The Kunneth formula, part 2 F Apr 15 Andrew Overview of Poincare duality Kyle Orientations M Apr 18 Patriot's Day, no class. W Apr 20 JJ The fundamental class, part 1 Rafael The fundamental class, part 2 F Apr 22 Andrew Direct limits Scott Cohomology with compact support M Apr 25 Aldo The cap product Danny Statement of Poincare duality W Apr 27 John Proof of Lemma 3.36 Noah Proof of Poincare duality F Apr 29 Andrew Closing lecture
May
M May 2 John Sheaf cohomology Noah Morse theory W May 4 JJ Hopf fibrations Aldo The Gauss—Bonnet theorem F May 6 Scott K-theory Umut De Rham cohomology M May 9 Danny The Hurewicz isomorphism Rafael Orbifold fundamental groups W May 11 Kyle Simplicial sets
