Presentation topics, Algebraic topology, Math 4152/9052, Fall 2011

Possible Topics: These are suggestions, but you can also propose other topics. Topics need to be discussed with me and approved. When you meet with me, I can give more information about the topics and can suggest further references. I will encourage the graduate students to choose slightly more challenging topics.

• Applications of homology, Hatcher 2.B. Jordan curve theorem, Alexander horned sphere, invariance of domain, division algebras.
• Applications of homology, Hatcher 2.B. Borsuk-Ulam and the transfer sequence. (There's also a book, Using the Borsuk-Ulam Theorem, by Jiri Matousek.)
• Introduction to cohomology, Hatcher Ch 3. Universal coefficient theorem, ring structure (cup product).
• K(G,1) spaces, Hatcher 1.B.
• Hurewicz Thm, Hatcher 2.A and p. 366-?. Sketch proof of 2A.1, several examples, statement of 4.32, partial converse, more examples.
• Vector fields and Euler characteristic. See, for example, these notes.
• The Lefschetz fixed-point theorem, Hatcher 2C.3. Proof relies on simplicial approximation, so may need to cover that as well.
• Long exact sequence of homotopy groups for a fibration, Hatcher p.375-?. Define fibration, state result, sketch proof, give examples. Can give a direct proof instead of using relative homotopy groups.
• The fundamental groupoid. E.g. 1971 book by Higgins, Categories and groupoids, QA171.H57, and 2006 book by Brown, Topology and groupoids. See also online notes by Baez. Could discuss the generalized van Kampen theorem.
• Classification of surfaces, Massey, GTM 127, Ch 1. (A lot of material, but can be surveyed.)
• Brown representability for generalized cohomology. Hatcher 4.E, but there are probably better sources too. Connections to K(G,1)'s.
• Vector bundles, universal bundles and Grassmanians, e.g. from the first chapter of Hatcher's book on Vector Bundles & K-Theory.
• K-theory as a generalized cohomology theory.
• Introduction to knot theory, e.g. Alexander/Conway polynomial, Jones/HOMFLY polynomial, etc. Kauffman, On Knots; Carlson, Topology of Surfaces, Knots and Manifolds.
• Applications of covering spaces to other topics, such as complex analysis.
• Introduction to group (co)homology, i.e. (co)homology of K(G,1).

Duration: 45-55 minutes for grad students, 40-50 for undergrads. The presentations are not long, so you will need to carefully select the appropriate amount of material to present.

Grading: The presentations will be worth 1/3 of the overall mark in the course. They will be graded on:

• knowledge of material. Be prepared to answer questions.
• organization of material: what you choose to cover, and how you choose to organize it. Don't forget to include motivating examples, history (e.g. attributions and years), and context.
• clarity and style of presentation: speaking clearly, looking at audience, giving clear explanations, etc.
• blackboard use: use boards in order, don't erase what you've just written, don't stand in front of what you've written, use coloured chalk when appropriate, etc.
• duration: if you end within the time span given, you get full marks for this category; otherwise, you lose marks. You might want to build some flexibility into the end of your presentation so you can adjust on the fly.

Timeline:

• In mid-October, look over topics and read about a couple of them.
• Meet with me by Nov 4 to discuss topics and select a date.
• Finalize choice of your topic ≥ 3 weeks ahead of your date.
• Give me an outline (1 to 2 pages) ≥ 2 weeks ahead of your date.
• Give me a draft of the whole talk ≥ 1 week ahead of your date. Indicate which parts you will say and which parts you will write on the board.