Presentation topics, Algebraic topology, Math 4152/9052

All students are expected to attend all presentations and to arrive on time.

Possible Topics: These are suggestions, but you can also propose other topics. Topics need to be discussed with me and approved. You should choose a topic that is not something you already know about. When you meet with me, I can give more information about the topics and can suggest further references. You should also do some research about the topics. Graduate students should choose 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 + exercises. 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), applications.
• 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.
• Relationship between singular cohomology and de Rham cohomology. E.g., John M. Lee, Introduction to smooth manifolds, Chapters 17 and 18.
• Vector fields and Euler characteristic: the Poincare-Hopf theorem. Hatcher explains how Euler characteristic can be computed using homology. These notes give a brief intro into vector fields and Euler char. Milnor's book Topology from the differential viewpoint explains why all vector fields have the same total index. Milnor's book Morse theory, and other sources, explain why a certain vector field has as its index the Euler char. Sections 11 and 12 of From Calculus to Cohomology, by Madsen and Tornehave, also explain the Poincare-Hopf theorem.
• 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 and the van Kampen theorem. E.g. 1971 book by Higgins, Categories and groupoids, QA171.H57, and 2006 book by Brown, Topology and groupoids. These notes by Camarena give a nice overview, but don't give full details.
• 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, or from Characteristic Classes, by Milnor and Stasheff. Or a talk about vector bundles and characteristic classes, based on either book. Hatcher gets to the construction more directly, and without relying on Steenrod squares.
• K-theory as a generalized cohomology theory.
• Introduction to knot theory, e.g. Alexander/Conway polynomial, Jones/HOMFLY polynomial, etc. Kauffman, On Knots, and 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). Weibel, An introduction to homological algebra, Ch 6 (especially Sections 6.1 and 6.10), and Benson, Representations and cohomology, vol II, Sections 2.1, 2.2.

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. You should focus on the key ideas, with illustrative examples, motivation, necessary background, and history (e.g. attributions and years). You aren't expected to prove everything, but should give one or two short proofs. It should be regarded more like a seminar talk than a course lecture.

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

• Outline and draft: Well-organized; appropriate choice of topics and amount of material; done on time.
• Knowledge of material. Be prepared to answer questions.
• 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, use the side board for things you want to leave up, 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. And take into account that there may be questions during your talk.

Timeline:

• In mid-February, look over topics and read about a couple of them.
• Meet with me after that to discuss topics and select a date. Bring two possible choices of topic when we meet. Talks will take place during the last two weeks of classes.
• Finalize choice of your topic ≥ 3 weeks ahead of your date.
• Give me a brief 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.