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. You should choose a topic that is not something you already know about. I must approve all choices. Please ask if you want more references about a topic. 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 or ring structure (cup product), applications. Hatcher proves the UCT directly in about five pages. Other sources, such as the book by Davis and Kirk, first set up the framework to make the argument more conceptual.
• 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. Also, Bott-Tu, Differential forms in algebraic topology, introduces de Rham cohomology in a very direct way in Sections 1 and 2, and proves the Poincare lemma and homotopy invariance in Section 4. Not a great source for the de Rham theorem, though.
• 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, Sections 1-8. (A lot of material, but can be surveyed.)
• Riemann surfaces and covering spaces. E.g. Chapter XVI of Complex Analysis, Gamelin.
• 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.
• 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.
• Brown representability for generalized cohomology. Hatcher 4.E, but there are probably better sources too. Connections to K(G,1)'s.
• H-spaces, division algebras, and Hopf invariant one. There's a brief discussion in Hatcher, 4.B, and a nice chart in On the non-existence of elements of Hopf invariant one, Adams, 1960. I also have notes I can share.
• Introduction to knot theory, e.g. Alexander/Conway polynomial, Jones/HOMFLY polynomial, etc. C. Adams, The Knot Book; Crowell and Fox, Introduction to Knot Theory; S. Carlson, Topology of Surfaces, Knots and Manifolds, Chapter 6; L. Kauffman, On Knots.
• 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 organization: Well-organized; appropriate choice of topics and amount of material; outline handed in 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.
• In late-February we will discuss the choices of topics in class. Each student should bring two possible choices of topic.
• Give me a brief outline (1 to 2 pages) at least 1 week ahead of your date.
• Talks will take place during the last two weeks of classes.