**Possible Topics:**
These are suggestions, but you can also propose other topics.
Topics need to be discussed with me and approved. 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, Borsuk-Ulam, ...
- Introduction to cohomology, Hatcher Ch 3. Universal coefficient theorem, ring structure (cup product).
- K(G,1) spaces, Hatcher 1.B.
- Classification of surfaces, Massey, GTM 127, Ch 1.
- Hurewicz Thm, Hatcher 2.A and maybe p. 366-?. Sketch proof of 2A.1, several examples, statement of 4.32, more examples.
- Long exact sequence of homotopy groups for a fibration, Hatcher p.375-?. Define fibration, state result, sketch proof of one part, give examples.
- Fundamental groupoid, generalized van Kampen thm. Higgins, "Categories and Groupoids". QA171.H57.
- Brown representability for generalized cohomology. Hatcher 4.E, but there are probably better sources too. Connections to K(G,1)'s.
- Phantom maps and univeral phantom maps. See, e.g., this paper by Gray and McGibbon and McGibbon's article in the Handbook of Algebraic Topology by I. James.

**Duration:**
40-50 minutes for grad students, 35-45 for undergrads.
The presentations are not long, so you will need to carefully
select a small 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
- organization of material: what you choose to cover, and how you choose to organize it.
- 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
- 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:**

- Choose your date before reading week. Choices are April 7, 9, 14, 16 (one on Mondays, two on Wednesdays)
- During reading week, look over topics and read about a couple of them.
- Meet with me during reading week or the week after to discuss topics.
- Choose 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 talk ≥ 1 week ahead of your date.