Introduction to Algebraic and Geometric Topology (5520), spring 2009
Here are the papers from your research projects:
-Introduction to 3-manifolds by Nik Aksamit, with bibliography.
-Indra's Pearls by Miles Fore.
-The Poincare-Hopf theorem by Sita Gakkhar.
-The degree-genus formula by Alex Pruss.
-Triangle reflection groups by Kyle Steffen.
-Gauss-Bonnet theorems by Sam Stephenson.
Homework assignment for this week:
-Week 10: There is no homework due this week. The second Midterm Exam will be on Tuesday 4/7, and will cover fundamental groups, free groups and free products, Van Kampen's theorem.
-Research projects: During the Spring Break you should start thinking about your topic for the research project. If you have an idea, please come discuss it with me after the break. If you don't, you can look at some of the books in the updated bibliography and this list (the items with ... are very wide and include a number of possible topics).
-Here is a schedule for the different parts of the project.
-Week 9:(due Tuesday 3/24) Read p. 43-46 of Hatcher and do exercises # 3,8,9 p. 52-53.
-Week 8:(due Tuesday 3/10) Read p. 40-42 of Hatcher and p. 97-105 of Massey (where free groups and free products are defined by what we called the Universal Property). Do exercises #4.1, 4.4, 4.6 and 5.1, 5.2, 5.3 of Massey (p. 100, 101 and 104).
-Weeks 6 and 7: (due Tuesday 3/3) Our main reference for this part is Hatcher's book (chapters 0 and 1). Read p.1-4 and 21-37 of Hatcher (there are several applications on p.31-33 that we didn't see in class). Hand in exercises #1,2 p. 18 and 5,8,10,12,16 p. 38-39 (see p. 10 for wedge sum notation). Further problems: if you are interested you can look at #5,6,7 p. 18 and #7,17,18 p. 38-39.
-Week 5: There is no homework due this week. The first Midterm Exam will be on Thursday 2/19 during (part of) class time; it will cover all the material from the first 4 weeks.
-Week 4: (due Tuesday 2/10) Read sections 4.6 and 5.1-5.4 of Kinsey. Do Exercises 5.2, 5.3, 5.10, 5.11, and 5.15. Bonus problems: 5.16, 5.18.
-Week 3: (due Tuesday 2/3) Read sections 4.4 and 4.5 of Kinsey. Do Exercises 4.14,4.16,4.17,4.19,4.21,4.22. Bonus problem: find a triangulation of the torus with 14 triangles (we'll see that this is the minimal number).
The proof of the classification theorem that we saw in class comes from this paper by Ed Burgess in the American Mathematical Monthly.
-Week 2: (due Tuesday 1/27) Read section 4.3 of Kinsey. Do Exercises 4.9,4.10,4.12. In section 4.5, do Exercise 4.18 and determine which surfaces correspond to the planar diagrams in figures 4.46, 4.51, and 4.52. You don't need to know the steps of the proof of the classification theorem (which we'll see next week), only the process that we used in class to prove that K2=P2#P2 and T2#P2=P2#P2#P2.
-Week 1: Read sections 4.1 and 4.2 of Kinsey. Do Exercises 4.1, 4.4, 4.5, 4.6 and 4.7 (be careful about the word "regular" in 4.6), and turn them in in class on Tuesday 1/20.
Lectures: Tuesdays and Thursdays 12:25-1:45 pm in LCB 222.
Material: Topics and bibliography. The books in this bibliography are on reserve in the Math department library (you can borrow them for a day, or 2 hours for Kinsey and Massey).
Homework:
Assignments will be given on-line each week, usually by Thursday, and will be due in class the following Tuesday.
Exams:
There will be two Midterm exams, possibly during weeks 6 and 12, but in place of a final exam you will be asked to complete a research project consisting in a written paper (5 to 10 pages) and an oral presentation (20 minutes). More details will be given, but you can already start thinking about topics which you like (see for instance the above bibliography for ideas). In time I will write a list of potential topics if you have trouble finding one, but you may choose one outside the list.
Grading: Homework, exams and the research project will each count for a quarter of the final grade.
Office hours: Tuesdays 2-3 pm in my office, JWB 209. If you can't make it, write me an email at paupert@math.utah.edu, or call 801-581-6846 to schedule an appointment.