Math 40740: Topology

Math 40740: Topology (Fall 2018)

Professor: Mark Behrens
12:50-1:40, MWF DeBartolo 206
Office hours: Tuesdays 10:00-11:45 or by appt, Hurley 287
Textbook: Introduction to Topology and Geometry, 2nd ed, by Stahl and Stenson
Prerequisite: Math 20630, Introduction to Mathematical Reasoning

Coverage: Time permitting, we will start with chapter 1, followed by a more rigorous treatment of general topology in chapter 10. After briefly discussing graphs (Ch 2), we will move onto surfaces, including the classification of surfaces (Ch 3). We will then discuss knots (Ch 5), including the Jones polynomial. We will then discuss the fundamental group, and study the fundamental group of surfaces (AKA 2-manifolds) and knot complements (ch 9). We will end the class with 3-manifolds.

Grade: Homework 50%, Take home midterm 20%, Final exam 30%.

Homework: Homework will be typically assigned on Wednesdays, due the following Wednesday. Homework may be done individually or in small groups, with the caveat that (1) you list your collaborators, and (2) you write up your own solutions, independently of your collaborators. Some problems assigned have answers in the back of the book - these still should be turned in, and you should attempt them before checking the back of the book. Full explanations/proofs (as appropriate) are expected for all problems. Late assignments, unless there is a legitimate reason (e.g. medical), but I will drop your lowest homework score (i.e. the homework where the most points were deducted).

Exams: There will be a take-home midterm exam and a final exam.No collaboration is allowed on either. Both are open-book, open-note.

Honor code: All students are expected to adhere to the Notre Dame Honor code.

Assignments (you may have to refresh your browser to get this page to update)

Pset 1 (assigned 8/22, due 8/29): Read Chapter 1. Do exercises 1.1, problems 1-4, plus BONUS PROBLEM
Pset 2 (assigned 8/29, due 9/5): Do exercises 10.1: problems 5,8,11,14 and exercise 10.2: problem 10 (in problem 10, you may assume T is a metric space if you like).
Pset 3 (assigned 9/5, due 9/12): Read Chapter 10.3. Do exercises 10.3: problems 1,2,9, 13, 15
- For 10.3(9), I found it useful to show that the sets of z with f(z) < f(-z) and f(z) > f(-z) are both open.
- For exercise 10.3(15), use problem 13: if R^2 and R are homeomorphic, then R^2 minus a point is homeomorphic to R minus a point - what goes wrong here?
Pset 4 (assigned 9/12, due 9/19): Read Chapters 10.4, 2.1, 2.4. Do exercises 10.4, problems 1, 11, 14 (in problem 14, the function is surjective) and 2.4 (1) b,d,e,f.
- Give an example of a non-compact space for which problem 11 fails.
Pset 5 (assigned 9/19, due 9/26): Read Chapter 3.1. Do 2.4 (7), 2.5 (2),(5), 3.1: (1)(4).
- For problem 2.5 (2), unlike in pset1, I expect you to be mathematically precise.
- For problem 2.5(5), you need only write down 1 adjacency matrix, not all 6.
Pset 6 (assigned 9/26, due 10/3): Read Chapters 3.2 and 3.3. Do 3.2 (1)(d,e), (2)(d,e), (5).
- Extra problem: compute the euler characteristic and determine orientability of each of the surfaces (d,e) in problem (1).
- For problem (5), you may use the classification of surfaces.
Pset 7 (assigned 10/10, due 10/24): Read Chapters 5.1, 5.2, and 5.3. Do 5.1 (3)(5), 5.2 (3)(9), 5.3 (1) [Note that L_1,n has n crossings in the "main braid"]
Pset 8 (assigned 10/24, due 11/2): Read Chapter 5.4, do 5.4 (2)(a), (3)(a), 5.4 (9), (10) (Maybe wait until after monday's lecture to do problem 10!)
Pset 9 (assigned 11/7, due 11/14): Read Chapter 9.1 and the definition of homotopy from section 10.2. Do 9.1 (2), (6),(7), (11), (15) [For (6),(11) and (15), just describe what you think the answer will be, and give an intuitive reason]
Pset 10 (assigned 11/14, due 11/28): Read Chapter 9.2. Do 9.1 (8), (18), 9.2 (1), (4)(a)(b)
Pset 11 (assigned 11/29, due 12/5): Read Chapter 9.3. Do 9.3 (2),(24),(27),(30). (For problem (2), I only want you to determine if it is a manifold, and if it is, I want you to determine a presentation of its fundamental group - you do not have to determine the link of each vertex)