2024 March 18,

Math 354: Topology

Carleton College, Winter 2024, Dr. Joshua R. Davis, , CMC 324, x4095

Introduction

It's hard to describe what topology is in a simple paragraph, but here's an attempt. Topology is the basic mathematics of space. By "space" I don't mean outer space, although topology is used in physics and astronomy. In topology, we define a notion of topological space. There are infinitely many topological spaces, although we tend to focus on a few kinds that appear frequently throughout math and its applications. By "basic" I don't mean easy; rather, I mean fundamental. Topology is the math of space, upon which other maths of space are built. A topological space has just enough structure to define what a continuous function is. Topology is the study of continuous functions.

Topology comes in three flavors: point-set (which is the core of the subject), algebraic (which connects point-set topology to abstract algebra), and differential (which connects to both calculus/analysis and abstract algebra). The first half of this course surveys the most important topics in point-set topology. The second half surveys the traditional first topic of algebraic topology, which is the fundamental group. We might briefly glimpse differential topology, but we might not.

The prerequisite for this course is Math 236: Mathematical Structures. We prove a lot of theorems about sets and functions. I am happy to admit students who have taken CS 202: Math of Computer Science instead. Talk to me if you are concerned about your background.

Responsibilities

The College's accreditation says that a 6-credit course is 150 hours of work. That's about 15 hours per week or 5 hours per class meeting. Those 5 hours break down into about 1 hour for class itself and 4 hours for homework, reading, studying, etc. If you find yourself spending much more time, then talk to me.

Participation

Our course meets in CMC 319 during period 5A. That's Monday 1:50-3:00, Wednesday 1:50-3:00, Friday 2:20-3:20. You are expected to attend every class meeting promptly. You are expected to take notes on paper or a tablet. (Typing on a keyboard is inadequate, because of all of the pictures needed.) You are expected to participate in discussion and group work.

Some students don't like participating in class, because they are shy, they are not confident about their English or math, etc. I urge those students to participate anyway. They can also compensate for a deficiency in class participation by collaborating with others in office hours.

Homework

Although class meetings may seem like the core of the course, homework assignments are actually where you learn the material. It is essential that you attempt each homework promptly, before the next class meeting. For then you better understand that next class meeting!

On homework, you are encouraged to figure out the problems with other students. However, you should always write/type your solutions individually, in your own words. You may not copy someone else's work — that includes artificial intelligence products — or allow them to copy yours. Presenting someone else's work as your own is an act of academic dishonesty. The College requires me to report you, if I suspect that you have not upheld its Academic Integrity standards.

Writing is not just for literature and history majors. Written and oral communication skills are essential to every academic discipline and are highly prized by employers. In this course, your written work is evaluated both for correctness and for presentation. Compose your solutions as if the intended audience is your fellow students. By doing so, you show enough detail that the grader can ascertain whether you yourself understand the material.

Homework is assigned at nearly every class meeting. Although you are expected to attempt the problems immediately, they are usually collected two meetings after they were assigned. This schedule tries to give you some flexibility.

But we all have bad weeks, where we can't get everything done, right? If you need to submit an assignment late, then do so. Put your paper in a separate pile from the on-time papers. If the grader hasn't graded the assignment yet, then they can grade your paper with the others for full credit. If the grader has graded the assignment already, then you might not get credit. There are limitations to how much delay and complication a grader can handle.

Please mark each assignment with the day that it was assigned (e.g., "Day 11"). Please staple multi-page packets. Paper clips don't work well in a stack of papers.

Depending on time constraints in any given week, perhaps not all of your homework will be graded. In order to ensure full credit, do all of the assigned problems by their due date.

Exams

We have three exams. Some are in-class, and some are take-home with time limits. They are given roughly 1/3, 2/3, and 3/3 of the way through the term. The last exam is given during our official final exam period, which is Monday March 11 8:30-11:00 AM in CMC 319.

Are the exams cumulative? No and yes. No, in that each exam is focused on the material that has not been tested yet. Yes, in that much of the material is inherently cumulative. Also, the last exam might have some questions that explicitly address material from earlier in the course.

Grading

For better or worse, we are required to measure your learning using grades. Your numerical grade is based on the responsibilities above: participation 5%, homework 10%, Exam A 25%, Exam B 25%, Exam C 35%.

Numerical grades are converted to letter grades only at the end of the term. Grades are not curved, so students are not in competition with each other. Grades are also not based on predetermined percentages (90%, 80%, 70%, etc.), because Math 354 problems are difficult to tune so precisely. (I could accidentally write a difficult exam and wreck everyone's percentages.) Rather, I assign grades by comparing the students to the course goals. For example:

• A student who earns an A is not perfect, but they can speak fluently about nearly all of the course content.
• A student who earns a B learns most of the big concepts pretty well, but they are missing some concepts. They lack the depth of understanding needed to appreciate subtleties in proofs, etc.
• A student who earns a C gets some of the material but has significant gaps in their understanding of major concepts or their ability to prove important theorems.

Talk to me, if you are concerned about your grade.

Resources

I want all of my students to work hard and learn a lot. I try to give them all of the resources that they need. For starters, let's list some documents:

• Our textbook is Topology by Munkres, 2nd edition. This is a deservedly popular introductory topology textbook. It was also used in the last offering of Math 354, two years ago. It gives a thorough overview of point-set topology and the fundamental group. It covers more material than we have time for. It will serve you well in the future, if you continue in mathematics.
• Here are our exams from this term.
  • Exam A, Solutions (quartiles: 46/48, 42/48, 38/48)
  • Exam B, Solutions (quartiles: 36/39, 31/39, 28/39)
  • Exam C, Solutions (quartiles: 45/48, 42/48, 37/48)
• I like to give my students some old exams, so that they have an idea of how I write exams. But the last time I taught this course was a while ago. So the problems might be pretty different from what I would write now. Anyway, here they are.
  • Exam 1, Answers (quartiles: 52/56, 48/56, 45/56)
  • Exam 2, Answers (quartiles: 31/34, 27/34, 25/34; question 3 is missing its images)
  • Exam 3 (quartiles and answers missing)
• Groups tutorial

Remember that I encourage you to solve problems with classmates (even if the work that you submit must be your own). If you want help in finding a study partner, then e-mail me, perhaps describing some of your habits: working in the middle of the night, not waiting until deadlines, etc.

I hold several office hours per week. Consult my weekly schedule. Office hours are essentially optional extra class meetings, where you pick the topic of conversation — usually homework problems. No appointment is needed for office hours; just drop in! If you want to visit office hours but the times don't work, then consult my weekly schedule and e-mail me, listing several possible meeting times.

If a health condition or other personal matter affects your participation in class, homework, exams, etc., then please let me know as soon as possible. Depending on the situation, we might want to confer with Accessibility Resources, Assistive Technology, Student Health and Counseling, or Sexual Misconduct Prevention and Response. When you ask me to help, I do my best to help. :)

Schedule

To help you decode the schedule below, here is an example. Wednesday January 3 is Day 01 of the 28-day course. During class, I give an intuitive overview of topology. You have a Day 01 homework assignment, which you should attempt immediately, but which is due at the start of class on Day 03. You are expected to skim Sections 1-7 of the textbook after (or before) class.