## Elementary Topology

### MA 571- Fall 2021, Prof. Manuel Rivera, Purdue University

**Course description:** This is a first graduate course on topology. We will emphasize those aspects of the subject that are fundamental to modern mathematics. The topics will include the basics of point set topology and a brief introduction to some of the ideas of algebraic topology.

**Instructor:** Prof. Manuel Rivera (manuelr at purdue dot edu), Office: Mathematics Building 708

**Textbook: **Topology by J. Munkres

**Course schedule: **Monday, Wednesday, Friday 3:30-4:20pm at Recitation Building 307

**Office hours: **TBD

**Homework: **Working out exercises by yourself is essential for learning and understanding mathematics. A homework problem set will be posted weekly. Each problem set will contain exercises from the book as well as exercises not in the book. Solutions to the exercises can probably be found online.* Do not look them up; try each problem as hard as you can and if you get stuck discuss with me or your collegues*. Then write clear and complete solutions to all of them. Collaboration is encouraged but you must write up your own solutions and indicate who you worked with.

*I highly suggest working on every single exercise in those sections from Munkres’ book corresponding to the topics discussed in class. This will also prepare you for the topology qualifying exam, if you are planning to take it. *

Here are some “Comments on style” written by J. Munkres and here are some “Guidelines for good mathematical writing” written by Francis Su. Please read these during the first week of the course and keep them in mind while writing your homework and exam solutions.

**Grader:** Zachary Himes

**Exams: **We will have a midterm and a final exam. The midterm will be a take home exam. You will have three days to work on the midterm. Consulting sources other than Munkres’ book will not be permitted. The final exam will be in real time and the format will be similar to that of a qualifying exam.

**Grading scale: **Homework 30 %, midterm 30%, final exam 40%.

**Weekly schedule:** We will roughly follow the following weekly schedule

Week 1: Definition of topology, basis, examples: order, product, and subspace topologies

Week 2: Closed sets and limit points, continuous functions, more on product topology

Week 3: Metric topology and quotient topology

Week 4: Connectedness

Week 5: Compactness

Week 6: Simplicial complexes

Week 7: Introduction to function spaces and the compact open topology

Week 8: Homotopy, homotopy equivalence, fundamental group, covering spaces

Week 9: Fundamental group of the circle, spheres, and applications

Week 10: Van Kampen’s theorem

Week 11: Classification of surfaces

**Special accommodations**: Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247.

In this mathematics course accommodations are managed between the instructor, the student and DRC Testing Center. If you have been certified by the Disability Resource Center (DRC) as eligible for accommodations, you should contact your instructor to discuss your accommodations as soon as possible. Here are instructions for sending your Course Accessibility Letter to your instructor: https://www.purdue.edu/drc/students/course-accessibility-letter.php

**Homework problems** (updated weekly)

- Problem set 1– due September 8
- Problem set 2– due September 24
- Problem set 3– due October 8
- Midterm– due October 22
- Problem set 4– due November 11
- Problem set 5– due November 24