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: firstname.lastname@example.org 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)