Math 592
Algebraic Topology
Course Information
Lecture: Monday, Wednesday, Friday 10:00am–10:50am
Classroom: East Hall 3096 (Map)
Professor: Jenny Wilson
Email: jchw@umich.edu
Office Hours: Tuesdays 12pm–1:30pm and Thursdays 8pm–9:30pm, on Zoom
Office: East Hall 3863 (Map)
Course Description:
This course covers the fundamentals of algebraic topology. Topics include fundamental group, covering spaces, simplicial complexes, graphs and trees, applications to group theory, singular and simplicial homology, Eilenberg-Steenrod axioms, Brouwer’s and Lefschetz’ fixed-point theorems. This is one of the basic courses for students beginning study towards the Ph.D. degree in mathematics, and preparation for the corresponding qualifying exam.
See the LSA course listings for more details.
Homework will be submitted through Gradescope.
Grades will be made available through Canvas.
Previous years’ webpages:
(2021) http://www.math.lsa.umich.edu/~jchw/2021Math592.html
Textbook: Algebraic Topology (Allen Hatcher)
Prerequisites / Should I take this course? Math 592 requires strong foundations in both topology and abstract algebra. The course will assume students are fluent with basic point-set topology, as well as group theory, linear algebra, and some abstract algebra. I recommend that all students first complete a course in topology, a course in the theory of R-modules (at the level of Math 593), and a course in group theory. Although Math 591 and Math 593 are recommended, I may admit some students who have not taken these courses. To gauge whether you have appropriate preparation for the course, please look at the following material. Please speak to me if you plan to enroll in the course without the recommended background.
A list of topics where I assume students have mastery:
A mock assignment to test your understanding:
| Grading Scheme: |
| Homework | 30% | |
| Midterm I | 20% | (Thursday 24 February 7pm–8:30pm in Angell Hall G127) |
| Midterm II | 20% | (Thursday 24 March 7pm–8:30pm in Angell Hall G127) |
| Final Exam | 30% | (Tuesday 26 April 1:30pm–3:30pm) |
Homework overview: Homework assignments will be posted to the course webpage. Homework is due at 8pm on Fridays, and collected through Gradescope. See the Gradescope instructions below. Your homework solutions should be neat and legible.
Each student's two lowest homework scores will be dropped.
Homework collaboration policy: You may work in groups and discuss homework problems with other students, but your solutions must be written up independently and in your own words. You must list all your collaborators at the top of your assignment.
You are welcome to use other texts and online resources to review the mathematical theory or computational techniques we cover. You may not, however, seek out solutions to specific homework problems. Outside sources should be used to improve your understanding of the material, not as a shortcut to finish assignments with an incomplete understanding. Use your discretion, and if in doubt ask Jenny what is allowed.
Unless otherwise stated in the problem, you are expected to put away any notes from discussions with classmates or other sources while you write up your homework solutions, to ensure you fully understand and can reproduce the arguments.
The homework is your foremost resource for practice with the course material, and for feedback on your work. Doing the homework thoughtfully is essential to your success in this class.
Late homework policy: Each student is permitted to drop their two lowest homework scores to allow for circumstances such as the occasional illness or overscheduled week. Late homework is generally not accepted, except under extenuating circumstances such as medical issues that impact more than two weeks of the semester.
Quizzes:
There will be frequent short in-class quizzes throughout the semester. Quizzes will typically be about 10-12 minutes. I will give advance notice about each quiz and hints about what it will cover.
The quizzes are intended to encourage the class to regularly review the material,
to provide practice for the exams, and to give feedback (both to you the student, and to me the teacher) about
your progress, early on and in a lower stakes setting than the exams.
Academic integrity: Students are expected to know and to uphold the LSA Community Standards of Academic Integrity.
Students with documented disabilities: If you might need an academic accommodation based on the impact of a disability, please get in touch with Jenny as soon as possible. Requests for accommodations by persons with disabilities may be made by contacting the Services for Students with Disabilities (SSD) Office located at G664 Haven Hall. The SSD phone number is 734-763-3000 and their website is ssd.umich.edu. Please note that under most circumstances University Policy is two weeks’ prior notice for any academic accommodation.
COVID-19: The coronavirus pandemic has created unprecedented challenges in our personal and professional lives. Your health and well-being are top priority. See below for a list of campus resources.
Homework
| Homework 0 | Due: Friday 7 January 2022 at 5pm | |
| Homework 1 | Due: Friday 14 January 2022 at 8pm | |
| Homework 2 | Due: Friday 21 January 2022 at 8pm | |
| Homework 3 | Due: Friday 28 January 2022 at 8pm | |
| Homework 4 | Due: Friday 4 February 2022 at 8pm | |
| Homework 5 | Due: Friday 11 February 2022 at 8pm | |
| Homework 6 | Due: Friday 18 February 2022 at 8pm | |
| Homework 7 | Due: Saturday 26 February 2022 at 8pm | |
| Happy March Break | ||
| Homework 8 | Due: Friday 11 March 2022 at 8pm | |
| Homework 9 | Due: Friday 18 March 2022 at 8pm | |
| Midterm Week | ||
| Homework 10 | Due: Friday 1 April 2022 at 8pm | |
| Homework 11 | Due: Friday 8 April 2022 at 8pm | |
| Homework 12 | Due: Friday 15 April 2022 at 8pm |
Quizzes
| Quiz 1 | Friday 14 January 2022 | (Solutions) | |
| Quiz 2 | Friday 28 January 2022 | (Solutions) | |
| Quiz 3 | Friday 11 February 2022 | (Solutions) |
Exams
The course will have two midterms exam and a final exam. The exams are closed-book.The midterm exams will be held Thursday 24 February and Thursday 24 March at 7pm.
Our final exam will be held Thursday 26 April from 1:30pm–3:30pm. It is a comprehensive exam, covering material from the entire semester, but with more emphasis on material since the midterms.
| Midterm I Review Package (Updated Feb 2022) |
| Midterm I (2021) | (Solutions) |
Our Midterm I and solutions are now available:
| Midterm I (2022) | (Solutions) |
| Midterm II Review Package (updated Mar 2022) |
| Midterm II (2021) | (Solutions) |
Our Midterm II and solutions are now available:
| Midterm II (2022) | (Solutions) |
| Final Exam Review Package (updated April 2022) |
| Final Exam (2021) | (Solutions) |
Our Final Exam and solutions are now available:
| Final Exam (2022) | (Solutions) |
Gradescope Instructions
Gradescope is an online platform for grading homework and exams. Your work is still being graded by a human on the Math 592 instructional team, but Gradescope streamlines the process. Gradescope is designed around grading best-practices, for example, the solutions are anonymized for the grader, points are assigned according to a rubric that we set, and Gradescope allows the grader to give more detailed feedback more efficiently.
You can find instructions and trouble-shooting advice at the Gradescope student centre and Gradescope help page.
How to set up a Gradescope account. Gradescope synchronizes with Canvas to create our course roster. Students should receive an email from Gradescope with information on how to create their log-in credentials. If you have not received this email by Thursday 21 January, contact Jenny.
How to upload an assignment or test. To upload an assignment, you must:
Note: This final step is important! The grader will not see your solution to a question if it is not properly selected.
This video explains the homework submission process.
Resubmission. If you find a mistake in your solution, it is possible to resubmit it anytime before the deadline passes. In Gradescope, click on your assignment, and you will see a "resubmit" button in the bottom right corner. Unfortunately, to make any changes to your solution you must re-upload your whole solution and repeat the page selection process.
How to produce a pdf of your homework. If you write your homework solutions by hand, you can "scan" your solutions to create a pdf. Gradescope has recommendations for apps you can use to produce a pdf with a smartphone. To create legible scans, it is best to write with a dark pencil or pen.
Please preview your scan before you upload it to ensure it is clearly readable.
You can also complete your homework on a computer, using software such as LaTeX. If you do use LaTeX, it may be easiest to draw figures separately by hand and scan or photograph them. You can add graphics to a LaTex document by using the graphicx package, or use software to collate the pdf files after your LaTeX document is complete.
Viewing your graded assignment. Once your work is graded and the grades are "published", you will be able to log into Gradescope to see your graded solutions. Click on the name of your assignment to see a problem-by-problem breakdown of your score. Click on an individual problem to see your solution, the complete grading rubric, and any comments from the grader.
Optional Reading
The following reading is strictly optional: it is not related to the course material and will not be discussed in the course. These are articles on math education and learning psychology which may be of interest to math students.
Dweck - Beliefs about intelligence (Nature.com)
Kimball and Smith - The myth of 'I'm bad at math' (The Atlantic)
Tough - Who gets to graduate (New York Times Magazine)
Paul - How to be a better test-taker (New York Times)
Boaler - Timed tests and the development of math anxiety (Education Week)
Parker - Learn math without fear (Stanford Report)
Steele - Thin ice: stereotype threat and black college students (The Atlantic)
Vedantam - How stereotypes can drive women to quit science (NPR)
Stroessner and Good - Stereotype threat: an overview (University of Arizona)
Lockhart - A mathematician's lament (Mathematical Association of America)
Campus Resources for Wellbeing
As a student, you may experience personal challenges that impacts your ability to participate or impacts your academic performance in our class. These could include anxiety, depression, interpersonal or sexual violence, difficulty eating or sleeping, loss, and/or alcohol or drug problems. The University of Michigan provides a number of resources available to all enrolled students.- Counseling and Psychological Services (CAPS)
- University Health Service 24-hour Nurse Line (734-764-8320)
- Psychiatric Emergency Service (PES) (734-936-5900)
- Sexual Assault Prevention and Awareness Center (SAPAC)
(24-hour Crisis Line: 734-936-3333) - University Psychology Clinic (evaluation and therapy)
- Services for Students with Disabilities (SSD)
- Campus Mind Works (mental health education and drop-in wellness groups
for students with a diagnosed mental health disorder) - CAPS Grief and Loss
UMHS and Ann Arbor resources for grief and loss - Resources for eating disorders
- Resources for alcohol and other drugs
- Addiction Center and Collegiate Recovery Program
- Wolverine Support Networks
(peer-facilitated group meetings and community events) - UHS Wellness Coaching
- MiTalk
- Helping a friend
- Resources for financial needs and medical assistance
- National Suicide Prevention Lifeline (1-800-273-8255)
- Substance Abuse and Mental Health Services Administration Helpline (1-800-662-4357)
- Trevor Project Lifeline (24-hour chat or phone
1-866-488-7386),
and the Trans Lifeline (877-565-8860)
- UM Psychiatry resources for COVID-19, including advice for college students
- Advice from UCSF on coping during COVID-19