Math 696
Algebraic Topology II
Homological stability


Course Information

Classes: MWF 9:00am–9:50am at East Hall Room 2866
Professor: Jenny Wilson
Email: [email protected]
Office Hours: Mondays 10am–11:30am and Fridays 2:30pm–4pm
Office: East Hall Room 3863

Course Webpage: https://websites.umich.edu/~jchw/2026Math696.html

Textbook: This course has no assigned textbook. Suggested references are listed below.

Course Material: We will study some foundational topics in algebraic topology, geared toward covering the background needed to introduce the subfield of homological stability. This course will include a combination of lectures and small group work on guided worksheets.

We will cover some of the following topics, depending on time and student interest.
  • cohomology: review of basic properties and computational tools
  • category theory: review of major concepts
  • group cohomology: basic properties and computational methods
  • spectral sequences: basic structure and computational methods
  • (semi)simplicial objects: definitions, examples, and computational tools
  • homotopy (co)limits, the bar construction
  • applications to (co)homological stability

Prerequisites: Math 592 or equivalent. Math 695 recommended.

IBL: Our course will use an Inquiry-Based Learning (IBL) format. For a portion of each class, students will work on exercises together in small groups. Development of collaboration and mathematical communication skills is an overarching goal of the course.

Worksheet solutions: For each worksheet, I will select one or more problems for formal write-up. For each selected problem, I will assign one student to write a solution. We will collaborate on these solutions using Overleaf.
Note: I would like to be able to use these worksheets again in future, so please do not publicly share any class solutions.

Explainer videos: As a project component of the class, each student will be asked to plan and create a 20-30 minute ``explainer video'' on the topic of their choosing. The videos should be an introduction to one of the topics in a course, aimed at a student who has completed a first course in algebraic topology, at the level of Math 592. We will post these videos as a resource for rising topology students.

Grading Scheme:
Class Participation    40%
Worksheet Solutions    40%
Explainer video    20%

Textbook feedback: Cary Malkiewich is writing an introductory textbook on spectra and stable homotopy theory (current draft). Students can obtain bonus credit by doing the following. Choose one (or more!) subsections of the book; they are typically about 10 pages. Read the subsection carefully, and attempt the corresponding exercises. Email Cary (and cc me) with feedback. Did you find any typos or errors? Did you find any mathematical points confusing or misleading? Include any other comments you have.

Attendance policy: Because in-class group work is a major component of the class, attendance counts toward the 'participation' component of the grade. Starting on Wednesday 7 Jan (or the first class after a student registers), students can miss three 'unexcused' lectures without penalty. Please let me know if you have a reason to be absent; 'excused absences' (such as illness, academic travel, job interviews, religious observances, certain university-sponsored events, etc) typically do not count toward the missed classes.

Class conduct: Class discussions and small group work are major components of this course. Students are expected to be active participants in the classroom, and are expected to conduct themselves with professionalism and respect for their classmates. Our goal is to create a supportive class environment where students are comfortable testing ideas, questioning each others' ideas, having their ideas challenged, and working together to reach a solution.

The student 'participation' grade is based on the following expectations. Students should ...
  • attend class and participate in a group discussion
  • present ideas and contribute to the discussion
  • ensure their groupmates have equal opportunity to contribute
  • make a genuine effort to engage with their groupmates' ideas
  • treat their groupmates with patience and encouragement
  • take responsibility for speaking up when they are confused
  • take responsibility for ensuring their groupmates are included and are understanding the discussion.

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. Once your eligibility for an accommodation has been determined, this information will be reflected in SSD's Accommodate system. Please note that under most circumstances University Policy is two weeks’ prior notice for any academic accommodation.



Worksheets

Worksheet 1     Review: Foundations of cohomology           
Worksheet 2     Review: (Co)homology with coefficients           
Worksheet 3     Review: The Kunneth formula and the cup product           
Worksheet 4     Review: Poincare duality           
Worksheet 5     Review: Tor and Ext           
Worksheet 6     Group (co)homology via Tor and Ext           
Worksheet 7     Group (co)homology via K(G,1) spaces           
Worksheet 8     Induction and coinduction in group (co)homology           
Worksheet 9     The structure of a spectral sequence           


Student Explainer Videos

The Math 696 students' Algebraic Topology Explainer Videos are available at this link.



References

This list will be updated throughout the course.

Primary references:
Brown, Kenneth. Cohomology of groups. Springer Graduate Texts, 1982.
Hatcher, Allen. Algebraic Topology. Cambridge University Press, 2002.
Malkiewich, Cary. Spectra and stable homotopy theory. Textbook draft, 2025.
May, Peter. A concise course in algebraic topology. University of Chicago press, 1999.

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.

Some university resources:

Some non-university resources:

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