Math 797
Methods in Algebraic Topology


"The fact that wedges of spheres can, in fact, be identified by [such simple] numerical data partly explains why the main theorem of many papers in combinatorial topology is that a certain simplicial complex is homotopy equivalent to a wedge of spheres. Namely such complexes are the easiest to recognize. However, that does not explain why so many simplicial complexes that arise in combinatorics are homotopy equivalent to a wedge of spheres. I have often wondered if perhaps there is some deeper explanation for this."

– Robin Forman, A user's guide to discrete Morse theory

"It almost seems like a metatheorem in this area that any naturally-defined complex is either contractible or homotopy equivalent to a wedge of spheres."

– Allen Hatcher, MathOverflow, 2010

Course Information

Classes: MWF 3:00pm–3:50pm at East Hall 3866
Professor: Jenny Wilson
Email: jchw@umich.edu
Office Hours: Wednesdays 10am–11am, Thursdays 9:30am–11:30am
Office: East Hall 3863

Course Material: We will study some general tools in algebraic topology, with a focus on combinatorial methods with simplicial complexes. This course will include a combination of lectures and small group work on guided worksheets.

Tentatively, we plan to cover:
  • Topology foundations: CW complexes, delta and simplicial complexes. Cellular and simplicial approximation theorems. Higher homotopy groups. Hurewicz's theorem. Whitehead's theorem.
  • Combinatorial topology: tools to understand the homotopy type of simplicial complexes, including Quillen's lemmas, link arguments, flow arguments, shellability, nerve lemmas, discrete Morse theory
  • Applications, depending on time and student interest.

Prerequisites: Math 592 or equivalent.

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 be the "writer" and one or two students to be the "editors". 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.

Course Webpage: http://www.math.lsa.umich.edu/~jchw/2024Math797.html
Additional course information is posted to Canvas.

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

Grading Scheme:
Class Participation    50%
Worksheet Solutions    50%

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 Friday 12 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: The quotient topology           
Worksheet 2     Review: CW complexes           
Worksheet 3     Review: The homotopy extension property           
Worksheet 4     Higher homotopy groups           
Worksheet 5     Whitehead's theorem           
Worksheet 6     Cellular approximation           
Worksheet 7     Hurewicz's theorem           
Worksheet 8     (Generalized) simplicial complexes           
Worksheet 9     Abstract simplicial complexes           
Worksheet 10     Posets and order complexes           
Worksheet 11     Some families of simplicial complexes           
Worksheet 12     Joins           
Worksheet 13     Barycentric subdivision           
Worksheet 14     Simplicial group actions           
Worksheet 15     Cones and near-cones           
Worksheet 16     Shellable complexes           
Worksheet 17     EL Shellability           
Worksheet 18     PL Morse theory           
Worksheet 19     "Badness" arguments           
Worksheet 20     Discrete Morse Theory           


References

This list will be updated throughout the course.

Primary references:

Hatcher, Allen. Algebraic Topology. Cambridge University Press, 2002.
Kozlov, Dimitry. Combinatorial algebraic topology. Vol. 21. Springer Science & Business Media, 2008.

Other works cited:

Björner, Anders. ”A cell complex in number theory.” Advances in Applied Mathematics 46.1-4 (2011): 71-85.
Björner, Anders, and Gil Kalai. "An extended Euler–Poincaré theorem." Acta Mathematica 161 (1988): 279-303.
Bestvina, Mladen. "PL Morse theory." Mathematical Communications 13, no. 2 (2008): 149-162.
Bredon, Glen E. Introduction to compact transformation groups. Academic press, 1972.
Forman, Robin. "A user's guide to discrete Morse theory". Séminaire Lotharingien de Combinatoire, 48 (2002): Article B48c.
Hatcher, Allen, and Karen Vogtmann. "Tethers and homology stability for surfaces". Algebraic & Geometric Topology 17, no. 3 (2017): 1871-1916.
Milnor, John. "Construction of universal bundles, II." Annals of Mathematics 63, no. 3 (1956): 430-436.
Munkres, James R. Elements of algebraic topology. Addison-Wesley Publishing Company, 1984.
Munkres, James R. Topology (Second Edition). Prentice Hall, Incorporated, 2000.


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 non-university resources:

COVID-19 resources:



















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