Math 525 Algebraic topology, Spring 2024

Math 525 Algebraic topology, Spring 2024.

Course announcement.
Professor: Igor Mineyev.
Class time and place: 2:00pm-2:50pm, MWF, 329 Gregory Hall.
Textbook: Algebraic topology by Allen Hatcher. Freely available online. We will follow the numbering of the problems on that page, in case it is different from the book.
Office hours: for now are scheduled on Zoom for Monday and Wednesday at 5:00pm on Zoom to make it after the classes, if this is fine with the students. This way you can ask questions before the homework is due on Fridays. The meeting number and password is given in class. Please come close to the beginning of the hour, so that we can discuss things together with other students. In those rare situations when you need to talk to me privately, let me know during our Zoom meeting or before, then call me after the meeting is finished.

The best way to contact. My email is mineyev at illinois edu, but I have a slight disability that makes it hard for me to type, and generally to deal with email. Vocal real-time communication is much more preferred and very much appreciated. Please talk to me before/during/after the class and during the office hours. Face-to-face communication (virtual or otherwise) is also much more efficient for discussing things. I would appreciate if you use it as much as possible rather than communicate by email. I am happy to discuss any kind of math, or any other things that you are interested in. Math is the only possible meaning of life. And fun too. Take initiative, talk to me about it. Any suggestions are also welcome.

Prerequisites: The following background is expected for the course:

The standard concepts of point-set topology (metric spaces, open and closed sets, topological spaces, ...) Math 432 and Math 535 are good sources for this, or learn these notions from any standard textbook on general topology, for example, James Munkres, Topology or John Kelley, General Topology.
The standard concepts of abstract algebra (groups, group actions, fields, vector spaces, classification of finitely generated abelian groups, free groups ...) Math 417 is a source for some of this, or use any of the many textbooks on abstract algebra, for example, Derek Robinson, An introduction to Abstract algebra.

Tentative syllabus.
Fundamental group and covering spaces.
(1) Definition of the fundamental group.
(2) Covering spaces and lifts of maps.
(3) Computing the fundamental group via covering spaces.
(4) Applications, such as the Fundamental Theorem of Algebra and the Brouwer fixed point theorem in 2d.
(5) Deforming spaces: retraction and homotopy equivalence.
(6) Quotient topology and cell complexes.
(7) Homotopy extension property and applications to homotopy equivalence.
(8) Fundamental groups of CW complexes.
(9) Van Kampen's Theorem.
(10) Covering spaces and subgroups of the fundamental group.
(11) Universal covers.
(12) The definitive lifting criterion, classification of covering spaces.
(13) Covering transformations and regular covers.
Homology.
(14) Delta complexes and their cellular homology.
(15) Singular homology.
(16) Homotopic maps and homology.
(17) The long exact sequence of the pair.
(18) Relative homology and excision.
(19) Equality of cellular and singular homology.
(20) Applications, such as degree of maps of spheres, invariance of dimension, and the Brouwer fixed point theorem.
(21) Homology of CW complexes.
(22) Homology and the fundamental group: the Hurewicz theorem.
(23) Euler characteristic.
(24) Homology with coefficients.
(25) Intro to categories and axiomatic characterization of homology theories.
(26) Further applications, such as the Jordan curve theorem, wild spheres, invariance of domain.

Expectations. Without looking in the textbook, a student should be able to provide a solution to any problem similar to the homework problem, to know concepts covered, and to provide proofs to any statements that were proved in class and in the homework. It is one thing to hear a proof and to understand every step. It is quite another thing to be able to provide a proof without looking in a textbook. To check whether you have actually learned a particular material a good practice test is to close the textbook/notes and write a proof of any definition/statement covered in class/homework.

Grading policy. Homework 30%, two midterm exams 20% each, final exam 30%. The students are expected to attend the class most of the time. Missing one or two classes for some substantial reasons is understandable, but missing too many and with no strong reason can negatively affect your grade for the class. Grading will use the standard scale: A-,A,A+ are 90 to 100; B-,B,B+ are 80 to 90; C-,C,C+ are 70 to 80; D-, D,D+ are 60 to 70. One lowest homework assignment will be dropped at the end of the course. The grades will be posted on Learn@illinois Moodle: look for "MATH 525 F1 SP24: Algebraic Topology I (Mineyev, I)".

Exams. The midterm exams will be in the regular classroom at the regular time: Exam 1 on Friday, February 23, 2024, and Exam 2 on Friday, April 5, 2024. The final exam will be in the regular classroom at 7:00-10:00 p.m., on Monday, May 6, 2024, as required by Final Exam Scheduling Guidelines.

Homework will be posted here. Starting from homework 5, it is due handwritten on Learn@illinois Moodle before the beginning of the class. Here are links to the study explaining how handwriting is good for you: handwriting1, handwriting2. Write the class number and the homework number. By hand. :)

The online evaluation forms for this class should be available some time, probably between April 19 and May 1, 2024. You either receive this information by email or directly log in on the website https://go.illinois.edu/ices-online. I very much encourage you to fill out the evaluation forms. Since there is a deadline, please try not to miss it.

Various links related to topology, and mathematics in general.

As part of the ongoing IG³OR'S group, please feel free to join our weekly mini-seminar on Zoom in Spring 2024 (the meeting number and password were given in class). Xianhao An, Jihong Cai and Leslie Hu will read and present several articles by Akio Kawauchi that claim to have solved several long-standing open problems in topology/geometric group theory. Feel free to read the articles as well and try to find mistakes, if any. We meet on Tuesdays at 9.30 am.
IML (Illinois Mathematics Lab, formerly known as Illinois Geometry Lab) here at UIUC . The earlier you start doing research in math, the longer you will do it. :)
ColorTaiko!: the IML project that I am running starting in Spring 2024. The fun game that we will create in that project is secretly related to several long-standing open problems about group algebras called Kaplansky conjectures. My article showing the connection between the Kaplansky conjectures and the ColorTaiko! game will appear at some point on my website.
U of I Math seminars.
All kinds of pictures of topological spaces at Sketches of topology.
A list of resources about geometric group theory by Jon McCammond.
Open problems in low-dimensional topology.
Open problems listed by AIM, including some from geometric group theory.
Questions in geometric group theory by Mladen Bestvina.
Problems on boundaries of groups and kleinian groups by Misha Kapovich.
Unsolved problems in algebra.
Unsolved problems in group theory.
Open problems in combinatorial and geometric group theory by Baumslag-Myasnikov-Shpilrain. Here is an archived online version of this list. It does not seem to be updated, but still is a good source.

Back to Igor Mineyev's math page.