Fall  2020

Algebraic Topology, MAT 215A


CRN: 38848

When and where: MWF, 14:10-15:00 PM, Zoom. 

Zoom invitations will be sent via a Canvas announcement. 


Instructor: Professor Michael Kapovich, MSB 3147, kapovich@ucdavis.edu

Office hours: MF, 15:10-16:00 PM, Zoom. 

Textbook:  "Algebraic Topology" by A. Hatcher. This textbook is freely available from Hatcher's webpage: 

https://pi.math.cornell.edu/~hatcher/AT/AT.pdf


I will also supplement Hatcher's book with "A Basic Course in Algebraic Topology" by W. Massey. There is no need for you to buy Massey's book (unless you really like it). 


Course overview: This is the first quarter of a year-long Algebraic Topology sequence. As you will see, the central theme of Algebraic Topology is to develop a theory of algebraic invariants of topological spaces, translating topological problems into algebraic ones.


We will be covering Chapters 0 and 1 of Hatcher's book (Chapters 2, 3, 4 and 5 of Massey's book): Fundamental groups and covering spaces. Depending on the timing, I will also cover Chapter 4.1 of Hatcher's book, dealing with higher homotopy groups. We will start with definitions of homotopy between maps and of fundamental groups of topological spaces. Then we establish the main tool for computation of fundamental groups: Seifert - Van Kampen theorem, which allows one to compute fundamental groups of spaces "inductively," starting from simpler spaces and then dealing with more complex spaces obtained by combining simpler pieces. After that, we will introduce and study covering spaces which will allow us to interpret fundamental groups as groups of symmetries. Once we are done with this material, we will discuss cell complexes and higher homotopy groups.


The main prerequisites for MAT-215A are General (aka Point-Set) Topology (MAT-147) and Group Theory (MAT-250). However, unlike the basic abstract algebra you took, we will hardly ever deal with finite groups: Most groups appearing in the class will be infinite and, frequently, far from commutative. We will use several notions, constructions and tools that you learned in MAT-250: Free groups, free products, group presentations. There will be one more theme in this class reminiscent of MAT-250: Just as the Galois Theory is based on the correspondence between field extensions and groups (as automorphism groups of such extensions), the theory of covering spaces will be based on the correspondence between covering maps and spaces (analogues of field extensions) and subgroups of the fundamental group. These subgroups will reappear as automorphism groups of covering maps. (If you decide to take an Algebraic Geometry class, you will see that this similarity between the Galois Theory and the theory of covering spaces is more than just a mere formality.)   

 

Grading: Your grades will be based entirely on the homework that I will assign weekly, on Wednesdays (on Canvas); the homework will be due on Wednesday of the following week. You will be submitting the homework online, by sending it to my @ucdavis email address. 



Important dates:

First day of classes: Wednesday, September 30, 2020.

Holidays: Veterans Day (Monday, November 11) and Thanksgiving break (November 26-27). Last day of classes: Friday, December 11, 2020.



ADA Statement:


The Americans with Disabilities Act requires that reasonable accommodations be provided for students with physical, sensory, cognitive, systemic, learning and psychiatric disabilities. Please contact me at the beginning of the quarter to discuss any such accommodations for the course.