Topology (Math GU4501)

Instructor: Mike Miller Eismeier
Email: smm2344@columbia.edu
Webpage: here! homework will be posted to Courseworks
Classroom/time: Classroom TBA, on TR 2:40pm-3:55pm
Office: Math 427.

Office hours: TBA. I will hold two office hours per week, but see below about the Discord server.  If you can't make it to either of these but have questions you want to talk about, just get in touch with me!

Teaching assistant(s): TBA.

Topics: The course is separated into two halves: first, point-set topology (essentially, the fundamental tools of topology which are commonly used here and elsewhere in math); second, algebraic/geometric topology, where we apply the tools from the first half of the course to situations where we can actually see a picture of what's going on.

Prerequisites: You should be comfortable with mathematical logic and proofs, as well as be relatively adept at proving various statements about set-theoretic operations. I suggest taking a look at (and/or getting some practice with) Chapter 1 of Munkres if you feel uncomfortable. The most important pre-requisite by far is Modern Analysis I or equivalent. You should know what a metric space is, how to prove that a particular map between metric spaces is continuous in practice, and have some intuition for metric spaces. Much of topology is about taking this intuition and transporting it to more general and flexible contexts. We will use groups extensively in the latter third of the course, so you should know what they are and how to work with them by then.

Textbooks: We will be eclectic in our choice of references. Further, I will post typed notes for each class which cover (somewhat more than) what we do in class; my hope is that these typed notes will be posted at least 2 weeks before each class, with most of the notes posted before the semester begins.

The first half will reasonably closely follow Hatcher's notes on point-set topology, though we'll talk about some topics that he doesn't write about. I will also point you to the corresponding sections in Munkes, if that is useful.

References for the second half will be more scattered. Our discussion of fundamental groups will refer to portions of Chapter 1 of Hatcher's algebraic topology book. The proof of the Jordan curve theorem will follow Maehara's argument.

Homework: There will be a total of ten homework assignments. Homework will be assigned on Tuesday and due the following Tuesday. To submit your homework, you need to upload it on Courseworks. You have two options: write neatly, and scan using a good scanner (or scanner app on your phone / tablet), or learn to TeX your homework. If your homework is not readable, it will not be accepted. 

I encourage working with your classmates on your homework, but your submissions must be your own work. You can work on ideas and problems with each other, but ultimately, you should submit something where you have written down your own thought process / argument in your own words. Copying homework solutions, from classmates or online, is considered academic dishonesty and will be treated as such. I no longer have any patience for this.

Your lowest homework score will be dropped.

Late homework will not be accepted.

Curios: Now and again I will post a "curio", which is a long guided exercise towards some interesting result which is too complicated or out-of-the-way to cover in class or homework. These will be for your own interest, and you will not be expected to submit any. If you send me a writeup I will look over it and give comments on your arguments and your exposition; this can be a good way to get additional feedback.

Discord: This semester there will be a class Discord channel, where you can talk to your classmates about the material, the homework, or ask me questions. I'm imagining this like a less clunky Piazza, that hopefully people already have installed. 

I will direct that most math questions you ask me be posted to the Discord, so that my response is a matter of the public record (instead of stuck on an email chain between the two of us, that nobody else gets to see.) I will respond to questions there.

The link to the Discord will be posted in a Courseworks announcement. You will be asked to post your UNI so I can cross-reference. Only one Discord account per UNI will be allowed into the server.

Tests: There will be one midterm exam (set to cover material from the first half of the course) but no final exam. The midterm is a take-home exams, scheduled for one full day; you will have the full day to work on it. 

(tentative) Midterm: October 30 (12AM-11:59PM)

Project:  In lieu of a final exam, you will write an approximately 15 page expository paper on a topic in (or closely related to) topology. Some students from last year have made their projects available here. You will receive two rounds of detailed feedback before your final submission, and it will be graded on correctness, depth, and expository clarity. You will likely begin work on the project about half-way through the term.

Because this is a project I see little reason to make it due on the date of the final exam (though I will schedule feedback rounds so that you can submit it on that day, if you want); the final due date is 10PM on 12/22, at the end of the final exam period.

The final project is comparable to a final exam, and so unlike the homework is not cooperative.

Now that the semester is complete, some of the student projects are available here.

Grading: The final course grade is weighted as:

Homework: 40%
Midterm: 25%
Project: 35%

Your bottom homework score will automatically be dropped.

Students with disabilities: To receive accomodations for exams (or otherwise), you must register with the Disability Services office and present an accomodation letter.



Tentative schedule

Date Book Section(s) Homework Notes
9/9 Review of metric spaces    
9/14 Topological spaces    
9/16 Closure, separability, and bases    
9/21 Subspaces, products, and disjoint unions HW1 due  
9/23 Connectedness and path-connectedness    
9/28 Equivalence relations and components HW2 due  
9/30 Compact spaces    
10/5 Hausdorff spaces and compactness HW3 due. Curio 1-2 posted  
10/7 Proper maps and the 1-point compactification    
10/12 Quotient spaces I HW 4 due Drop date: Barnard, CC, GS, SPS
10/14 Quotient spaes II: Examples galore Curio 3 posted  
10/19 Orbit spaces (quotients by group actions) HW 5 due  
10/21 Case study: the real projective plane RP^2    
10/26 Countability axioms    
10/28 Review HW6 due Friday 10/29 at midnight  
11/2 Academic holiday    
11/4 Homotopies and homotopy equivalences Curio 4 posted  
11/9 Deformation retractions and homotopy rel A    
11/11 The fundamental group    
11/16 Induced maps and basepoints HW7 due; Curio 5 posted  
11/18 The fundamental group of the circle Curio 6 posted P/F deadline. Drop deadline: GSAPP; SEAS.
11/23 The Brouwer fixed-point theorem and applications

 

  
11/25 Academic holiday    
11/30 The Jordan curve theorem HW 8 due  
12/2 The topology of manifolds Curio 8 posted  
12/7 Classifying surfaces: the statement HW 9 due  
12/9 Every surface is standard    
[TBA] Bonus lecture (sketch): Standard surfaces are distinct    


Image of an excited mathematician by Ryan Armand