Math 431 / 531 Fall 2021

Instructor: Robert Lipshitz

Basic course information

Time: MWF 12:00–12:50 p.m.
Place: 106 University Hall.
Textbook: John Lee, Introduction to topological manifolds, second edition.
Office hours: In-person: Mondays 3-4 p.m. and Wednesdays 4-5 p.m. Online (Zoom): Mondays 8-9 p.m.. Subject to change.
Teaching assistant: Aurel Malapani.
TA office hourse: Thursdays 2:00-2:50, by Zoom.
Final exam: per Registrar's schedule.

Prerequisites

Math 431 requires Math 317 (Elementary Analysis). Math 531 is open to any beginning mathematics graduate student, though a knowledge of elementary analysis will be assumed. Anyone except a math graduate student seeking to enroll in 531 should obtain the instructor's permission.

Description and goals

This couse is about a fairly general notion of spaces and continuous functions. These turn out to be concepts that make sense even when more refined notions from calculus, like differentiability or integrability, do not. One goal in introducing such general spaces is to understand what special properties the "nice" spaces arising in calculus and differential geometry have. Another goal is to be able to deal with more pathological spaces arising in modern algebraic geometry and number theory.

Math 431 and 531 are a bridge course, between the levels of undergraduate and graduate mathematics. In particular, students should expect to find Math 4/531 and other bridge courses substantially more challenging than earlier undergraduate courses.

Particular "learning objects" for the course include:

  • Being able to work with, and write proofs about, metric spaces, general topological spaces, and continuous maps between them.
  • Developing an intuition for a wide range of topological spaces and maps between them.
  • Being able to construct topological spaces in various ways.
  • Understanding various properties that topological spaces and continuous maps may have (such as compactness, connectedness, path connectedness, the Hausdorff property, properness), and consequences of these properties, and being able to write proofs using these properties and their consequences.

Policies

Grading

Homework 35%
Midterm 25%
Final exam 40%

 

Textbook

The textbook for the class is Introduction to Topological Manifolds, second edition, by John Lee. Another textbook that may be useful to read along with this one is Topology by James Munkres. (Section numbers below are to the second edition.) For the first two weeks, Principles of Mathematical Analysis by Walter Rudin will be helpful. These lecture notes by Laura Person are another good resource.

Students are expected to read the sections in the textbook before coming to class each day. The relevant sections are listed in the syllabus below.

Homework

Homework is due on Wednesdays most weeks. The first homework assignment is due on Friday of the first week. At least initially, you will turn in the homework by uploading it to Canvas.

You may discuss the problems with other students, but must write up your solutions to the problems by yourself. Any resources you use other than the textbook must be cited in your homework. You may not use electronic resources (e.g., Google) other than the textbook and recommended textbook. Failure to follow this policy constitutes cheating; if you are caught cheating on the homework you will receive a 0 for the homework portion of the class and will be reported to the administration.

Requirements for students in Math 431 and 531 are slightly different. In addition to the requirements for Math 431, students in Math 531 are expected to:

  • Solve at least one of the "challenge" homework problems per week.
  • Type the solutions to their homework assignments in LaTeX. (See below for some resources.)

Students with disabilities

I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.

Assignments, handouts, and other resources

Homework

First day handout.

LaTeX resources

  • TeXlive. This is the LaTeX distribution I have found easiest to work with. It's available for Linux, Mac, and Windows. The Mac version goes under the name MacTeX. This also installs TeXworks (Windows) and TeXshop (Mac); if you're new to using LaTeX, these are probably good editors to start with. TeXworks is also available for Mac and Linux.
  • A (Not So) Short Introduction to LaTeX2ε. This is the document I learned TeX from.
  • Overleaf is an online LaTeX editor.

 

Schedule

This schedule is tentative, and subject to change (with or without notice).

L: Lee's book. M: Munkres's book. R: Rudin's book. +: material beyond that covered in the textbooks.

Week Date Topic Textbook sections
1 9/27 Overview. Sets, metric spaces. L. Chapter 1
M. §1–7
R. §2.2
  9/29 Open and closed sets in metric spaces.  
  10/1 Continuous functions on metric spaces. Homework 1 due

The last day to drop the class without a W is 10/3		 R: §4.1, 4.2
2 10/4 Sequences in metric spaces, completeness.
 R: §3.1 - 3.3
  10/6 Topological spaces, closed and open sets. Homework 2 due. L: §2.1. M: §12
  10/8 More examples of topological spaces. M: §14
3 10/11 Continuity. L: §2.2
  10/13 More continuity. Homework 3 due.  
  10/15 Hausdorff spaces. L: §2.3
4 10/18 Bases for topologies. L: §2.4. M: §13
  10/20 More examples of bases. Homework 4 due.  
  10/22 Countability axioms. L: §2.4. M: §30
5 10/25 Manifolds. L: §2.5
  10/27 Subspaces. Homework 5 due. L: §3.1. M: §16
  10/29 Product spaces, disjoint unions. L: §3.2, 3.3. M: §15
6 11/1 Midterm exam.  
  11/3 Quotient spaces, adjunction spaces. L: §3.4, 3.5. M: §22
  11/5 More quotients and gluings. Homework 6 due.  
7 11/8 Connectedness. L: §4.1. M: §23, 24, 25
  11/10 Path connectedness. Homework 7 due.  
  11/12 Compactness.

The last day to withdraw from the class is 11/13.		 L: §4.2. M: §26, 27
8 11/15 Sequential compactness. M: §28
  11/17 Local compactness, local connectedness, paracompactness. Homework 8 due. L: §4.3, 4.4. M: §29
  11/19 Proper maps. L: §4.5
9 11/22 CW complexes I. L: §5.1, 5.2
  11/24 Classification of 1-manifolds. Homework 9 due. L: §5.3
  11/26 Thanksgiving  
10 11/29 CW complexes II: more examples, properties.  
  12/1 Classification of surfaces I. Homework 10 due. L: §6.1, 6.2, 6.3
  12/3 Classification of surfaces II. L: §6.4, 6.5

 

Advice

  • Read the sections in the textbook before class, and again after class. Read with a pencil and paper in hand. Note down points that confuse you, and come back to them later to make sure you understand them. Work through the examples. When the author says something is "clear" that means it is not obvious -- make sure you understand why it is true.
  • Read the suggested sections in the secondary textbook, either before or after class (or both).
  • Get help as soon as you are confused. Your best options for help are my office hours, other students in the class, and other students not in the class.
  • Start on the homework problems as soon as we have covered the corresponding material in class.
  • Start working on the homework by yourself. After you have spent at least half an hour on each problem, try describing your solutions or where you are stuck to other students in the class. (Discussing the problems is helpful in both cases.) Then, go back and write (or type) your solutions nicely by yourself. If you work in a group, it is easy to think you've learned the material when you have not.
  • Do not look for hints or solutions on the internet. I guarantee you will not learn the material that way, even if you feel like you do.
  • Solve extra problems.
  • Think about the material, and how different parts of the material relate to each other, and examples of the different concepts, and counter-examples if you drop hypotheses from theorems, and how the material relates to material from other classes, constantly -- whenever there is a pause in your day.