Overview: This course is the second part of a two semester introduction to topology. In the first few weeks, students will learn a few advanced topics of general topology. The majority of the course will cover basic topics and examples in algebraic topology. Topology studies shapes and surfaces, sometimes in higher dimensions, along with the continuity properties of functions between two such shapes. Algebraic topology translates difficult questions about spaces into algebraic questions that can be answered. We begin with advanced topics in general topology, including the Tietze extension theorem, Tychonoff theorem, and Čech-Stone compactification. The majority of the course will focus on algebraic topology, including the fundamental group and covering spaces, the Seifert-van Kampen theorem, and the classification of surfaces.
Here is the catalog course description, and we also refer students, especially graduate students, to the first year exam topics.
Here is the webpage for the first semester prerequisite course, MTG 4302/5316.
Goals: Students will become fluent with the main ideas and the language of topology and algebraic topology, and will be able to communicate these ideas to others. Students will learn how to write rigorous mathematical proofs and how to construct counterexamples. A goal of the class is to teach the foundations of rigorous argument through proving claims built on axioms.
Syllabus: Here is the course syllabus.
Book: The course book is Topology by James R. Munkres, Second Edition. You are expected to read the relevant sections, and to come to class with questions.
Notes
Henry’s lecture notes are split by chapter (of Munkres):
Lecture notes for Chapter 9, Sections 51-56
Lecture notes for Chapter 9, Sections 57-60
Lecture notes for Chapter 4
Lecture notes for Chapter 11
Lecture notes for Chapter 12
Homework
The clarity of your solutions is as important as their correctness. Working in groups on homework and to study is encouraged! However, your submitted homework should be written up individually, in your own words, and without consulting anyone else’s written solutions of any form.
Homework 1 (LaTeX source) is due Wednesday, January 24.
Homework 2 (LaTeX source) is due Monday, February 5.
Homework 3 (LaTeX source) is due Wednesday, February 28.
Homework 4 (LaTeX source) is due Monday, March 25.
Homework 5 (LaTeX source) is due Friday, April 5.
Homework 6 (LaTeX source) is due Friday, April 19.
Homework solutions are posted on Canvas.
Exams
The exams will be in-class. You will only be able to use your brain and a pen or pencil – no notes, books, or electronic devices. The exams will be cumulative.
Here is a Practice Midterm.
Here is the Midterm.
Here is a Practice Final.
Here is the Final.
Exam solutions are posted on Canvas.
Schedule
| Date | Class Topic | Remark |
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| Jan 8 | Course overview | |
| Jan 10 | §51: Homotopy of paths | |
| Jan 12 | §51: Homotopy of paths | |
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| Jan 15 | Martin Luther King Jr. Day | |
| Jan 17 | §52: Fundamental group | |
| Jan 19 | §52: Fundamental group | |
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| Jan 22 | §53: Covering spaces | |
| Jan 24 | §54: Fundamental group of the circle | HW1 due |
| Jan 26 | §54: Fundamental group of the circle | |
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| Jan 29 | §54: Fundamental group of the circle | |
| Jan 31 | §55: Retractions and fixed points | |
| Feb 2 | §35: Tietze extension theorem | |
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| Feb 5 | §35: Tietze extension theorem | HW2 due |
| Feb 7 | §38: Čech-Stone compactification | Class taught by Dana Bartošová |
| Feb 9 | §38: Čech-Stone compactification | Class taught by Dana Bartošová |
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| Feb 12 | §37: Tychonoff theorem | Class taught by Jeremy Booher |
| Feb 14 | §37: Tychonoff theorem | Class taught by Jeremy Booher |
| Feb 16 | §56: The fundamental theorem of algebra | Class taught by Philip Boyland |
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| Feb 19 | §57: The Borsuk-Ulam theorem | |
| Feb 21 | §57: The Borsuk-Ulam theorem | |
| Feb 23 | §57 and §58 | |
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| Feb 26 | §58: Deformation retract and homotopy type | |
| Feb 28 | §58: Deformation retract and homotopy type | HW3 due |
| Mar 1 | §59: Fundamental group of spheres | |
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| Mar 4 | Practice Midterm Review | |
| Mar 6 | Midterm | |
| Mar 8 | §60: Fundamental group of some surfaces | YouTube Video |
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| Spring Break | ||
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| Mar 18 | §60: Fundamental group of some surfaces | |
| Mar 20 | §67: Direct sums of abelian groups | |
| Mar 22 | §68: Free products of groups | |
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| Mar 25 | §69: Free groups | HW4 due |
| Mar 27 | §70: The Seifert-van Kampen theorem | |
| Mar 28 | Dinner at Henry and Ewo’s! | |
| Mar 29 | §70: The Seifert-van Kampen theorem | |
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| Apr 1 | §71: The fundamental group of a wedge of circles | |
| Apr 3 | §72: Adjoining a two-cell | |
| Apr 5 | §73: The fundamental groups of the torus and the dunce cap | HW5 due |
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| Apr 8 | §74: Fundamental groups of surfaces | |
| Apr 10 | §75: Homology of surfaces | |
| Apr 12 | §76: Cutting and pasting | |
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| Apr 15 | §77: The classification theorem | Notes |
| Apr 17 | §77: The classification theorem | Notes |
| Apr 19 | An introduction to applied topology | HW6 due, Slides |
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| Apr 22 | Higher homotopy groups | Video1, Video2 |
| Apr 24 | Practice Final Exam Review | |
| Apr 26 | Reading Day | |
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| Apr 30 | Final Exam, 3:00-5:00pm | Little Hall 233 |