Overview: This is the first part of a two semester course on topology. Topics covered include general topology, algebraic topology, homotopy theory, and the topology of manifolds. We begin with a brief review of homotopy equivalences and an introduction to CW complexes. We then proceed to an advanced treatment of the fundamental group, covering spaces, van Kampen’s theorem, and covering spaces. The main focus of the course is the theory of homology, including simplicial homology, singular homology, homotopy invariance, exact sequences, excision, the Mayer-Vietoris sequence, and the Lefschetz fixed point theorem.
Here is the catalog course description. We also refer students, especially graduate students, to the PhD exam topics and to the past PhD exams.
Here is the webpage for the prerequisite course, MTG 4303/5317.
Goals: Students will become fluent with the main ideas of algebraic topology, and will be able to communicate these ideas to others. Algebraic topology involves abstract machinery, which students will learn. Students will also ground their knowledge by applying the tools of algebraic topology to solve concrete problems and to construct counterexamples.
Syllabus: Here is the course syllabus.
Book: The course book is Algebraic Topology by Allen Hatcher, which is freely available online, and also available as a paperback. You are expected to read the relevant sections, and to come to class with questions.
Notes
Henry’s lecture notes are split by chapter or section of Hatcher’s book:
Lecture notes for Chapter 0
Lecture notes for Chapter 1
Lecture notes for Section 2.1, part a
Lecture notes for Section 2.1, part b
Lecture notes for Section 2.2
Some of my notes may borrow from the content at CSU Math 570 notes (which also covered point-set topology and which used a different book) and at CSU Math 571 notes (which, like this class, skipped point-set topology and used Hatcher’s book).
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 Friday, September 6.
Homework 2 (LaTeX source) is due Friday, September 20.
Homework 3 (LaTeX source) is due Monday, October 7.
Homework 4 (LaTeX source) is due Wednesday, October 30.
Homework 5 (LaTeX source) is due Friday, November 8.
Homework 6 (LaTeX source) is due Friday, November 22.
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.
Here is a Practice Exam 1.
Here is Exam 1.
Here is a Practice Exam 2.
Here is Exam 2.
Exam solutions are posted on Canvas.
Schedule
Week 1: Chapter 0: Some Underlying Geometric Notions
Weeks 2-5: Chapter 1: The Fundamental Group
– Week 2: §1.1: Basic Constructions
– Week 3: §1.2: Van Kampen’s Theorem
– Week 4: §1.3: Covering Spaces
– Week 5: §1.A and §1.B: Additional Topics
Weeks 6-14: Chapter 2: Homology
– Weeks 6-8: §2.1: Simplicial and Singular Homology
– Weeks 9-11: §2.2: Computations and Applications
– Week 12: §2.3: The Formal Viewpoint; §2.A: Homology and Fundamental Group
– Week 13: §2.B: Classical Applications
– Week 14: §2.C: Simplicial Approximation
| Date | Class Topic | Remark |
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| Aug 23 | Course overview | |
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| Aug 26 | Chp 0: Homotopy and homotopy type | |
| Aug 28 | Chp 0: Cell complexes, Complex projective space | Hopf fibration video |
| Aug 30 | Chp 0: Operations on spaces | |
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| Sep 2 | Labor Day | |
| Sep 4 | Chp 0: Two criteria for homotopy equivalence, §1.1: Basic constructions | |
| Sep 6 | §1.1: Fundamental group of the circle | HW1 due |
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| Sep 9 | Class cancelled: Henry was sick | |
| Sep 11 | §1.2: Van Kampens theorem | |
| Sep 13 | §1.2: Applications to cell complexes | |
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| Sep 16 | §1.3: Covering spaces | |
| Sep 18 | §1.3: Classification of covering spaces | |
| Sep 20 | §1.3: Classification of covering spaces | HW2 due |
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| Sep 23 | §1.3: Deck transformations and group actions | |
| Sep 25 | §1.3: Deck transformations and group actions | |
| Sep 27 | Class cancelled: Hurricane Helene | |
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| Sep 30 | Chp 2: Homology | |
| Oct 2 | §2.1: Simplicial homology for simplicial complexes | |
| Oct 4 | §2.1: Simplicial homology for simplicial complexes | |
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| Oct 7 | §2.1: Simplicial homology (for Δ-complexes) | HW3 due |
| Oct 9 | Class cancelled: Hurricane Milton | |
| Oct 11 | §2.1: Simplicial homology (for Δ-complexes) | |
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| Oct 14 | §2.1: Singular homology | |
| Oct 16 | §2.1: Singular homology: Homotopy invariance | |
| Oct 18 | No class: Homecoming | |
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| Oct 21 | Exam 1 | |
| Oct 23 | §2.1: Singular homology: Homotopy invariance | |
| Oct 25 | §2.1: Singular homology: Exact sequences and excision | |
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| Oct 28 | §2.1: Singular homology: Exact sequences | |
| Oct 30 | §2.1: Singular homology: Relative homology groups | HW4 due |
| Nov 1 | §2.1: Singular homology: Excision | |
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| Nov 4 | §2.1: Singular homology: Excision, Five lemma | |
| Nov 6 | §2.1: Equivalence of simplicial and singular homology | |
| Nov 8 | §2.2: Computations and applications: Degree | HW5 due |
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| Nov 11 | Veterans Day | |
| Nov 13 | §2.2: Cellular homology | |
| Nov 15 | §2.2: Cellular homology | |
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| Nov 18 | §2.2: Mayer-Vietoris sequences | |
| Nov 20 | §2.2: Homology with coefficients | |
| Nov 22 | §2.2: Euler characteristic | HW6 due |
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| Thanksgiving | ||
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| Dec 2 | Practice Exam 2 Review | |
| Dec 4 | Exam 2 — Dinner at Tayo, Ewo, and Henry’s! | |