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 section (of Hatcher):
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, August 30.
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.
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
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| Aug 23 | Chp 0: Homotopy and homotopy type | |
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| Aug 26 | Chp 0: Cell complexes | |
| Aug 28 | Chp 0: Examples, Complex projective space | |
| Aug 30 | Chp 0: Operations on spaces | HW1 due |
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| Sep 2 | Labor Day | |
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| Sep 13 | HW2 due? | |
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| Sep 27 | HW3 due? | |
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| Oct 4 | Dinner at Henry and Ewo’s? | |
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| Oct 11 | HW4 due? | |
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| Oct 14 | Practice Midterm Review | |
| Oct 16 | Midterm | |
| Oct 18 | Homecoming? | |
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| Nov 11 | Veterans Day | |
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| Thanksgiving | ||
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| Dec 2 | ||
| Dec 4 | Practice Final Exam Review | |
| Dec 6 | Reading Day | |
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| Dec ? | Final Exam | |