Overview: This is the second part of a two semester course on topology. Topics covered include homology, the Borsuk-Ulam theorem, cohomology, the cup product and cap product, the universal coefficient theorem, the Künneth formula, manifolds, orientation, homology and cohomology of manifolds, and Poincaré duality.
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 6346.
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 course overview, Section 2.3, Section 2.C.
Lecture notes Section 3.1.
Lecture notes Section 3.2.
Some of my notes may borrow from the content 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, January 31.
Homework 2 (LaTeX source) is due Monday, February 10.
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.
Exam solutions are posted on Canvas.
Schedule
Week 1: §2.3: The Formal Viewpoint
Week 2: §2.C: Simplicial Approximation, Lefschetz Fixed Point Theorem
Weeks 3-11: Chapter 3: Cohomology
– Weeks 3-5: §3.1: Cohomology Groups, The Universal Coefficient Theorem
– Weeks 6-8: §3.2: Cup Product, A Künneth Formula
– Weeks 9-11: §3.3: Poincaré Duality
Weeks 12-13: Chapter 4: Homotopy Theory
– Weeks 12-13: §4.1: Homotopy Groups
| Date | Class Topic | Remark |
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| Jan 13 | Course overview | |
| Jan 15 | §2.3: The formal viewpoint, categories and functors | |
| Jan 17 | §2.3: Natural transformations; §2.C: Simplicial approximation | |
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| Jan 20 | MLK Day | |
| Jan 22 | §2.C: Simplicial approximation, Lefschetz fixed point theorem | |
| Jan 24 | §2.C: Lefschetz fixed point theorem | |
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| Jan 27 | Chapter 3: Cohomology | |
| Jan 29 | §3.1: Cohomology groups | |
| Jan 31 | §3.1: Cohomology of spaces | HW1 due |
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| Feb 3 | §3.1: Cohomology of spaces | |
| Feb 5 | §3.1: Universal coefficient theorem | |
| Feb 7 | §3.1: Universal coefficient theorem | |
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| Feb 10 | §3.1: Universal coefficient theorem | HW2 due |
| Feb 12 | §3.2: Cup product | |
| Feb 14 | §3.2: Cup product | |
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| Feb 17 | §3.2: Cup product | |
| Feb 19 | §3.2: Cohomology ring | |
| Feb 21 | §3.2: Cohomology ring, §4.B: Hopf invariant | |
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| Feb 24 | Practice Exam 1 Review | |
| Feb 26 | Exam 1 | |
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| Spring Break | ||
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| Apr 21 | Practice Exam 2 Review | |
| Apr 23 | Exam 2 | |
| Apr 25 | Reading Day | |