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.
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 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.
Exam solutions are posted on Canvas.
Schedule
| Date | Class Topic | Remark |
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| Jan 13 | Course overview | |
| Jan 15 | §2.3: The formal viewpoint | |
| Jan 17 | §2.C: Simplicial approximation | |
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| Jan 20 | MLK Day | |
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| Feb 24 | Practice Exam 1 Review | |
| Feb 26 | Exam 1 | |
| Feb 28 | ||
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| Spring Break | ||
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| Apr 21 | Practice Exam 2 Review | |
| Apr 23 | Exam 2 | |
| Apr 25 | Reading Day | |