Basic course information
Time: MWF 11:00-11:55 a.m.
Place: Zoom. Zoom link posted in Canvas.
Textbook: Allen Hatcher, Algebraic Topology. Available for download here.
Office hours: TBD and by appointment. By Zoom. Subject to change.
Teaching assistant: Wei Zhang.
Final exam: take home.
Prerequisites
Math 635.
Resources for this online-synchronous class
We will use:
- Zoom for the lectures and office hours. I prefer, but do not require, that you have your video camera on during lecture.
- A OneNote "class notebook" as a second place you will have access to what I write during lecture. You will receive a link to the notebook by e-mail.
- Canvas for announcements, submitting homework, seeing feedback on homework, posting solutions, and tracking scores.
Description and goals
This course is focused mainly on homotopy groups of spaces and their applications, particularly to what Hatcher calls the homotopy interpretation of cohomology.
Particular "learning objects" for the course include:
- Being able to compute homotopy groups in special cases, using the Hurewicz theorem, long exact sequence for a fibration, and other elementary techniques.
- Understanding and applying Whitehead's theorem to show when spaces are homotopy equivalent.
- Reinterpreting cohomology as a representable functor, and understanding how this classifies cohomology operations.
Policies
Grading
| Homework | 35% |
| Midterm | 25% |
| Final exam | 40% |
Textbook
The textbook for the class is Algebraic Topology by Allen Hatcher. It is available online or in print. Some other useful textbooks include:
- Glen Bredon, Topology and Geometry.
- William Massey, Algebraic Topology: An Introduction and A Basic Course in Algebraic Topology.
- John McCleary, A User's Guide to Spectral Sequences.
- James Munkres, Elements of Algebraic Topology and Topology.
- Edwin Spanier, Algebraic Topology.
- J. Peter May, A Concise Course in Algebraic Topology. Apparently available for download from Peter May's website.
- Anatoly Fomenko and Dmitry Fuchs, Homotopical Topology. Available for download via SpringerLink, through the library.
Students are expected to read the sections in the textbook before coming to class each day. The relevant sections are listed in the syllabus below.
Homework
Homework is due at 11:59 p.m. on Fridays most weeks. You may discuss the problems with other students, but must write up your solutions to the problems by yourself. Any resources you use other than the textbook must be cited in your homework. You may not use electronic resources (e.g., Google) other than the textbook and recommended textbook. Failure to follow this policy constitutes cheating; if you are caught cheating on the homework you will receive a 0 for the homework portion of the class and will be reported to the administration.
Students with disabilities
I, and the University of Oregon in general, are committed to an inclusive learning environment. If you have a disability which may impact your performance on exams, please contact the Accessible Education Center to discuss appropriate accommodations. If there are other disability-related barriers to your participation in the course, please either discuss them with me directly or consult with the Accessible Education Center.
Assignments, handouts, and other resources
Homework
- Homework 1. Due April 2.
- Homework 2. Due April 9.
- Homework 3. Due April 17.
- Homework 4. Due April 23.
- Homework 5. Due April 30.
- Homework 6. Due May 7.
- Homework 7. Due May 14.
- Homework 8. Due May 21.
- Homework 9. Due May 28.
- Homework 10. Due June 4.
Handouts
Schedule
Sections refer to Hatcher's book. + denotes material beyond that covered in these sections of the textbook.
| Week | Date | Topic | Textbook sections |
| 1 | 3/29 | More on intersection product. |
Bredon, VI.11 |
| 3/31 | More on intersection product. |
||
| 4/2 | More on intersection product. Saturday is last day to drop class without a W. |
||
| 2 | 4/5 | Homotopy groups. Relative homotopy groups. |
pp. 339-346, 421-426 |
| 4/7 | Whitehead's theorem. |
pp.346-348 | |
| 4/9 | Cellular approximation. |
pp. 348-357 | |
| 3 | 4/12 | CW models |
|
| 4/14 | More on CW models |
||
| 4/16 | Freudenthal suspension theorem. |
pp. 360-366 | |
| 4 | 4/19 | More on Freudenthal suspension theorem and its proof. |
|
| 4/21 | Hurewicz isomorphism theorem. |
pp. 366-375, + | |
| 4/23 | More Hurewicz. |
||
| 5 | 4/26 | Fiber bundles and fibrations. |
pp. 375-384 |
| 4/28 | Long exact sequence for a fibration. |
||
| 4/30 | Hopf invariant. |
pp. 427-428 | |
| 6 | 5/3 | Topology on function spaces. Homotopy revisited. |
pp. 529-532, + |
| 5/5 | More examples of fibrations. |
||
| 5/7 | Stable homotopy groups. |
pp. 384-388 | |
| 7 | 5/10 | Eilenberg-MacLane spaces and cohomology. |
pp. 393-405 |
| 5/12 | More on Eilenberg-MacLane spaces and cohomology. |
||
| 5/14 | More on fibrations. |
pp. 405-409 | |
| 8 | 5/17 | Postnikov towers. |
pp. 410-415 |
| 5/19 | More Postnikov towers. |
||
| 5/21 | Filtered complexes and spectral sequences |
+ | |
| 9 | 5/24 | Serre spectral sequence: statement, first examples |
+ |
| 5/26 | Sketch of proof of Serre spectral sequence |
+ | |
| 5/28 | Statement and proof of Serre's mod-C Hurewicz theorem |
+ | |
| 10 | 5/31 | Memorial day - no class. |
+ |
| 6/2 | Applications of the mod-C Hurewicz theorem |
+ | |
| 6/4 | Cohomology of Eilenberg-MacLane spaces and the Steenrod algebra |
+ |