Basic course information
Time: MWF 3:00–3:50
Place: Deady 205
Textbook: Allen Hatcher, Algebraic Topology. Available for download here.
Office hours: M 12:00-1:00, F 1:45-2:45 in Fenton 303. Subject to change.
Teaching assistant: Dan Raies
TA office hourse: T 3:00-4:00, F 11:30-12:50. Subject to change.
Final exam: per Registrar's schedule.
Prerequisites
Math 634.
Description and goals
This course continues where Math 634 left off, studying properties, computations, and applications of singular homology and cohomology. The second half of the course focuses on the (co)homology of manifolds and various forms of Poincaré duality, as well as further computations and applications.
Particular "learning objects" for the course include:
- Being able to compute and apply homology and cohomology, including the ring structure on cohomology.
- Understanding and applying Poincaré, Lefschetz, and Alexander duality for the (co)homology of manifolds.
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:
- William Massey, Algebraic Topology: An Introduction and A Basic Course in Algebraic Topology.
- 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 the beginning of class on Wednesdays 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 January 16.
- Homework 2. Due January 23.
- Optional homological algebra worksheet.
- Homework 3. Due January 30.
- Homework 4. Due February 6.
- Homework 5. Due February 13.
- Homework 6. Due February 20.
- Homework 7. Now due March 6 (originally February 27).
- Homework 8. Now due March 13 (originally March 6).
Handouts
Schedule
Sections refer to Hatcher's book. + denotes material beyond that covered in these sections of the textbook.
Updated Schedule
| Week | Date | Topic | Textbook sections |
| 1 | 1/7 | Degree. |
pp. 134-137 |
| 1/9 | Cellular (co)homology. |
pp. 137-146 | |
| 1/11 | More cellular (co)homology. |
||
| 2 | 1/14 | (Co)Homology with coefficients. Euler characteristics. |
pp. 146-147, 153-155 |
| 1/16 | Universal coefficient theorem. |
§3.1, 3.A | |
| 1/18 | More on universal coefficient theorem. |
||
| 3 | 1/21 | Martin Luther King day (no class) |
|
| 1/23 | Axioms for homology. |
§2.3 | |
| 1/25 | The 1-dimensional Hurewicz theorem. |
§2A | |
| 4 | 1/28 | Invariance of dimension, invariance of domain. |
§2B |
| 1/30 | Application to division algebras. Borsuk-Ulam theorem. |
§2B | |
| 2/1 | Simplicial approximation, Lefschetz fixed point theorem |
§2C | |
| 5 | 2/4 | Cohomology of projective spaces. |
pp. 211-217 |
| 2/6 | The cohomology ring. |
||
| 2/8 | More properties of cup product. |
||
| 6 | 2/11 | Catching up |
|
| 2/13 | Cross product |
pp. 218-223, §3.B | |
| 2/15 | Kunneth theorem. |
||
| 7 | 2/18 | More Kunneth theorem. |
|
| 2/20 | Poincaré duality: motivation. Orientations. |
pp. 233-239 | |
| 2/22 | More on orientations. |
||
| 8 | 2/25 | Snow - no class. |
|
| 2/27 | Snow - no class. |
||
| 3/1 | More on orientations. Compactly supported cohomology. |
pp. 242-252 | |
| 9 | 3/4 | Poincaré duality |
pp. 245-252 |
| 3/6 | Lefschetz duality |
pp. 252-257 | |
| 3/8 | Alexander duality |
||
| 10 | 3/11 | Smooth manifolds, transversality |
+ |
| 3/13 | Vector bundles, Thom isomorphism theorem |
+ | |
| 3/15 | Intersection product |
+ | |
Original Schedule
(For archival purposes; see updated schedule above.)
| Week | Date | Topic | Textbook sections |
| 1 | 1/7 | Degree. |
pp. 134-137 |
| 1/9 | Cellular (co)homology. |
pp. 137-146 | |
| 1/11 | More cellular (co)homology. |
||
| 2 | 1/14 | (Co)Homology with coefficients. Euler characteristics. |
pp. 146-147, 153-155 |
| 1/16 | Universal coefficient theorem. |
§3.1, 3.A | |
| 1/18 | More on universal coefficient theorem. |
||
| 3 | 1/21 | Martin Luther King day (no class) |
|
| 1/23 | Axioms for homology. |
§2.3 | |
| 1/25 | The 1-dimensional Hurewicz theorem. |
§2A | |
| 4 | 1/28 | Invariance of dimension, invariance of domain. |
§2B |
| 1/30 | Application to division algebras. Borsuk-Ulam theorem. |
§2B | |
| 2/1 | Simplicial approximation, Lefschetz fixed point theorem |
§2C | |
| 5 | 2/4 | Cohomology of projective spaces. |
pp. 211-217 |
| 2/6 | The cohomology ring. |
||
| 2/8 | More properties of cup product. |
||
| 6 | 2/11 | Midterm exam. | |
| 2/13 | Cross product |
pp. 218-223, §3.B | |
| 2/15 | Kunneth theorem. |
||
| 7 | 2/18 | Orientations |
pp. 233-239 |
| 2/20 | Compactly supported cohomology |
pp. 242-245 | |
| 2/22 | Poincaré duality |
pp. 245-252 | |
| 8 | 2/25 | Lefschetz duality |
pp. 252-257 |
| 2/27 | Alexander duality |
||
| 3/1 | Applications of duality |
||
| 9 | 3/4 | Smooth manifolds, transversality |
+ |
| 3/6 | Vector bundles |
+ | |
| 3/8 | Thom isomorphism theorem |
+ | |
| 10 | 3/11 | Intersection product |
+ |
| 3/13 | Massey products |
+ | |
| 3/15 | Review. |
||
Advice
- Read the sections in the textbook before class, and again after class. Read with a pencil and paper in hand. Note down points that confuse you, and come back to them later to make sure you understand them. Work through the examples. When the author says something is "clear" that means it is not obvious -- make sure you understand why it is true.
- Read the suggested sections in the secondary textbook, either before or after class (or both).
- Get help as soon as you are confused. Your best options for help are my office hours, other students in the class, and other students not in the class.
- Start on the homework problems as soon as we have covered the corresponding material in class.
- Start working on the homework by yourself. After you have spent at least half an hour on each problem, try describing your solutions or where you are stuck to other students in the class. (Discussing the problems is helpful in both cases.) Then, go back and write (or type) your solutions nicely by yourself. If you work in a group, it is easy to think you've learned the material when you have not.
- Do not look for hints or solutions on the internet. I guarantee you will not learn the material that way, even if you feel like you do.
- Solve extra problems.
- Think about the material, and how different parts of the material relate to each other, and examples of the different concepts, and counter-examples if you drop hypotheses from theorems, and how the material relates to material from other classes, constantly -- whenever there is a pause in your day.