| Instructor | Joj Helfer | TA | Chenchen Zhao |
| jhelfer (at) usc (dot) edu | zhao109 (at) usc (dot) edu | ||
| Office Hours | MF 12-1pm, W 2-3pm, by appointment (in KAP 464-B) | Office hours | M 10-11am, Tu 1-3pm (in the Math Center, KAP 263) |
All announcements will be posted on Blackboard.
(The latest announcement was on October 31, and is about the final project.)
The lecture notes can be accessed here and will be updated continuously as the course proceeds.
(Last update: November 28, 2023.)
Here are the notes for just Lecture 37
Homework 12 has been posted. It is due on Monday, November 27.
Click on the link above to see older homework assignments.
This course is an introduction to topology, and specifically to the two most important tools in this field: the fundamental group (and higher homotopy groups), and the homology groups. We will also see various applications of these tools.
In addition, we will begin with an introduction to category theory, which is an important tool in algebraic topology and most other fields of mathematics.
For a list of topics to be covered, please see the lecture plan (by clicking on the button above).
We will not be following any specific text. There will be course notes, that will be updated continuously througout the semester.
Here are some standard textbooks which I recommend and which I may reference:
For an excellent history of the subject, I also recommend
These books are all made available online either by their author or through the USC library.
The prerequisites are an introductory course’s worth of topology and algebra.
Most information and material related to the course will be posted on this page.
The exceptions are:
Homework assignments will be assigned every week.
Homework is to be submitted on Gradescope (you should see the course there if you are enrolled; if you have any problems, please inform the instructor or TA). When you submit on Gradescope, please don’t forget to match your scanned pages with the problems.
Homework must be submitted by the posted due dates. If you expect to have issues submitting the homework on time, or if you are having difficulties with gradescope, please write to the instructor or TA as soon as possible, and attach a scanned copy of your submission (this is a wise practice for any class).
You are encouraged though not required to type up your homework using LaTeX. If you do not have experience with LaTeX and would like to learn how to use it, please ask the instructor or TA.
Both exams are closed book, closed notes exams, with no calculators or other electronic aids permitted. The final exam will cover all topics from the semester, but will have greater emphasis on topics developed after the midterm.
A portion of your class grade will be based upon a project exploring an aspect of topology beyond the topics covered in class. Concretely, with a small group of 2-3 students, you will be asked to write a short expository article (around 4-6 pages, typed), and give an in-class 20 min presentation. The topic of study will be chosen in consultation with the instructor.
The TA’s and instructor’s office hours are TBA.
Any student requesting academic accommodations based on a disability is required to register with Disability Services and Programs (DSP) each semester A letter of verification for approved accommodations can be obtained from DSP. Please be sure the letter is delivered to me (or to TA) as early in the semester as possible. DSP is located in GFS 120 and is open 8:30 a.m.–5:00 p.m., Monday through Friday. Website for DSP (https://dsp.usc.edu/) and contact information: (213) 740-0776 (Phone), (213) 740-6948 (TDD only), (213) 740-8216 (FAX) dspfrontdesk@usc.edu.
USC seeks to maintain an optimal learning environment. General principles of academic honesty include the concept of respect for the intellectual property of others, the expectation that individual work will be submitted unless otherwise allowed by an instructor, and the obligations both to protect one’s own academic work from misuse by others as well as to avoid using another’s work as one’s own. All students are expected to understand and abide by these principles. SCampus, the Student Guidebook, contains the University Student Conduct Code (see University Governance, Section 11.00), while the recommended sanctions are located in Appendix A.
USC seeks to maintain an optimal learning environment. General principles of academic honesty include the concept of respect for the intellectual property of others, the expectation that individual work will be submitted unless otherwise allowed by an instructor, and the obligations both to protect one’s own academic work from misuse by others as well as to avoid using another’s work as one’s own. All students are expected to understand and abide by these principles. SCampus, the Student Guidebook, contains the University Student Conduct Code (see University Governance, Section 11.00), while the recommended sanctions are located in Appendix A.
In case of a declared emergency if travel to campus is not feasible, USC executive leadership will announce an electronic way for instructors to teach students in their residence halls or homes using a combination of Blackboard, teleconferencing, and other technologies. See the university’s site on Campus Safety and Emergency Preparedness.
This syllabus is not a contract, and the Instructor reserves the right to make some changes during the semester.
Problem sets will be posted here each week. Please notify the instructor or TA if you have trouble accessing the problem sets.
| Due date | Assignment |
|---|---|
| Wed, Aug. 30 | Homework 1 (Solutions) |
| Wed, Sep. 6 | Homework 2 (Solutions) |
| Wed, Sep. 13 | Homework 3 (Solutions) |
| Wed, Sep. 20 | Homework 4 (Solutions) |
| Wed, Sep. 27 | Homework 5 (Solutions) |
| Wed, Oct. 4 | Homework 6 (Solutions) |
| Wed, Oct. 11 | Homework 7 (Solutions) |
| Wed, Oct. 25 | Homework 8 (Solutions) |
| Wed, Nov. 1 | Homework 9 (Solutions) |
| Wed, Nov. 8 | Homework 10 (Solutions) |
| Wed, Nov. 15 | Homework 11 (Solutions) |
| Wed, Nov. 27 | Homework 12 |
Note: this lecture plan is tentative and will be continuously updated to adapt to the pace of the course.
| Week | Date | Material | SOTD |
|---|---|---|---|
| Mon, Aug 21 | Categories | ||
| 1 | Wed, Aug 23 | Isomorphisms | Spheres |
| Mon, Aug 28 | Products | Manifolds | |
| Fri, Aug 25 | Coproducts and duality | Tori | |
| 2 | Wed, Aug 30 | Pushouts and pullbacks | Connected sum |
| Fri, Sep 01 | Functors | Graphs | |
| Mon, Sep 04 | Labor Day. No class! | ||
| 3 | Wed, Sep 06 | Natural transformations | CW complexes |
| Fri, Sep 08 | Diagrams and limits | Wedge product | |
| Mon, Sep 11 | Fundamental group | Covering spaces | |
| 4 | Wed, Sep 13 | Fundamental groupoid | Contractible spaces |
| Fri, Sep 15 | Functoriality of the fundamental group(oid) | Presentation complex | |
| Mon, Sep 18 | Fundamental group of a product | Cayley complex | |
| 5 | Wed, Sep 20 | π1 and homotopies | Real projective space |
| Fri, Sep 22 | Homotopy equivalence | Complex projective space | |
| Mon, Sep 25 | π1 and covering spaces | Lens space | |
| 6 | Wed, Sep 27 | The lifting criterion | Simplicial complexes |
| Fri, Sep 29 | Consequences of the lifting crtierion | Abstract simplicial complexes | |
| Mon, Oct 02 | Seifert-Van Kampen | ||
| 7 | Wed, Oct 04 | Consequences of Seifert-Van Kampen | Algebraic varieties |
| Fri, Oct 06 | Classification of covering spaces | Projective varieties | |
| Mon, Oct 09 | Higher homotopy groups | Infinite projective spaces | |
| 8 | Wed, Oct 11 | Semi-simplicial sets | Semi-simplicial sets |
| Fri, Oct 13 | Fall recess. No class! | ||
| Mon, Oct 16 | Midterm in class | ||
| 9 | Wed, Oct 18 | Simplicial chains | |
| Fri, Oct 20 | Simpicial homology | Topological groups | |
| Mon, Oct 23 | Singular homology | Loop space | |
| 10 | Wed, Oct 25 | H0 and H1 | Mapping cylinder |
| Fri, Oct 27 | Chain complexes | Suspensions | |
| Mon, Oct 30 | Relative homology | Smash product | |
| 11 | Wed, Nov 01 | The Eilenberg-Steenrod axioms | Vector bundles |
| Fri, Nov 03 | Applications of homology of the sphere | Grassmannians | |
| Mon, Nov 06 | Mayer-Vietoris | Homogeneous spaces | |
| 12 | Wed, Nov 08 | More on homotopy groups | Fibre bundles |
| Fri, Nov 10 | Veterans Day. No class! | ||
| Mon, Nov 13 | Euler characteristic | ||
| 13 | Wed, Nov 15 | Cellular homology | |
| Fri, Nov 17 | Homology of manifolds | ||
| Mon, Nov 20 | Poincaré duality | ||
| 14 | Wed, Nov 22 | Thanksgiving break. No class! | |
| Fri, Nov 24 | Thanksgiving break. No class! | ||
| Mon, Nov 27 | Project presentations | ||
| 15 | Wed, Nov 29 | Project presentations | |
| Fri, Dec 01 | Project presentations |
