Math 636 Spring 2024

Basic course information

Time: MWF 11:00-11:55 a.m.
Place: McKenzie Hall 347
Textbook: Allen Hatcher, Algebraic Topology. Available for download here.
Office hours: Monday 2:00 – 3:00, Thursday 2:00 – 3:00. Subject to change.
Teaching assistant: Ka Fai Wong
Final exam: per Registrar's final exam schedule

Prerequisites

Math 635.

Description and goals

The first weeks of the course will discuss Poincaré duality, a key tool for understanding the topology of manifolds.

The rest of the 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:

• Understanding and applying the notion of homology orientations and Poincaré duality
• 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

Reason-Neutral Missed Homework and Exam Policy

The University of Oregon now requires reason-neutral policies for missed work or deadlines. That is, students who miss an exam because of an illness or family crises must be treated the same as students who simply choose not to show up to the exam. So:

• The lowest homework score will be dropped, to account for illnesses or other temporary obstacles to participating in the course.
• If you have a conflict with the midterm exam and contact me well in advance of it, you will have the opportunity to take the midterm early, without penalty.
• If you miss the midterm exam you will be given the opportunity to take an oral midterm exam as a make-up, by the end of week 7.
• If you miss the final exam and have a passing score in the rest of the course you will receive an Incomplete in the course, and be allowed to take a makeup final exam during the first week of the Summer Quarter to resolve the Incomplete. If you do not take the makeup exam will receive a 0 on the exam.
• If you miss the final exam and do not have a passing score in the rest of the course, you will fail the course.
• In all cases of conflicts or missed exams, you should contact me as soon as practically possible to make arrangements.

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 Mondays most weeks, by Canvas upload. 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, ChatGPT) 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. There is homework due in week 10.

In addition to solving the homework, studens will give anonymous peer feedback on other students' homework each week. Providing peer feedback is graded for completion (in good faith).

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 8.
• Homework 2. Due April 15.
• Homework 3. Due April 22.
• Homework 4. Due April 29.
• Homework 5. Due May 6.
• Homework 6. Due May 13.
• Homework 7. Due May 20.

Handouts

• First day handout.

Schedule

Sections refer to Hatcher's book. + denotes material beyond that covered in these sections of the textbook.