Course Syllabus
Content: This class is an introduction to point-set and algebraic topology. Topics in point-set topology will include: topological spaces, connectedness, compactness, metric spaces. In algebraic topology, we will mostly focus on fundamental groups, homotopy, quotients and gluing, and covering spaces. See detailed list of topics below.
Textbook: J. Munkres, Topology (2nd edition). The international edition is acceptable.
Course staff:
- Prof. Denis Auroux (auroux@math.harvard.edu): office hours Mondays and Tuesdays 11:00-12:45 in SC 539. (also Thu 12/7, 9am-11am).
- Gregory Li: office hours Thursdays 3-5pm in Lowell Dining Hall
- Nico Miller: office hours Thursdays 6-8pm, Math Dept. Lounge
- Quinn Brussel: office hours Sundays 2-4pm, Mather Dining Hall
- Emma Cardwell: office hours Saturdays 3-5pm in Quincy Dining Hall
- Dora Woodruff: office hours Sundays 8-10pm in Quincy Dining Hall
- Eric Shen: office hours Mondays 9:15-11:15 PM in Leverett Dining Hall (Math night)
Lectures will be held on Tuesdays and Thursdays from 9:00 to 10:15am, starting Tuesday September 5, in Science Center 507.
Assessments: There will be weekly (or near-weekly) homework assignments, a midterm (take-home, open between October 17-23), and a final exam (take-home, December 8-15). Approximate grading weight: homework 50%, midterm 20%, final 30%. One homework score will be dropped for everyone, so you may miss one assignment without penalty, but you are still responsible for working through the material.
Academic integrity policy:
- You are encouraged to discuss and collaborate with each other on the homework assignments; the use of external resources (including generative AI) is acceptable. However, make sure that you are able to work through the problems yourself. Additionally, you must write up your answers in your own words and in a manner that reflects your own understanding, and you must credit any sources or resources that were used. This is not only a matter of academic integrity, but also crucial for properly learning the material and the problem-solving and mathematical writing skills that this course aims to cover.
- For exams, collaboration or consultation of sources other than those explicitly permitted is not allowed.
Homework assignments
Direct links to the homework assignments will be posted here as PDF files. Go to "Assignments" to see the LaTeX source and to submit your solutions.
You are encouraged to discuss the homework problems with other students, and you are allowed to consult external sources. However, the homework that you hand in should reflect your own understanding of the material and be written in your own words. You are NOT allowed to just copy solutions from other students or other sources (including generative AI).
No late homeworks will be accepted (unless circumstances genuinely warrant an extension). However, we will drop your lowest homework score, so you are allowed to miss one assignment without a penalty.
All homework submissions should be uploaded to Canvas. Handwritten work is welcome, but please upload a readable scan.
- Homework 1 (due Tue Sept 12) and solutions
- Homework 2 (due Tue Sept 19) and solutions
- Homework 3 (due Tue Sept 26) and solutions
- Homework 4 (due Tue Oct 3) and solutions
- Homework 5 (due Tue Oct 10) and solutions
- Homework 6 (due Tue Oct 17) and solutions
- Homework 7 (due Tue Oct 31) and solutions
- Homework 8 (due Thu Nov 9) and solutions
- Homework 9 (due Tue Nov 21) and solutions
- Homework 10 (due Tue Dec 5) and solutions
Lecture topics
The following syllabus is approximate -- we may end up deviating slightly from this list of topics or timetable. To the extent possible, (usually handwritten) lecture notes will be posted for each lecture.
- Lecture 1 (Tue Sept 5): Introduction; metric spaces as a motivating example. (Handout)
- Lecture 2 (Thu Sept 7): Topological spaces and bases; examples. (Munkres §12-13)
- Lecture 3 (Tue Sept 12): Examples. Definition of continuity. (Munkres §15-16, 18)
- Lecture 4 (Thu Sept 14): Continuity, homeomorphisms; closed sets and limit points. (Munkres §17-18)
- Lecture 5 (Tue Sept 19): Limits of sequences, Hausdorff spaces; products. (Munkres §17, 19)
- Lecture 6 (Thu Sept 21): Topologies on infinite products. (Munkres §19-20)
- Lecture 7 (Tue Sept 26): Connected spaces. (Munkres §23-24)
- Lecture 8 (Thu Sept 28): Path-connectedness; compact spaces. (Munkres §24, 26-27)
- Lecture 9 (Tue Oct 3): Compactness continued; compactness in metric spaces. (Munkres §26-27)
- Lecture 10 (Thu Oct 5): Compactness in metric spaces. (Munkres §28)
- Lecture 11 (Tue Oct 10): Compactifications and local compactness. (Munkres §29)
- Lecture 12 (Thu Oct 12): Countability and separation axioms; Urysohn's theorem. (Munkres §30-34)
- Lecture 13 (Tue Oct 17): Midterm review (CAs). (Practice midterm and solutions)
- Lecture 14 (Thu Oct 19): time off for midterm (midterm solutions)
- Lecture 15 (Tue Oct 24): Quotients and gluing; examples; category language. (Munkres §22)
- Lecture 16 (Thu Oct 26): Categories and functors; paths and homotopy. (Handout, Munkres §51)
- Lecture 17 (Tue Oct 31): The fundamental group. (Munkres §52)
- Lecture 18 (Thu Nov 2): Covering spaces and path lifting. (Munkres §53-54)
- Lecture 19 (Tue Nov 7): Path lifting, π1(S1), and the Brouwer fixed point theorem. (Munkres §54-55)
- Lecture 20 (Thu Nov 9): Brouwer and Borsuk-Ulam. (Munkres §55, 57)
- Lecture 21 (Tue Nov 14): Deformation retracts, homotopy equivalence. (Munkres §58)
- Lecture 22 (Thu Nov 16): Introduction to van Kampen, fundamental group calculations. (Munkres §59-60)
- Lecture 23 (Tue Nov 21): Equivalence of covering spaces. (Munkres §79)
- Lecture 24 (Tue Nov 28): Universal covering; Free products of groups. (Munkres §80,68-69)
- Lecture 25 (Thu Nov 30): Seifert-Van Kampen theorem; further examples. (Munkres §70-73)
- Lecture 26 (Tue Dec 5): Review session (list of topics, practice problems)
Course Summary:
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