Spring 2026, Prof. Manuel Rivera, Purdue University
This course will be an introduction to algebraic topology. The main topics to be covered will be:
1) singular homology and Eilenberg-Steenrod axioms
2) computational methods based on cellular homology of CW complexes and homological algebra
4) Cohomology ring
5) Poincaré duality of manifolds
Along the way will introduce some basic category theory, simplicial techniques, and develop the necessary homological algebra. These tools were born in the context of algebraic topology but are widely used throughout modern mathematics.
Instructor: Manuel Rivera (manuelr at purdue dot edu)
Textbooks: The lectures will not strictly follow any textbook. They will be roughly based on these notes written by Haynes Miller:
http://math.mit.edu/~hrm/papers/lectures-905-906.pdf
There are lots of other textbooks that treat algebraic topology more or less at the level of this course.
Bredon, Glen E. Topology and Geometry
Davis, James F., and Paul Kirk. Lecture Notes in Algebraic Topology
Dold, Albrecht. Lectures on Algebraic Topology
Hatcher, Allen. Algebraic Topology. (Chatty)
May, J. Peter. A Concise Course in Algebraic Topology
Munkres, James R. Elements of Algebraic Topology
Shastri, Anant R. Basic Algebraic Topology
Spanier, Edwin H. Algebraic Topology
Vick, James W. Homology Theory: An Introduction to Algebraic Topology
Pre-requisites: We will assume familiarity with basic point set topology, covering spaces, and the fundamental group, as discussed in MA571. We will also assume some basic results in algebra such as the classification theorem of finitely generated abelian groups.
Grading: Homework 75 %, one exam 25%
Students who get at least 97% of the total points in this course are guaranteed an A+, 93% guarantees an A, 90% an A-, 87% a B+, 83% a B, 80% a B-, 77% a C+, 73% a C, 70% a C-, 67% a D+, 63% a D, and 60% a D-; for each of these grades, it’s possible that at the end of the semester a somewhat lower percentage will be enough to get that grade.
Homework: Working out exercises by yourself is essential for learning and understanding mathematics. There will be about six problem sets they will be posted on the course website. Each problem set will contain exercises from the book as well as exercises not in the book. Solutions to the exercises can probably be found online. Do not look them up; try each problem as hard as you can and if you get stuck discuss with me or your colleagues. Then write clear and complete solutions to all of them. Collaboration is encouraged but you must write up your own solutions and indicate who you worked with. Homework should be sent directly through email to the grader and the instructor. These can be either hand written or tex’ed. Exercises will be taken from Haynes Miller’s lecture notes unless otherwise stated.
Problem set 1 – due January 27: 1.8, 2.3, 3.7, and 3.8. We worked through 3.7 in class, but you should write up the solution independently.
Problem set 2 – due February 3: Read again sections 5 and 6. Exercise 6.3.
Problem set 3 – due February 10: 5.15, 8.8, 9.11
Problem set 4 – due February 17: 9.9. 10.10
Problem set 5- due February 24: 11.7, 11.8, 12.2, 13.6
Exams: This course will have a written in-class final exam during finals week.
Ethical Use of Generative AI in this Course:
One of the fundamental goals of higher education is to develop one’s understanding of a subject and ability to express original thoughts. While Generative AI (GAI) tools, such as ChatGPT, Google Gemini, DALL-E 2, and others, can aid in providing information and understanding, students are reminded that the value of their education comes from developing their own voice and original ideas. Using AI tools should not overshadow the importance of personal intellectual growth.
In this course, you may utilize GAI as an editor, translator, data visualization tool, or to improve grammar and spelling. GAI can be a powerful learning tool, but like all tools, it has limitations and weaknesses that you should be aware of. Do not use GAI to find solutions to homework exercises. GAI can fabricate seemingly credible data and generate wholly inaccurate content that is nonetheless highly persuasive. This is especially true when asking it for references, quotations, citations, and calculations. Therefore, it is imperative that you carefully read and verify GAI-generated content when incorporating it into your learning experience. Purdue Libraries offers a guide and resources on the use of GAI for research and other uses.
Special accommodations:
Purdue University strives to make learning experiences accessible to all participants. If you anticipate or experience physical or academic barriers based on disability, you are encouraged to contact the Disability Resource Center at: drc@purdue.edu or by phone: 765-494-1247, as soon as possible.
If the Disability Resource Center (DRC) has determined reasonable accommodations that you would like to utilize in this class, you must send your Course Accommodation Letter to the instructor. Instructions on sharing your Course Accommodation Letter can be found by visiting: https://www.purdue.edu/drc/students/course-accommodation-letter.php. Additionally, you are strongly encouraged to contact the instructor as soon as possible to discuss implementation of your accommodations.
riveramanuel.com 