MAT8021 – Algebraic Topology, Spring 2026

Instructor

朱一飞 ZHU Yifei

CoS-M705

zhuyf@sustech.edu.cn

Office hours: Tuesdays 2–3:50 pm and by appointment

Grader: 殷春双 YIN Chunshuang

Prerequisites

Topology (MA323) or consent of the Department.

Objectives

Algebraic topology, especially in the form of homotopical and categorical methods, plays an increasingly vital role in many of today's central areas of study in mathematics and science, including number theory, data science, and condensed matter physics. This is a half of the graduate compulsory courses in geometry and topology, the other half being MAT8024 Differentiable Manifolds. The main topics are covering spaces, homology, and cohomology.

For undergraduate students, this is a sequel to MA323 Topology, developing further algebraic (and computable) machinery beyond the fundamental group to analyze spaces qualitatively.

References

The main reference: William S. Massey, A basic course in algebraic topology, Chapter V onwards

• For this iteration of the course, why a textbook written 40 years ago?

  Besides previous experience and shortcomings, a motivation for this choice came from an interview with Yuri Manin from 2008, in which he shared his perspectives on what does not change much and what keeps evolving throughout the development of mathematics, on a relatively large scale.

  Part of this interview that is relevant to the role of algebraic topology (and the "homotopical and categorical methods" appearing in the Objectives above) was quoted in a talk given by John Baez, The rise and spread of algebraic topology, at a conference on applied algebraic topology in 2017.

  The author and mathematician, William Massey, passed away that same year. Besides well-known for introducing Massey product, which you may hopefully understand and appreciate by the end of this course, and which is widely applied in contemporary research (e.g., recently here), he is well remembered by the algebraic topology community including reminiscences from Peter May, Peter Landweber (mentioning the precursor of this book), and Martin Tangora.

Some additional references:

• Allen Hatcher, Algebraic topology (online version available here along with additional exercises and content items), Chapter 1 Section 3, Chapters 2 and 3
• Haynes Miller, Lectures on algebraic topology (online version available here), Chapters 1–3
• 姜伯驹，《同调论》
• John M. Lee, Introduction to topological manifolds, second edition (Chapter 5, Theorem 7.21, Chapters 11 and 12 are recommended, group actions on manifolds being an active research direction such as here and here)
• William Fulton, Algebraic topology – A first course (offers non-algebraic-topologist as well as historical perspectives of the subject)
• Jiacheng Liang's blog posts (a former Master's student and TA for the course) are yet another source of informal, personal contemplation on some of the topics in the process of learning. You should not read them unless you really want to.

Exams

There will be one final exam worth 50% of your final grade.

Homework

The assigned problems for each week are due each Tuesday in-class at 10:20 am, listed at the top of this page. Homework is worth 50% of your final grade. You must make arrangements in advance if you will not be handing in homework on time. You may alternatively send an electronic version of your homework (TeX'd up or scanned properly, in PDF format) to the grader at the mouse-over email address from above.

We encourage you to discuss homework problems with your classmates, including strategies for solving different kinds of problems. However, when you actually write up your solutions, you must do this on your own.