MAT8021 – Algebraic Topology, Spring 2025

• Assignment 1 due Friday, Feb. 28

Instructor

朱一飞 ZHU Yifei

CoS-M705

zhuyf@sustech.edu.cn

Office hours: Thursdays 10:20 am–12:10 pm and by appointment

Grader: 李航 LI Hang

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 homology, cohomology, and covering spaces. This includes acyclic models in proving the graded commutativity of cup products on cohomology, calculations with homology and cohomology of Grassmannians, an honest account of the machinery needed in the proof of Poincaré duality, and an introduction to persistent homology as a method for recognizing the shape of data.

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

• Allen Hatcher, Algebraic Topology (online version available here along with additional exercises and content items), 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 Friday in-class at 10:20 am, listed at the top of this page. Homework is worth 50% of your final grade. Students must make arrangements in advance if they will not be handing in homework on time. You may 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.