MAT8021 – Algebraic Topology, Spring 2024

  • The topics course next semester will cover homotopy theory roughly corresponding to Chapter 4 of Hatcher, along with supplementary materials such as fiber bundles, characteristic classes, spectral sequences, rational homotopy theory, cohomology operations, bordism, and Thom spaces.

  • The final exam will take place on Thursday, June 13, 2–4 pm, in Lecture Hall 1, Room 305.

  • Office hours during the final exam weeks: Thursday, June 13, 10:20 am–12:10 pm.

  • More (non)examples of regular coverings:
    • The example of the double cover $S^1\vee S^1\vee S^1\to S^1\vee S^1$ given in the last lecture is in fact regular. This corresponds to the algebraic fact that any subgroup of index 2 is normal (and arises in a not-so-satisfying topological classification for certain quantum mechanical systems, see Fig. 1 here). Here is an example of a covering space that is not.
    • A normal covering with the symmetric group as the deck transformation group: configuration spaces and the phenomenon of homological stability. Here is a portrait of the "fat diagonal" from the article in the case of $C_4({\mathbb C})$. See here for Arnold's original paper.

  • Assignment 15

  • Assignment 14 due Tuesday, May 28

  • The spectral cover associated to a Higgs bundle (Figure 6 here) and, speaking of branched covers, higher-dimensional versions of the Hurwitz formula?

  • Assignment 13 due Tuesday, May 21

  • Assignment 12 due Tuesday, May 14

  • Assignment 11 due Tuesday, May 7

  • Assignment 10 due Tuesday, Apr. 30

  • Assignment 9 due Tuesday, Apr. 23

  • Assignment 8 due Tuesday, Apr. 16 (the cohomology ring of a lens space)

  • $\mathbb{O}\mathbb{P}^n$?

  • Assignment 7 due Tuesday, Apr. 9

  • As we've introduced the notion of a generalized homology theory, here is a nice article explaining the methodology of algebraic topology which you may begin to read.

  • Assignment 6 due Tuesday, Apr. 2 (a corrigendum)

  • Assignment 5 due Tuesday, Mar. 26

  • Assignment 4 due Tuesday, Mar. 19

  • Additional topic: Persistent homology and topological data analysis
    • A brief introduction to persistent homology: Sec. S.2.2 here (the paper also contains a section of a general review of literature on topological data analysis and its applications)
    • An informal introductory article emphasizing applications
    • An earlier discovery made by persistent homology: the Klein-bottle distribution of image data, pp. 69–111 here
    • A subsequent advance in deep learning of image and video data, confirming and enhanced by the Klein-bottle topology: a research paper; a survey article; an expository lecture

  • Assignment 3 due Tuesday, Mar. 12

  • Assignment 2 due Tuesday, Mar. 5

  • Assignment 1 due Tuesday, Feb. 27

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: 吴一凡 WU Yifan

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 maps. 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

The main references will be Hatcher's Algebraic Topology (online version available here along with additional exercises and content items) and 姜伯驹《同调论》.

This semester we will cover homology and cohomology roughly corresponding to Chapters 2 and 3 of Hatcher, along with a quick recap of homotopy and the fundamental group followed by discussion on covering maps (and fiber bundles, if time permits). Jiang's Chinese text is also quite handy and helpful at times.

Other useful references include Haynes Miller's Lectures on algebraic topology (2022) and John M. Lee's Introduction to topological manifolds, second edition (a main textbook for some other iterations of this course). William Fulton's Algebraic topology – A first course offers non-algebraic-topologist as well as historical perspectives of the subject (to quote, "The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topologists—without, we hope, discouraging budding topologists.") Jiacheng Liang's blog posts (he was last year's MAT8021 & MA323 grader) 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 on 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.