MAT7064 – Topics in Geometry and Topology, Fall 2024

  • Here are recent memorial tributes to Waldhausen from the algtop-l mailing list, who "advocated the philosophy that (higher) algebra over the sphere spectrum should simultaneously be easier and contain more information than (ordinary) algebra over the integers."

  • Here are the classic papers by Adams and by Adams and Atiyah on the Hopf invariant, as well as a recent talk by Xu on the Kervaire invariant. There is also an interesting post on MathOverflow about the phenomenon of "early" stabilization of homotopy groups of spheres which is related to the Hopf invariant.

  • As we proceeded from spectral sequences to cohomology operations, again here is Greenlees's nice article explaining the methodology of algebraic topology (and an announcement of the inaugural Elias M. Stein Prize for Transformative Exposition awarded to Hatcher for his book Algebraic Topology).

  • Assignment 10 due Friday, Dec. 20

  • Assignment 9 due Friday, Dec. 6 (updated Dec. 3 with correction on Question 1 thanks to Yifan Wu)

  • Assignment 8 due Friday, Nov. 29

  • The computations of $\pi_4S^3$ and $\pi_5S^3$ are just the beginning. If you are eager to learn more about computations in homotopy theory, here is a survey (or “how to become an expert”) on the topic.

  • Assignment 7 due Friday, Nov. 22

  • In the process of proving Hurewicz's theorem, we introduced the $\Delta$-complex ${\rm Sing}(X)$ associated to any topological space $X$ and the adjunction between the geometric realization functor $|-|$ and the singular complex functor ${\rm Sing}(-)$. These constructions have morphed into elements of contemporary homotopy-coherent mathematics, with $\infty$-categories interpolating algebra and topology (see Chapter 1 of Lurie's online textbook). Yet, one should keep in mind its humble, intuitive origin just for the sake of a genuine understanding.

  • An alternative proof of the Blakers–Massey theorem in the context of homotopy type theory

  • Assignment 6 due Friday, Nov. 8

  • Assignment 5 due Friday, Oct. 25

  • Assignment 4 due Friday, Oct. 18

  • Assignment 3 due Friday, Oct. 11

  • Assignment 2 due Friday, Sep. 27

  • Assignment 1 due Friday, Sep. 20

    • Instructor

    朱一飞 ZHU Yifei

    CoS-M705

    zhuyf@sustech.edu.cn

    Office hours: Official times are Thursdays 10:20 am–12:10 pm for questions about the course material or geometry and topology in general. If you have more immediate concerns or cannot come during ordinary office hours, feel free to contact me by email or in person.

    Grader: 孙运毫 SUN Yunhao

    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 iteration of the topics course will cover homotopy theory roughly corresponding to Chapter 4 of Hatcher (see below), along with supplementary materials such as fiber bundles, characteristic classes, spectral sequences, rational homotopy theory, cohomology operations, bordism, and Thom spaces. It also serves as the second half of a one-year course in modern algebraic topology, following MAT8021, though the order can be switched.

    Prerequisites

    MAT8021 Algebraic Topology is not required. Indeed, homotopy theory is relatively independent of homology and cohomology (see here, for example). However, as the course progresses, it will be helpful to incorporate the latter into the discussion, even if taking them as blackboxes, such as when we reach spectral sequences.

    Textbook

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

    Other useful references include Haynes Miller's Lectures on algebraic topology (2022).

    Homework

    There will be weekly problem sets to be handed in during class each Friday. The first problem set is due Friday, September 20. Your lowest problem set score will be dropped from your final grade. Homework will be posted online on the course webpage.

    Late homework will be docked by 15% per day (or portion thereof) up to a maximum of 45%, unless you have made prior arrangements.

    You are allowed (and encouraged) to work with other students while trying to understand the homework problems. However, the homework that you hand in should be your work alone.

    Exams

    There will be none.