MAT8021 – Algebraic Topology, Spring 2026

  • Supplementary notes on exact sequences and diagram chasing

  • Assignment 5 due Tuesday, Apr. 14 (not Apr. 7, but you are encouraged to read the textbook, including a colored version of Figure 7.1, and work on these in advance, as we won't go into details of Sections VII.5–7 in class)
    • Exercise 5.1
    • Prove Proposition 5.2.
    • Exercises 5.2, 5.4, 5.5 (not 5.6!), 6.2
    • Prove Lemma 7.1 if you haven't done it before.

  • In the References section below, you can now find Greenlees's article mentioned in class on how algebraic invariants detect and distinguish spaces and maps progressively. Cf. Notes at the end of Chapter VII on extraordinary homology/cohomology theories.

  • Assignment 4 due Tuesday, Mar. 31
    • Prove the identity (7.2.2).
    • Chapter VII, Exercises 2.1, 3.3
    • Suppose that we defined the (cubical singular) homology groups without “normalization,” i.e., using the chain complex $Q_\bullet(X)=\{Q_n(X),\partial_n\}$ instead of $C_\bullet(X)$ with $C_n(X)=Q_n(X)/D_n(X)$. Compute $H_*({\rm pt})$. You should compare this to [Hatcher, Proposition 2.8] or [Lee, Proposition 13.7], where using simplices instead of cubes one does not need to deal with degeneracies $D_n(X)$, and at least get an idea of how the "simplicial" approach works.

  • In visual connection to Massey's "cubical" approach to homology (vs. simplicial, cf. Hatcher, Lee, and many others), here is a representative painting of Cubism by Picasso, more of whose artworks you may encounter in person during the upcoming fairs of Art Central and Art Basel in Hong Kong later this week.

  • Assignment 3 due Tuesday, Mar. 24
    • Chapter V, Exercises 8.5, 9.1, 9.2, 9.3, 10.1 (cf. [Lee, Theorem 12.29])
    • Given the example of the Hawaiian earring on p. 142, construct an example of a space that is semilocally simply connected but not locally simply connected. Can you further ensure that it is not simply connected either?
    • Read Chapter VI (motivations needed in next week's lectures).

  • Notes
    • For a relationship between regular covering spaces and Galois extensions, see Example 1.1 (d) and (a) of John Rognes's Galois extensions of structured ring spectra (see [Hatcher, Section 4.F] for the definition of a spectrum and the example of the sphere spectrum $S$).
    • Related to Massey's note on branched covering spaces at the end of Chapter V, Hurwitz spaces of branched covers have recently found spectacular applications to arithmetic, such as here and here.

  • Assignment 2 due Tuesday, Mar. 17
    • Chapter V, Exercises 5.1, 5.2, 5.4, 6.2, 6.3, 6.4, 7.1
    • In class, we constructed new covering spaces by taking products of known ones (cf. Example 2.4). Suppose $X$ consists of more than one point. Given two covering spaces $(\widetilde{X}_1,p_1)$ and $(\widetilde{X}_2,p_2)$ of $X$, is ${\rm Aut}(\widetilde{X}_1\times\widetilde{X}_2,p_1\times p_2)$ isomorphic to ${\rm Aut}(\widetilde{X}_1,p_1)\times{\rm Aut}(\widetilde{X}_2,p_2)$? If yes, give a proof. If no, give an example with justification.

  • Assignment 1 due Tuesday, Mar. 10
    1. Find the genus of $\widetilde{S}$ in Chapter V, Example 2.6.
    2. Prove the assertion on p. 122: If $(\widetilde{X},p)$ is a covering space of $X$, and $V$ is a connected, open, proper subset of $\widetilde{X}$, then $p|_V$ is a local homeomorphism, but $(V,p|_V)$ is not a covering space of $X$.
    3. Exercise 2.4
    4. Exercise 4.1
    5. Read Appendix B (equivariant algebra needed in next week's lectures) and do Exercise 1.1.

Instructor

朱一飞 ZHU Yifei

CoS-M705

zhuyf@sustech.edu.cn

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

Graders: 文子杰 WEN Zijie and 殷春双 YIN Chunshuang (Yin works as a grader till the end of March)

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:

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.