MA323 – Topology, Spring 2024
 The final exam will take place on Monday, June 17, 10:30 am–12:30 pm, in Lecture Hall 3303.
 Office hours during the final exam weeks: Thursday, June 13, 10:20 am–12:10 pm.
 Here is the midterm exam.

Assignment 12, required but not collected: Sec. 4.3 #3; Sec. 4.4 #4, 8, 10, 11, 13; Sec. 4.5 #2, 3, 5, 8, 10.
Reading: [Y] Secs. 4.3–4.5, Appendix A; [BBT] Sec 6.7; [M] Sec. 55.
Optional reading: [Y] Appendix B; [M] Secs. 67–69 categorically, 56.

Assignment 11 due Tuesday, May 28: [Y] Sec. 4.2 #4, 5, 6; [M] Sec. 52 #7.
Reading: [Y] Sec. 4.2; [BBT] Sec 6.2.

Assignment 10 due Tuesday, May 21: [Y] Sec. 4.1 #2, 3, 4, 5, 6, 7, 8.
Reading: [M] Secs. 36, 46; [BBT] Secs. 5.1, 5.6.1, 6.1; [Y] Sec. 4.1.
Čech complexes (p. 51 onwards) applied in topological data analysis

Assignment 9 due Tuesday, May 7.
Reading: [Y] Secs. 3.3, 3.4; [B] Secs. 3.2, 3.4, 3.5.
Visualization of adding handles
Computational topology: Computational
aspects of surfaces can be found in Chapter II.
TOP vs DIFF vs PL
Optional reading: [B] Sec. 3.6.
 Midterm exam on Friday, Apr. 19 covers [Y] Secs. 1.1–1.3, 2.1–2.6, and 3.1–3.2.

Assignment 8 due Tuesday, Apr. 16: [Y] Sec. 2.5 #1, 5, 6, Sec. 2.6 #1, 3, 4, 5, Sec. 3.2 #8, 9, 14.
Reading: [Y] Secs. 2.5, 2.6, 3.1, 3.2; [BBT] Sec. 1.3.
Klein bottle as a distribution space for image data: discussion begins on page 96 of slides; and as reflected from
the topology of convolutional neural networks.
Hopf maps: Visualization of $S^3\to S^2$; Hopf "bundles" as eigenbundles for Hamiltonians in
quantum mechanics (Section 5 here, visualization over $\mathbb R$).

Assignment 7 due Tuesday, Apr. 9: [Y] Sec. 2.4 #4, 5, 8, 10.
Reading: [Y] Sec. 2.4.
Optional reading: [M] Sec. 24.

Assignment 6 due Tuesday, Apr. 2: [Y] Sec. 2.3 #1, 2, 4, 5, 6, 7, 9, 11, 16, 18, 19.
Reading: [Y] Sec. 2.3; [BBT] Sec. 5.5.
Examples of compact spaces: higherdimensional spheres
Optional reading: [M] Secs. 27, 37.

Assignment 5 due Tuesday, Mar. 26: [Y] Sec. 2.2 #1, 2, 3, 4; [M] Sec. 33 #3, Sec. 35 #7b.
Reading: [Y] Sec. 2.2; [M] Secs. 33–35.

Assignment 4 due Tuesday, Mar. 19: [Y] Sec. 2.1 #5, 6, 12, 14, 16, 18.
Reading: [Y] Sec. 2.1; [M] Secs. 30–32.

Assignment 3 due Tuesday, Mar. 12: [Y] Sec. 1.3 #3, 4, 6, 8, 11.
Reading: [Y] Sec. 1.3; [BBT] Secs. 1.4, 0.3.3, 0.3.4.
Optional reading: The structure of meaning in language: Parallel narratives in
linear algebra and category theory (interesting applications to Large Language Models begin in the section "Structures in the Real World" on page 6, with Figure 1,
but need Isbell adjunction introduced in the previous section in parallel with Singular Value Decomposition for Principal Component Analysis)

Assignment 2 due Tuesday, Mar. 5: [Y] Sec. 1.2 #2 (also show that the subspace topology is in fact the coarsest topology on $B$ for which the inclusion $i$ is
continuous), 4, 5, 6, 9, 10, 12, 13.
Reading: [Y] Sec. 1.2; [BBT] Secs. 0.2.1, 0.2.2.

Assignment 1 due Tuesday, Feb. 27: [Y] Sec. 1.1 #3, 7, 14, 6, 8, 13, 15.
Reading: [Y] Introduction (set operations, map, cartesian product, equivalence relation), Sec. 1.1.
Optional reading: [M] Secs. 20, 21, Exe. 20.8.
Instructor
朱一飞 ZHU Yifei
CoSM705
zhuyf@sustech.edu.cn
Office hours: Thursdays 10:20 am–12:10 pm and by appointment
Grader: 郑力豪 ZHENG Lihao
Prerequisites
Abstract Algebra (MA214/219) or consent of the Department.
Although strictly (and traditionally) speaking notions such as groups and group homomorphisms will not be used until the latter half (or even later) of this course on the
fundamental group(oid) of a topological space and its computation, familiarity with basic abstract algebra is necessary for acquiring a sophistication with writing proofs
and for supplying examples, intuitions, and analogies.
Objectives
This course introduces basic notions, examples, and applications of topology, a subject aimed to classify "spaces" according to those of their properties that are
invariant under continous deformations (in contrast to those rigid ones adhering to metrics).
With the classification theorem for closed surfaces as a main goal, we will cover pointset topology systematically together with elements of geometric and algebraic
topology, incorporating standard Chinese textbooks such as 尤承业《基础拓扑学讲义》, Chapters 1–4, as well as Munkres's Topology and Bradley, Bryson, Terilla's
Topology – A categorical approach (2020). This includes categories, functors and their adjunction, universal properties, and examples; compactification;
topological manifolds, partition of unity, and embeddings of manifolds into Euclidean spaces; simplicial complexes, Euler characteristic, and orientation; function
spaces, the compact–open topology, homotopy, and basic ideas of higher categories; projective spaces, lens spaces, Dehn surgery, and knots; glimpses of topological
classifications of physical systems and of topologyenhanced machine learning.
References
[Y] 尤承业，基础拓扑学讲义，北京大学出版社，1997.
(The main textbook, minimum requirement; good organization.)
[M] James R. Munkres, Topology,
Prentice Hall, Inc., Upper Saddle River, NJ, 2000, second edition of [MR0464128].
(A classic textbook; many indepth, elaborate discussions and examples; somewhat oldfashioned; mostly optional reading on selected topics and exemplary mathematical
exposition in general.)
[BBT] TaiDanae Bradley, Tyler Bryson, and John Terilla, Topology – A categorical approach, MIT Press, Cambridge, MA, 2020. MR4232168 (A new textbook; supplementary with categorical perspectives.)
[B] 包志强，点集拓扑与代数拓扑引论，北京大学出版社，
2013. (A relatively recent textbook; supplementary with selected topics.)
[A] Mark Anthony Armstrong, Basic
topology, Undergraduate Texts in Mathematics, SpringerVerlag, New YorkBerlin, 1983, corrected reprint of the 1979 original.
MR705632 (For additional perspectives.)
Exams
There will be one inclass midterm exam, on April 19, and one final exam. Each of these exams is worth 30% of your final grade.
Homework
The assigned problems for each week are due each Tuesday inclass at 2 pm, listed on this page. Homework is worth 40% 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 mouseover 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.
