MAT 215B: (Graduate) Topology






Course information

Class Meetings

CRN 44986
Lectures: MWF 10:00--10:50 AM, STORER 1342
Discussion: T 10:00--10:50 AM, PHYSIC 148

[MAT 215B Course Syllabus]

Updated 4.4.25

[MAT 215B Class Calendar]

The class calendar is my personal lesson planning calendar. Lesson plans are subject to change.

Course instructor: Melissa Zhang

Office hours: MSB 2142 2112, on Wednesdays, 3:00-5:00 pm. See below / Canvas announcements for modifications due to travel.

Week of April 2: Office hours will be 3-4 on Wednesday.
April 9: Office hours are on Zoom; see Canvas announcement.

Textbook: Algebraic Topology by Allen Hatcher [Online Version]


Homework

Homeworks are generally posted over the weekend and due Friday nights at 9 pm.

[HW03]

Edited 4/23: In Exercise 3(b), what originally said '\varepsilon' should have read '\alpha'.

[HW02]

Edited 4/15: Exercise 5 differential fixed.
Edited 4/16: Exercise 1 edited, Exercise 6 removed (will be modified and moved to next week's HW)

[HW01]

Edited 4/7, 7:30pm: Exercise 4 typo fixed.
Edited 4/9, 10:00am: Exercise 8 part (b) removed.

Here is my sample solution to HW01 Exercise 3: [HW01 Ex3 Soln]
The purpose is to give you a sense of how much detail is preferable for homework solutions in this course.


Exam Information

The Midterm Exam will be held on Friday, May 2, 2025, at our usual lecture time and location.

The Final Exam will be held on Wednesday, June 11, 2025 from 10:30AM--12:30PM, at our usual lecture location.


Lectures and Materials

Here you'll find a brief summary of each lecture / discussion, along with any additional reference materials.

Discussion 5

Midterm review / info: [midterm review]

[Week 4 notes]

There is an error in the notes about the LES on reduced homology; there should not be Z's in negative homological degrees!

Lecture 12

LES odds and ends, equivalence of simplicial and singular homology

Lecture 11

finish proof of excision

Discussion 4

barycentric subdivision of linear chains

Lecture 10

intro to excision, using excision to relate relative and absolute homology

[Week 3 notes]

Lecture 9

proof that the 'LES' induced by SES of chain complexes is exact
relative homology groups, not good pair vs. good pair example

Lecture 8

exact sequences, SESs of chain complexes, LESs, connecting homomorphism

Discussion 3

proof of homotopy invariance, prism operators
discussion about level of detail preferred in homeworks

Lecture 7

chain maps, induced maps on homology, chain homotopy

[Week 2 notes]

Lecture 6

singular homology
path components decomposition (proofs using singular homology)
reduced homology

Discussion 2 and Lecture 5

This is a 4-part collection of Youtube videos:

Lecture 4

recall definition of simplicial chain complex
simplicial chain complex for X = the 2-simplex
simplicial chain complex for two Delta-structures on S^1
simplicial homology of 2-sphere vs. RP^2

[Week 1 notes]

Here are my lecture notes from Week 1 (Lectures 1-3, Discussion 1). Please kindly expect the neatness of the notes to decline.

The following typos (write-os?) have been identified:

Lecture 3

chain complexes and homology (algebra review)
formal definition of simplicial chain complex

Lecture 2

explicit definition of a Delta-complex structure on a space
more examples: RP^2, Klein bottle, Dunce cap
intro toward translating Delta-complex structures into data of simplicial chain complex (1-simplex as example)

Discussion 1

orientations on faces of simplices come from the total order
examples of Delta-complexes: circle, torus

Lecture 1

intro to idea of homology of spaces
simplicies as subsets of Euclidean space, and abstractly as ordered sets
intro to Delta-complexes