MAT 215B: (Graduate) Topology

[Course Information]

[Homework]

[Exams]

[Lectures and Materials]

Course information

Class Meetings

CRN 44986
Lectures: MWF 10:00--10:50 AM, STORER 1342

[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 Canvas announcements for modifications due to travel.

Textbook: Algebraic Topology by Allen Hatcher [Online Version]

Homework

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

[HW04]

Edited 5/13: Added Fact for use in Exercise 4. Also, if you'd prefer to give a proof using CW approximation, that's ok with me too. Just note we won't be covering Section 4.1 in this course.

No homework the week of May 5

[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.

[Week 7 notes]

Lecture 22

(intended) examples of cup product using simplicial cohomology

Lecture 21

introduction to cup product on cohomology via cross products, direct definition of cup product

Lectures 19 and 20

[Week 6 notes]

Edited: Fixed typos in the proof that χ(C) = χ(H).

Lecture 18

a few remarks about homology classes in low dimensions
Künneth formula
homology with coefficients

Lecture 17

Mayer-Vietoris sequences

Discussion 6

lens spaces via identification of surface triangles of lens/lentil, L(2,1) = RP^2 example
Aside: Moore spaces
Euler characteristic

Lecture 16

cellular chain complex
examples: 2- and 3-torus

[Week 5 notes]

Lecture 15

(midterm exam)

Lecture 14

local degree
Lemma 2.34: some facts about CW complexes, setting up for CW homology

Discussion 5

Midterm review / info: [midterm review]

Lecture 13

Degree (of maps between spheres), some fun applications, antipodal map

[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:

On page 7, the purple and parallel black edge of the 2-simplex has the wrong orientation!
On page 8, the orientations draw on the blue and red halves of the sphere are both incorrect. (They should indeed disagree if I'm drawing them on the outside of the sphere, but the blue should be CCW and the red should be CW, viewed from the outside.)

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