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
Updated 4.4.25
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.
Edited 4/23: In Exercise 3(b), what originally said '\varepsilon' should have read '\alpha'.
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)
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]
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
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
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
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